A373291 Contribution to the OEIS

A373291 This is one of my favorite sequences. Least perfect power of n containing some decimal digit of n. +30 0

1, 32, 243, 64, 25, 36, 16807, 32768, 729, 100, 121, 144, 169, 196, 225, 256, 4913, 5832, 361, 400, 441, 234256, 529, 13824, 625, 676, 729, 784, 24389, 900, 961, 1024, 35937, 39304, 1225, 1296, 1369, 54872, 59319, 1600 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,2 COMMENTS "Perfect power of n" here means n^k with k>1. The sequence gives the value of n^k, not the value of k. - N. J. A. Sloane, May 31 2024 LINKS Table of n, a(n) for n=1..40. FORMULA a(n) = n^A253600(n). EXAMPLE For n=12, 12^2=144 contains digit 1 from n so that a(12) = 144. MATHEMATICA seq={}; Do[k=1; Until[ ContainsAny[IntegerDigits[n], IntegerDigits[n^k] ], k++ ]; AppendTo[seq, n^k] , {n, 40}]; seq PROG (PARI) a(n) = my(sd = Set(vecsort(digits(n))), k=2); while (#setintersect(sd, Set(vecsort(digits(n^k)))) == 0, k++); n^k; \\ Michel Marcus, May 31 2024 CROSSREFS Cf. A111442, A253600. KEYWORD nonn,base,new AUTHOR James C. McMahon, May 30 2024 STATUS approved

A373203 Contribution to the OEIS

 

A373203 a(n) = minimum k>1 such that n^k contains all distinct decimal digits of n. 0

2, 2, 5, 5, 3, 2, 2, 5, 5, 3, 2, 2, 3, 5, 4, 6, 5, 5, 5, 7, 5, 3, 4, 7, 3, 2, 8, 2, 5, 3, 5, 4, 3, 3, 3, 6, 6, 5, 4, 3, 3, 6, 7, 4, 3, 4, 4, 4, 4, 3, 2, 3, 7, 5, 3, 2, 3, 5, 5, 3, 2, 3, 5, 2, 2, 3, 2, 3, 4, 5, 5, 3, 3, 3, 2, 3, 2, 5, 5, 5, 5 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 0,1 LINKS Table of n, a(n) for n=0..80. FORMULA A253600(n) <= a(n) <= A045537(n). - Michael S. Branicky, May 28 2024 A111442(n) = n^a(n). EXAMPLE For n=12, a(12)=3 because 12^3=1728 contains all decimal digits of n. Compare to A253600(12)=2 because 12^2=144 contains any digit of n. MATHEMATICA seq={}; Do[k=1; Until[ContainsAll[IntegerDigits[n^k], IntegerDigits[n] ], k++]; AppendTo[seq, k] , {n, 0, 80}]; seq PROG (Python) from itertools import count def a(n): s = set(str(n)) return next(k for k in count(2) if s <= set(str(n**k))) print([a(n) for n in range(81)]) # Michael S. Branicky, May 27 2024 CROSSREFS Cf. A045537, A111442, A253600. Sequence in context: A005177 A357123 A253600 * A045537 A243941 A161622 Adjacent sequences: A373188 A373189 A373195 * A373207 A373208 A373210 KEYWORD nonn,base,new AUTHOR James C. McMahon, May 27 2024 STATUS approved

Guinness Beer and Statistics

 

(Image: https://www.iconfinder.com/search/icons?family=ionicons-fill )

Student's t-test is a common statistical test using the t-distribution developed by William Gosset. William Gosset. Gosset worked at the Guinness brewery in Dublin and his employer preferred that staff use pen names when publishing scientific papers, so in 1908 he published under the pseudonym, Student, when his work was published in the journal, Biometrika.

The t-distribution is used when one has small sample sizes. The distribution has a bell-shape like the normal distribution but is wider at the tails. 

In May, 2024, Scientific America published an opinion piece about this important contribution to the field of statistics: How the Guinness Brewery Invented the Most Important Statistical Method in Science: How the Guinness Brewery Invented the Most Important Statistical Method in Science.

Proving the Existence of God

 

(Image: https://www.iconfinder.com/ekarifqi001)

I recently read an article about Kurt Gödel's proof of the existence of God. This blog has included other posts about religious mathematicians, including: 

https://jamesmacmath.blogspot.com/2023/12/book-review-zero-biography-of-dangerous.html

https://jamesmacmath.blogspot.com/2023/10/how-does-math-formula-highlight-creator.html

https://jamesmacmath.blogspot.com/2023/09/anniversary-of-eulers-death.html

https://jamesmacmath.blogspot.com/2022/09/alec-wilkinson-staff-writer-for-new.html

https://jamesmacmath.blogspot.com/2022/02/perfect-number-generator.html

https://jamesmacmath.blogspot.com/2022/01/maths-true-source-man-or-nature.html

https://jamesmacmath.blogspot.com/2020/05/george-lemaitre-and-big-bang.html

https://jamesmacmath.blogspot.com/2020/05/thomas-bayes-attempt-to-prove-existence.html

 

I had a goal of writing a post that summarizes these many theories, however, during my research, I found an article by Manon Bischoff in Scientific American that already has a very good summary. It includes many of the theories of the scientists and mathematicians from the above links and the summary of Kurt Gödel's proposed proof. I recommend readers of this blog to his article.

200th - Triple Experience

 200

In one week, I had a triple  experience with the number 200: I made my 200^th post to my math blog ( https://jamesmacmath.blogspot.com), donated my 200^th  blood product, and made my 200^th contribution to the On-Line Encyclopedia of Integer Sequences OEIS-(https://oeis.org/search?q=James+C.+McMahon+&sort=created&go=Search ).

Unification of Different Fields of Mathematics

(Image: https://www.iconfinder.com/roywj)

During WWII, André Weil proposed an idea for linking three fields of mathematics: number theory, geometry, and finite fields.

For the full story, please see the following link it Quanta Magazine's  article describing the unification of three major branches of mathematics: A Rosetta Stone of Mathematics. 

Also related, is the work begun by Robert Langlands, known as the Langlands program.

A371030 Contribution to the OEIS

 

A371030 n written in compositorial base. 0

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 40, 41, 42, 43, 50, 51, 52, 53, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 123, 130, 131, 132, 133, 140, 141, 142, 143, 150, 151, 152, 153, 200, 201, 202, 203, 210, 211, 212, 213 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 0,3 COMMENTS Compositorial base is a mixed-radix representation using the composite numbers (A002808) from least to most significant. Places reading from right have values (1, 4, 24, 192, ...) = compositorial numbers (A036691). a(n) = concatenation of decimal digits of n in compositorial base. This concatenated representation is unsatisfactory for large n (above 172799), when coefficients of 10 or greater start to appear. LINKS Table of n, a(n) for n=0..55. EXAMPLE a(35)=123; 35 = 1*24 + 2*4 + 3*1. MATHEMATICA Table[FromDigits@ IntegerDigits[n, MixedRadix[Reverse@ ResourceFunction["Composite"]@ Range@ 8]], {n, 0, 55}] CROSSREFS Cf. A002808, A036691, A049345. Sequence in context: A301382 A288657 A055655 * A276326 A007090 A102859 Adjacent sequences: A371027 A371028 A371029 * A371031 A371032 A371033 KEYWORD nonn,base,less AUTHOR James C. McMahon, Mar 08 2024 STATUS approved

A370831 Contribution to the OEIS

 

A370831 Alternating sum of composites. 0

4, 2, 6, 3, 7, 5, 9, 6, 10, 8, 12, 9, 13, 11, 14, 12, 15, 13, 17, 15, 18, 16, 19, 17, 21, 18, 22, 20, 24, 21, 25, 23, 26, 24, 27, 25, 29, 26, 30, 27, 31, 29, 33, 30, 34, 31, 35, 33, 36, 34, 38, 36, 39, 37, 40, 38, 42, 39, 43, 41, 44, 42, 45, 43, 47, 44, 48, 45, 49, 46 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,1 COMMENTS Unlike equivalent sequence for primes, A008347, there are repeated terms. LINKS Table of n, a(n) for n=1..70. FORMULA a(n) = A002808(n) - a(n-1), for n>1. EXAMPLE a(4) = 9 - 8 + 6 - 4 = 3. MATHEMATICA Join[{4}, a[1]=4; a[n_]:=ResourceFunction["Composite"][n] - a[n-1]; Table[a[n], {n, 2, 70}]] (* or with signs *) R=70; a[1]=4; a[n_]:=a[n-1]-ResourceFunction["Composite"][n] *(-1)^n; Table[a[n], {n, 70}] CROSSREFS Cf. A002808, A008347. Sequence in context: A246879 A302794 A247361 * A297307 A163238 A097362 Adjacent sequences: A370827 A370828 A370830 * A370832 A370833 A370834 KEYWORD nonn AUTHOR James C. McMahon, Mar 02 2024 STATUS approved

A368805 Contribution to the OEIS

 

A368805 Primes whose digits are prime in both base 9 and base 10. 0

2, 3, 5, 7, 23, 227, 277, 2777, 5333, 5573, 23537, 23753, 25373, 225527, 25737557, 27775337, 27775357, 35275777, 35277233, 37333757, 227773753, 227775533, 232372577, 233752577, 252777737, 337777277, 25322233723, 25322237323, 25322237357, 25322237723, 25322327753, 25322327777, 25322532523 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,1 COMMENTS Subsequence of A019546. LINKS Table of n, a(n) for n=1..33. EXAMPLE 2777 is in this sequence because it is prime, all its digits are prime and 2777 in base 9 is 3725, whose digits are all prime. MATHEMATICA Select[Range[2.1*10^7], PrimeQ[#]&&AllTrue[IntegerDigits[#], PrimeQ]&&AllTrue[IntegerDigits[#, 9], PrimeQ]&] (* or *) seq1[dignum_, b_] := Module[{s = {}}, Do[s = Join[s, Select[FromDigits[#, b] & /@ Tuples[{2, 3, 5, 7}, k], PrimeQ]], {k, 1, dignum}]; s]; seq[maxdig9_] := Select[Intersection[seq1[maxdig9, 9], seq1[maxdig9, 10]], # <= 9^maxdig9 &]; seq[11] (* Amiram Eldar, Jan 06 2024 *) PROG (Python) from gmpy2 import digits, is_prime from itertools import count, islice, product def bgen(): yield from [2, 3, 5, 7] for d in count(2): for f in product("2357", repeat=d-1): for last in "37": yield int("".join(f)+last) def agen(): yield from (t for t in bgen() if is_prime(t) and set(digits(t, 9)) <= set("2357")) print(list(islice(agen(), 33))) # Michael S. Branicky, Jan 07 2024 CROSSREFS Cf. A000040, A019546. Sequence in context: A343834 A070029 A360497 * A262339 A110094 A088054 Adjacent sequences: A368802 A368803 A368804 * A368806 A368807 A368808 KEYWORD nonn,base AUTHOR James C. McMahon, Jan 06 2024 STATUS approved

A366693 Contribution to the OEIS

 

A366693 Minimal number of primorials or their negatives that add to n. 2

0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 2, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 5, 5, 4, 4, 3, 4, 4, 4, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 5, 6, 6, 6, 5, 6, 6, 6, 5, 5, 4, 5, 5, 5 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 EXAMPLE 5 = 6 - 1 (two primorials), so a(5) = 2. 27 = 30 - 2 - 1 (three primorials), so a(27) = 3. MATHEMATICA a[nthPrimorials_Integer?NonNegative (* Increase nthPrimorials to use more positive and negative primorials in sum *), numberOfPrimorials_Integer?NonNegative (* Increase numberOfPrimorials to increase cap of minimal number of primorials *)] := a[nthPrimorials, numberOfPrimorials] = Module[{A002110, f, h, s}, A002110[nthPrimorials] = Join[{1}, Denominator[Accumulate[1/Prime[Range[nthPrimorials]]]]]; A002110[n_] := A002110[n] = Join[{1}, Denominator[Accumulate[1/Prime[Range[n]]]]]; f[n_] := f[n] = Flatten[Table[p*r, {p, A002110[n - 1]}, {r, {1, -1}}]]; h[n_, u_] := h[n, u] = Sort[Select[DeleteDuplicates[Flatten[Table[Sum[p[j], {j, 1, u}], ##] & @@ Table[{p[j], f[n]}, {j, 1, u}]]], # > 0 &]]; s = Table[Infinity, {A002110[nthPrimorials][[-1]]}]; Monitor[Do[If[s[[k]] > k, s[[k]] = l], {l, 1, numberOfPrimorials}, {k, h[nthPrimorials, l]}], {l, k}]; s = Join[{0}, s]; If[MemberQ[s, Infinity], s[[1 ;; Position[s, Infinity][[1, 1]] - 1]], s]]; a[6, 6] (* Robert P. P. McKone, Oct 21 2023 *) CROSSREFS Cf. A002110, A276150, A366136. Sequence in context: A261915 A109037 A366136 * A147680 A192895 A210685 Adjacent sequences: A366690 A366691 A366692 * A366694 A366695 A366696 KEYWORD nonn,look AUTHOR James C. McMahon, Oct 16 2023 STATUS approved

A366136 Contribution to the OEIS

 

A366136 Minimal number of factorials or their negatives that add to n. 2

0, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 3, 2, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 4, 3, 4, 4, 4, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 5, 6, 5, 5, 4, 5, 5, 5, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 5, 6, 5, 5, 4, 5, 5 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 EXAMPLE 11 = 6 + 6 - 1 (three factorials), so a(11) = 3. 15 = 6 + 6 + 2 + 1 or 15 = 24 - 6 - 2 - 1 (four factorials), so a(15) = 4. MATHEMATICA seq[m_] := Module[{s = Table[m!, {m!}], d, b, sum}, Do[d = PadLeft[Most@ IntegerDigits[k, MixedRadix[Range[m, 1, -1]]], m]; Do[b = 2*PadLeft[ IntegerDigits[i, 2], m] - 1; sum = Total[b * d * Range[m, 1, -1]!]; If[0 < sum <= m!, s[[sum]] = Min[s[[sum]], Total[d]]], {i, 1, 2^m - 1}], {k, 1, 2*m!}]; Join[{0}, s]]; seq[5] (* Amiram Eldar, Oct 03 2023 *) CROSSREFS Cf. A000142, A034968, A366693. Sequence in context: A204018 A261915 A109037 * A366693 A147680 A192895 Adjacent sequences: A366133 A366134 A366135 * A366137 A366138 A366139 KEYWORD nonn,look AUTHOR James C. McMahon, Sep 30 2023 STATUS approved

A365472 Contribution to the OEIS

 

A365472 Numbers whose digits are either all primes or all nonprimes. 1

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 19, 22, 23, 25, 27, 32, 33, 35, 37, 40, 41, 44, 46, 48, 49, 52, 53, 55, 57, 60, 61, 64, 66, 68, 69, 72, 73, 75, 77, 80, 81, 84, 86, 88, 89, 90, 91, 94, 96, 98, 99, 100, 101, 104, 106, 108, 109, 110, 111, 114 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,3 COMMENTS Complement of A365589. Union of A046034 and A084984. LINKS Table of n, a(n) for n=1..65. MATHEMATICA a[n_Integer?NonNegative] := Select[Range[0, n], Module[{digits, primeDigits}, digits = IntegerDigits[#]; primeDigits = MemberQ[{2, 3, 5, 7}, #] & /@ digits; AllTrue[primeDigits, Identity] || AllTrue[primeDigits, Not]] &]; a[114] (* Robert P. P. McKone, Sep 13 2023 *) CROSSREFS Cf. A046034, A084984, A365589. Sequence in context: A252494 A030294 A125668 * A034894 A263564 A061430 Adjacent sequences: A365469 A365470 A365471 * A365473 A365474 A365475 KEYWORD nonn,base AUTHOR James C. McMahon, Sep 11 2023 STATUS approved

A365471 Contribution to the OEIS

 

A365471 Numbers whose digits are not all primes. 1

0, 1, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 29, 30, 31, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,3 COMMENTS Complement of A046034. Union of A084984 and A365589. LINKS Table of n, a(n) for n=1..67. MATHEMATICA a[n_Integer?NonNegative] := Select[Range[0, n], Not[AllTrue[MemberQ[{2, 3, 5, 7}, #] & /@ IntegerDigits@#, Identity]] &]; a[86] (* Robert P. P. McKone, Sep 13 2023 *) Select[Range[0, 100], AnyTrue[IntegerDigits[#], !PrimeQ[#]&]&] (* Harvey P. Dale, Dec 22 2023 *) CROSSREFS Cf. A046034, A084984, A365589. Sequence in context: A225668 A091985 A091984 * A228651 A045762 A320985 Adjacent sequences: A365468 A365469 A365470 * A365472 A365473 A365474 KEYWORD base,nonn AUTHOR James C. McMahon, Sep 11 2023 STATUS approved

A365589 Contribution to the OEIS

 

A365589 Numbers that have at least one prime digit and at least one nonprime digit. 2

12, 13, 15, 17, 20, 21, 24, 26, 28, 29, 30, 31, 34, 36, 38, 39, 42, 43, 45, 47, 50, 51, 54, 56, 58, 59, 62, 63, 65, 67, 70, 71, 74, 76, 78, 79, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,1 COMMENTS The first 44 terms coincide with A085556 (Number of prime digits = number of nonprime digits). a(45) = 102; A085556(45) = 1022. LINKS Table of n, a(n) for n=1..65. MATHEMATICA Select[Range[132], ContainsAny[IntegerDigits[#], {2, 3, 5, 7}] && ContainsAny[IntegerDigits[#], {0, 1, 4, 6, 8, 9}]&] (* James C. McMahon, Oct 08 2023 *) CROSSREFS Cf. A085556, A118950, A365472 (complement). Sequence in context: A096514 A074163 A076979 * A085520 A085556 A175225 Adjacent sequences: A365586 A365587 A365588 * A365590 A365591 A365592 KEYWORD nonn,base AUTHOR James C. McMahon, Sep 10 2023 STATUS approved

A364831 Contribution to the OEIS

 

A364831 Primes whose digits are prime and in nonincreasing order. 1

2, 3, 5, 7, 53, 73, 733, 773, 5333, 7333, 7753, 55333, 75533, 75553, 77773, 733333, 755333, 775553, 7553333, 7555333, 7775533, 7777753, 55555333, 55555553, 77755553, 555553333, 755555533, 773333333, 777555553, 777773333, 777775333, 777775553, 777777773 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,1 COMMENTS Intersection of A028867 and A019546. The subsequence for primes whose digits are prime and in strictly decreasing order has just six terms: 2 3 5 7 53 73 (see A177061). LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 MATHEMATICA Select[Prime[Range[3100000]], AllTrue[d = IntegerDigits[#], PrimeQ] && GreaterEqual @@ d &] PROG (Python) from itertools import count, islice, chain, combinations_with_replacement from sympy import isprime def A364831_gen(): # generator of terms yield 2 yield from chain.from_iterable((sorted(s for d in combinations_with_replacement('753', l) if isprime(s:=int(''.join(d)))) for l in count(1))) A364831_list = list(islice(A364831_gen(), 30)) # Chai Wah Wu, Sep 10 2023 CROSSREFS Cf. A009996, A019546, A177061, A028867. Sequence in context: A214179 A178237 A118842 * A244595 A244597 A237439 Adjacent sequences: A364828 A364829 A364830 * A364832 A364833 A364834 KEYWORD nonn,base AUTHOR James C. McMahon, Aug 09 2023 STATUS approved

A362678 Contribution to the OEIS

 

A362678 Primes whose digits are prime and in nondecreasing order. 1

2, 3, 5, 7, 23, 37, 223, 227, 233, 257, 277, 337, 557, 577, 2237, 2333, 2357, 2377, 2557, 2777, 3557, 5557, 22277, 22777, 23333, 23357, 23557, 25577, 33377, 33577, 222337, 222557, 223337, 223577, 233357, 233557, 233777, 235577, 333337, 335557, 355777 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 1,1 COMMENTS Intersection of A009994 and A019546. The subsequence for primes whose digits are prime and in strictly increasing order has just eight terms: 2 3 5 7 23 37 257 2357 (see A177061). LINKS Michael S. Branicky, Table of n, a(n) for n = 1..10000 MAPLE M:= 7: # for terms with <+ M digits R:= NULL: for d from 1 to M do S:= NULL: for x2 from 0 to d do for x3 from 0 to d-x2 do for x5 from 0 to d-x2-x3 do x7:= d-x2-x3-x5; x:= parse(cat(2$x2, 3$x3, 5$x5, 7$x7)); if isprime(x) then S:= S, x fi; od od od; R:= R, op(sort([S])); od: R; # Robert Israel, Jul 04 2023 MATHEMATICA Select[Prime[Range[31000]], AllTrue[d = IntegerDigits[#], PrimeQ] && LessEqual @@ d &] (* Amiram Eldar, Jul 07 2023 *) PROG (Python) from sympy import isprime from itertools import count, combinations_with_replacement as cwr, islice def agen(): yield from (filter(isprime, (int("".join(c)) for d in count(1) for c in cwr("2357", d)))) print(list(islice(agen(), 50))) # Michael S. Branicky, Jul 05 2023 (PARI) isok(p) = if (isprime(p), my(d=digits(p)); (d == vecsort(d)) && (#select(isprime, d) == #d)); \\ Michel Marcus, Jul 07 2023 CROSSREFS Cf. A009994, A019546, A177061. Sequence in context: A100552 A210566 A155873 * A106711 A235110 A048398 Adjacent sequences: A362675 A362676 A362677 * A362679 A362680 A362681 KEYWORD nonn,base AUTHOR James C. McMahon, Jul 03 2023 STATUS approved