Narcissistic Numbers

 (Image: Narciso (Jan Cossiers), CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons)

In Greek mythology, Narcissus was known for his beauty. He rejected advances from all suitors, both men and women, and instead fell in love with his own reflection. The legend gives us the source of the words narcissist and narcissistic. 

There are also numbers known as Narcissistic Numbers. In the On-Line Encyclopedia of Integer Sequences (OEIS), sequence A005188 defines such numbers to be:  m-digit positive numbers equal to the sum of the m-th powers of their digits.

For example, the number 153 has 3 digits. The sum of 1^3 + 5^3 + 3^3 = 153. The sequence of Narcissistic Numbers begins: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407...

Narcissistic Numbers.

Münchhausen Numbers

 

(Image: Statue of Baron Münchhausen riding half of a horse - https://commons.wikimedia.org/wiki/User:Franzfoto)

Baron Münchhausen is a fictional character known for making exaggerated stories (other spellings include Munchhausen and Munchausen). In addition to the fables of the Baron, the name is used to describe Münchhausen syndrome and Münchhausen syndrome by proxy, in which someone fabricates a story of their illness or the illness of one they care for to gain attention or sympathy.

There are also numbers known as Münchhausen Numbers. These are numbers that equal the sum of each of their digits raised to their own power. For example, in base ten, the numbers include 1 and 3435. 1^1=1, 3^3 +4^4+3^3+5^5=3435. This sequence of numbers is A046253 in the On-Line Encyclopedia of Integer Sequences (OEIS). Another sequence of other Münchhausen Numbers is A166623, which lists such numbers in multiple bases.

Achilles Numbers

 

(Photo: Closeup of Achilles thniskon in Corfu Achilleion  - Dr.K., CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons)

In Greek mythology, Achilles, a central figure in Homer's Iliad, was known as the greatest Greek warrior. By legend, Achilles was invulnerable in all of his body except for one heel. According to myth, when his mother Thetis dipped him in the river Styx as an infant, she held him by one of his heels, leaving it untouched by the waters and thus his only vulnerable body part. To this day, Achilles' heel refers to a point of weakness that can lead to downfall.

There is also a series of numbers named for Achilles. Sequence A052486 of the On-Line Encyclopedia of Integer Sequences (OEIS) describes Achilles numbers as "powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power."

This definition uses the names of other types of numbers. 

Powerful numbers:  if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers). This is sequence A001694 in the OEIS.

Perfect numbers: Numbers k such that k is equal to the sum of the proper divisors of k. See OEIS: A000396.

Perfect power: Numbers of the form m^k where m > 0 and k >= 2.  See OEIS: A001597.

The first few Achilles numbers are: 72, 108, 200, 288, 392.

This post was inspired by the comment left by a reader to a recent post about the number 108. He recognized that 108 is also an Achilles number.

50400: A Highly Composite Number

This blog recently had its 50400th visitor. The number 50400 is the 27th highly composite number. A highly composite number has more divisors than all previous highly composite numbers. In the case of 50400, it has 108 divisors.

The first 27 highly composite numbers and the number of their divisors are:

Order	Highly Composite Number	Number of Divisors
1	1	1
2	2	2
3	4	3
4	6	4
5	12	6
6	24	8
7	36	9
8	48	10
9	60	12
10	120	16
11	180	18
12	240	20
13	360	24
14	720	30
15	840	32
16	1260	36
17	1680	40
18	2520	48
19	5040	60
20	7560	64
21	10080	72
22	15120	80
23	20160	84
24	25200	90
25	27720	96
26	45360	100
27	50400	108

The highly composite numbers are designated in the On-Line Encyclopedia of Integer Sequences as A002182 , and the number of their divisors is the sequence A002183.

A385454 - Contribution to The OEIS: Difference of the largest and smallest semiperimeters of an integral rectangle with area n.

 A385454

Difference of the largest and smallest semiperimeters of an integral rectangle with area n.

0

0, 0, 0, 1, 0, 2, 0, 3, 4, 4, 0, 6, 0, 6, 8, 9, 0, 10, 0, 12, 12, 10, 0, 15, 16, 12, 16, 18, 0, 20, 0, 21, 20, 16, 24, 25, 0, 18, 24, 28, 0, 30, 0, 30, 32, 22, 0, 35, 36, 36, 32, 36, 0, 40, 40, 42, 36, 28, 0, 45, 0, 30, 48, 49, 48, 50, 0, 48, 44, 54, 0, 56, 0

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

OFFSET

1,6

COMMENTS

For all noncomposite n, a(n) = 0.

For each square k^2, a(k^2) = (k^2 + 1) - 2*k = (k-1)^2.

LINKS

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

FORMULA

a(n) = 1 + n - A063655(n).

EXAMPLE

The largest semiperimeter of an integral rectangle with area 9 is 10 (1 x 9 rectangle); the smallest semiperimeter is 6 (3 x 3 rectangle). The difference, a(9) = 4.

MATHEMATICA

a[n_]:=1+n-2Median[Divisors[n]]; Array[a, 73]

CROSSREFS

Cf. A063655.

Sequence in context: A274441 A213859 A330492 * A350534 A101336 A300001

Adjacent sequences: A385442 A385443 A385444 * A385459 A385460 A385461

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Jun 29 2025

STATUS

approved

A385288 - Contribution to the OEIS: Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent

 A385288

Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent.

0

4, 8, 9, 25, 27, 32, 49, 72, 108, 121, 125, 128, 169, 200, 243, 288, 289, 343, 361, 392, 500, 529, 675, 800, 841, 961, 968, 972, 1125, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1800, 1849, 2048, 2187, 2197, 2209, 2312, 2700, 2809, 2888, 3087, 3125, 3267, 3481

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

OFFSET

1,1

COMMENTS

a(n) = A114129(n) through n=25; then a(26) = 961 and A114129(26) = 864.

Subset of A056166.

Subset of A001694. - Michael De Vlieger, Jun 25 2025.

LINKS

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

EXAMPLE

200 = 2^3 * 5^2; 200 has a prime number of prime factors, counted with multiplicity (3 + 2 = 5), and exponents 3 and 2 are prime.

MATHEMATICA

Select[Range[10^4], AllTrue[Last/@FactorInteger[#], PrimeQ]&&PrimeQ[PrimeOmega[#]]&]

PROG

(PARI) isok(k) = my(f=factor(k)); isprime(bigomega(k)) && (sum(k=1, #f~, isprime(f[k, 2])) == omega(f)); \\ Michel Marcus, Jun 25 2025

CROSSREFS

Cf. A001248, A001694, A030078, A056166, A114129.

Sequence in context: A383211 A051676 A114129 * A383103 A053810 A076702

Adjacent sequences: A385283 A385284 A385286 * A385289 A385292 A385293

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Jun 24 2025

STATUS

approved

Online Encyclopedia of Integer Sequences, OEIS.

40,000 Views!

(Graphic: Grok)

Celebrating this blog's 40,000th view. Here are some interesting numbers close to 40,000 from Wikipedia (interestingly, all facts of these numbers were from the OEIS):

40001 to 40999

• 40320 = smallest factorial (8!) that is not a highly composite number
• 40425 = square pyramidal number
• 40585 = largest factorion^[1]
• 40678 = pentagonal pyramidal number
• 40755 = the smallest number >1 to be a Triangular number, Pentagonal number, and a Hexagonal number.^[2] Additionally, it's a 390-gonal, 4077-gonal, and 13586-gonal number.^[3][4][5]
• 40804 = palindromic square
  
  1.  "Sloane's A014080 : Factorions". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-15.
  2.  "A046180 - OEIS". oeis.org. Retrieved 2024-12-18.
  3. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4.  Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. Sloane, N. J. A. (ed.). "Sequence A000384 (Hexagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

 

A381246 Contribution to the OEIS (supporting a juggler sequence)

 A381246

Largest value in trajectory of n under the juggler map of A380891.

0

1, 2, 4, 4, 8, 6, 30, 8, 18, 10, 24, 12, 30, 14, 36, 16, 150, 18, 50, 20, 1320, 22, 43366048, 24, 41678, 26, 350, 28, 41678, 30, 234421146, 32, 2438232, 34, 114, 36, 5184, 38, 132, 40, 124026, 42, 150, 44, 160, 46, 934, 48, 1008, 50, 1084, 52, 43366048, 54, 1240

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

OFFSET

1,2

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)]]; a381246[n_]:=Max[Delete[FixedPointList[fj, n], -1]]; Array[a381246, 55]

PROG

(Python)

import sys

import gmpy2

sys.set_int_max_str_digits(0)

def floorJuggler(n):

a=n

max=n

while a > 1:

b=0

if a%2 == 0:

b1=gmpy2.iroot(a, 3)

b=b1[0]

else:

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

b=b1[0]

a=b

if a > max:

max = a

return max

maxcount=0

for i in range (1, 100):

print (i, floorJuggler(i))

CROSSREFS

Cf. A380891, A383135.

Sequence in context: A117295 A235999 A093820 * A095400 A237264 A265560

Adjacent sequences: A381243 A381244 A381245 * A381247 A381248 A381249

KEYWORD

nonn,new

AUTHOR

James C. McMahon and Vikram Prasad, Apr 17 2025

STATUS

approved

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