A383135 Contribution to the OEIS (supporting a juggler sequence)

 

A383135

a(n) = number of iterations that n requires to reach 1 under the x -> A380891(x) map, or -1 if it never does.

0

0, 1, 2, 1, 3, 1, 5, 2, 3, 2, 3, 2, 4, 2, 4, 2, 6, 2, 4, 2, 6, 2, 13, 2, 13, 2, 8, 3, 8, 3, 10, 3, 10, 3, 3, 3, 10, 3, 5, 3, 10, 3, 5, 3, 5, 3, 6, 3, 5, 3, 5, 3, 17, 3, 5, 3, 5, 3, 17, 3, 3, 3, 3, 2, 12, 2, 3, 2, 5, 2, 5, 2, 12, 2, 3, 2, 7, 2, 3, 2, 7, 2, 7, 2

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,3

COMMENTS

A380891(x) map is If x mod 2 = 0 then a(x) = floor(x^(1/3)) else a(x) = floor(x^(4/3)).

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.

MATHEMATICA

fj[n_]:=If[Mod[n, 2]==0, Floor[Surd[n, 3]], Floor[n^(4/3)]]; a383135[n_]:= Length[ NestWhileList[fj, n, # != 1 &]] - 1; Array[ a383135, 84]

PROG

(Python)

import sys

import gmpy2

sys.set_int_max_str_digits(0)

def floorJuggler(n) :

a=n

count=0

while (a > 1) :

b=0

if (a%2 == 0) :

b1=gmpy2.iroot(a, 3)

b=b1[0]

count=count+1

else :

b1=gmpy2.iroot(a**4, 3)

b=b1[0]

count=count+1

a=b

return count

for i in range (1, 100) :

print (i, floorJuggler(i))

CROSSREFS

Cf. A380891, A381246.

Cf. A006577.

Sequence in context: A277120 A104725 A289079 * A249810 A257111 A011129

Adjacent sequences: A383132 A383133 A383134 * A383136 A383137 A383138

KEYWORD

nonn,new

AUTHOR

James C. McMahon and Vikram Prasad, Apr 17 2025

STATUS

approved

A380943 Contribution to the OEIS (Saint Patrick Primes)

 A380943

Primes written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

+40
0

37, 73, 113, 173, 197, 311, 313, 317, 331, 337, 359, 367, 373, 593, 617, 673, 719, 733, 761, 797, 977, 1093, 1097, 1123, 1277, 1319, 1361, 1373, 1783, 1913, 1931, 1979, 1997, 2293, 2297, 2311, 2347, 2389, 2713, 2837, 2971, 3109, 3119, 3137, 3191, 3229, 3271

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,1

COMMENTS

Member primes could be named Saint Patrick primes because the date of Saint Patrick's Day, March 17 (3/17), produces the terms 173 and 317.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..1000

Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 40 and pi(q) <= 40, labeling p in red and q in blue.

Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 2000 and pi(q) <= 2000.

EXAMPLE

Primes 173 and 317 are members because they are formed by the concatenation of 17 & 3 and 3 & 17, respectively.

While the concatenation of 13 and 7 forms the prime 137, it is not a member because the concatenation of 7 and 13 is 713, which is not prime.

MATHEMATICA

lim=3300; plim=Max[FromDigits[Rest[IntegerDigits[lim]]], FromDigits[Drop[IntegerDigits[lim], -1]]]; p=Prime[Range[PrimePi[plim]]]; p2=Subsets[p, {2}]; fp[{a_, b_}]:=FromDigits[Join[IntegerDigits[a], IntegerDigits[b]]]; rfp[{a_, b_}]:=FromDigits[Join[IntegerDigits[b], IntegerDigits[a]]]; pabba=Select[p2, PrimeQ[fp[#]]&&PrimeQ[rfp[#]]&]; pp1=fp/@pabba; pp2=rfp/@pabba; Select[Union[Join[pp1, pp2]], #<=lim&]

PROG

(Python)

from sympy import isprime

def ok(n):

if not isprime(n): return False

s = str(n)

return any(s[i]!='0' and isprime(int(s[:i])) and isprime(int(s[i:])) and isprime(int(s[i:]+s[:i])) for i in range(1, len(s)))

print([k for k in range(3300) if ok(k)]) # Michael S. Branicky, Apr 05 2025

CROSSREFS

Subsequence of A019549 and A105184.

Cf. A133187.

KEYWORD

nonn,base,new

AUTHOR

James C. McMahon, Apr 03 2025

STATUS

approved

See post: https://jamesmacmath.blogspot.com/2025/03/saint-patrick-another-prime-day.html

Also, Michael De Vlieger made interesting graphics for this sequence: https://oeis.org/A380943/a380943.png and https://oeis.org/A380943/a380943_1.png

Smart Crows and Geometry

 

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

Crows are recognized as very intelligent. Listen to this from NPR: https://www.npr.org/2025/04/11/nx-s1-5355849/a-new-study-finds-crows-can-recognize-geometric-shapes

Or read the summary here: https://www.npr.org/2025/04/12/nx-s1-5359438/a-crows-math-skills-include-geometry

Also summarized in ScienceAdvances: https://www.science.org/doi/10.1126/sciadv.adt3718

Another post on math skills of animals: https://jamesmacmath.blogspot.com/2025/03/counting-critters-whos-got-numbers-game.html

Some researchers believe birds developed intelligence in a way independent of how humans and primates did.