A005188 -id:A005188 - cp's OEIS frontend

A256363 Numbers that are multiple-digit narcissistic numbers in exactly four bases.

4901, 8450, 21125, 33125, 41405, 42050, 47125, 71825, 90625, 117325, 142805, 142885, 151250, 184093, 244205, 272000, 325125, 361250, 520625, 535717, 546325, 638450, 690625, 777925, 861125, 874225, 903125, 982125, 990125, 1035125, 1053405

Offset: 1

Tim Johannes Ohrtmann, Mar 26 2015

			a(1) = 4901 because this is the first number that is a multiple-digit narcissistic number in exactly four bases (75, 99, 186 and 4831).

W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256359 (every number of bases).
Cf. A256360, A256361, A256362, A256364, A256365 (1, 2, 3, 5 and 6 bases).
Cf. A256459 (first occurrences).

for(n=3, 1000000, k=0; for(z=2, n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L, x, j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1, j, t+=L[p]^j); if(n==t, k++)); if(k==4, print1(n, ", ")))

A256364 Numbers that are multiple-digit narcissistic numbers in exactly five bases.

Original entry on oeis.org

274625, 465125, 528125, 710645, 912925, 983125

Offset: 1

Tim Johannes Ohrtmann, Mar 30 2015

			a(1) = 274625 because this is the first number that is a multiple-digit narcissistic number in exactly five bases (528, 603, 1318, 1958 and 4217).

W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256359 (every number of bases).
Cf. A256360, A256361, A256362, A256363, A256365 (1 to 4 and 6 bases).
Cf. A256459 (first occurrences).

for(n=3, 1000000, k=0; for(z=2, n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L, x, j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1, j, t+=L[p]^j); if(n==t, k++)); if(k==5, print1(n, ", ")))

A256365 Numbers that are multiple-digit narcissistic numbers in exactly six bases.

Original entry on oeis.org

36125, 190125, 444925

Offset: 1

Tim Johannes Ohrtmann, Mar 30 2015

			a(1) = 36125 because this is the first number that is a multiple-digit narcissistic number in exactly six bases (193, 212, 327, 423, 1057 and 7187).

W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256359 (every number of bases).
Cf. A256360, A256361, A256362, A256363, A256364 (1 to 5 bases).
Cf. A256459 (first occurrences).

for(n=3, 1000000, k=0; for(z=2, n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L, x, j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1, j, t+=L[p]^j); if(n==t, k++)); if(k==6, print1(n, ", ")))

A256459 Least number that is a multiple-digit narcissistic number in exactly n bases.

Original entry on oeis.org

5, 17, 125, 4901, 274625, 36125

Offset: 1

Tim Johannes Ohrtmann, Mar 30 2015

			a(3) = 125 because this is the least number that is a multiple-digit narcissistic number in exactly three bases (12, 23 and 57).

W. Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
Eric Weisstein's World of Mathematics, Narcissistic Number
Wikipedia, Narcissistic number

Cf. A005188.
Cf. A256359 (every number of bases).
Cf. A256360, A256361, A256362, A256363, A256364, A256365 (1 to 6 bases).

b=0; until(b==6, b++; n=2; until(k==b, n++; k=0; for(z=2, n, y=n; j=0; L=List(); until(y==0, x=y%z; j++; listinsert(L, x, j); while(!((y%z)==0), y--); y=y/z); t=0; for(p=1, j, t+=L[p]^j); if(n==t, k++)); if(k==b, print1(n, " "))))

A134703 Powerful numbers (2b): a sum of nonnegative powers of its digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 254, 258, 262, 263, 264, 267, 283, 308, 332, 333, 334, 347, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 472, 518, 538, 598, 629, 635, 653, 675, 730, 731, 732, 733, 734

Offset: 1

David W. Wilson, Sep 05 2009

Here 0 digits may be used, with the convention that 0^0 = 1. Of course 0^1 = 0, so one is free to use the 0 digit to get an extra 1, or not.

			43 = 4^2 + 3^3; 254 = 2^7 + 5^3 + 4^0 = 128 + 125 + 1.
209 = 2^7 + 0^1 + 9^2.
732 = 7^0 + 3^6 + 2^1.

David Wilson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074.

Different from A007532 and A061862, which are variations.

If n = d_1 d_2 ... d_k in decimal then there are integers m_1 m_2 ... m_k >= 0 such that n = d_1^m_1 + ... + d_k^m_k.

A281858 Curious cubic identities based on the Armstrong number 370.

Original entry on oeis.org

370, 336700, 333667000, 333366670000, 333336666700000, 333333666667000000, 333333366666670000000, 333333336666666700000000, 333333333666666667000000000, 333333333366666666670000000000, 333333333336666666666700000000000, 333333333333666666666667000000000000

Offset: 1

Wolfdieter Lang, Feb 08 2017

See a comment in A067275, and the analog to the Armstrong number 153 = A005188(10) treated in A281857, 370 = A005188(11).

			n=1: 370 =  3^3 + 7^3 + 0^3; n=2: 336700 = 33^3 + 67^3 + (00)^3; n=3: 333667000 = 333^3 + 667^3 + (000)^3.

Colin Barker, Table of n, a(n) for n = 1..333
Index entries for linear recurrences with constant coefficients, signature (1110,-111000,1000000).

Cf. A002277, A005188, A067275, A246057, A281857.

Table[FromDigits@ Join[ConstantArray[3, n], ReplacePart[ConstantArray[6, n], -1 -> 7], ConstantArray[0, n]], {n, 12}] (* Michael De Vlieger, Feb 08 2017 *)

Vec(10*x*(37 - 7400*x + 100000*x^2) / ((1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^30)) \\ Colin Barker, Feb 08 2017

a(n) = A002277(n)^3 + A067275(n+1)^3 + 0(n)^3, n >= 1, with 0(n) standing for n 0's.
From Colin Barker, Feb 08 2017: (Start)
G.f.: 10*x*(37 - 7400*x + 100000*x^2) / ((1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
a(n) = 10^n*(1 + 10^n + 100^n) / 3.
a(n) = 1110*a(n-1) - 111000*a(n-2) + 1000000*a(n-3) for n>3. (End)

A157714 Base-10 pseudo-altruistic numbers.

Original entry on oeis.org

136, 160, 217, 244, 259, 352, 496, 586, 664, 736, 853, 862, 1009, 2178, 2929, 3233, 3283, 4274, 4394, 6514, 6562, 7154, 10933, 13154, 18829, 50062, 58618, 59536, 73318, 76438, 124618, 282595, 312962, 329340, 376761, 537059, 578955, 681069

Offset: 1

Hans Havermann, Mar 04 2009

These integers reoccur (with a period greater than 1) upon the iteration of raising every digit to the power of the number's length and summing.

If the reoccurrence is immediate (period 1), the numbers are (instead) narcissistic (A005188).

			2929 is pseudo-altruistic because 2929 -> 13154 (2^4 + 9^4 + 2^4 + 9^4) -> 4394 (1^5 + 3^5 + 1^5 + 5^5 + 4^5) -> 7154 (4^4 + 3^4 + 9^4 + 4^4) -> 3283 (7^4 + 1^4 + 5^4 + 4^4) -> 4274 (3^4 + 2^4 + 8^4 + 3^4) -> 2929 (4^4 + 2^4 + 7^4 + 4^4).

Hans Havermann, Table of n, a(n) for n=1..265
E. Angelini, A Recurring Digital Invariant variant
E. Angelini, A Recurring Digital Invariant Variant [Cached, with permission]
H. Havermann, Self-serving altruistic numbers
H. Havermann, Self-serving altruistic numbers [Cached copy, with permission]

Cf. A005188. The "Recurring Digital Invariant Variant" is described in more detail in A151543.

A261433 k-digit integers equal to the sum of the k-th powers of the tens' complements of their digits.

Original entry on oeis.org

5, 378, 91882, 3762938, 46478818, 564426414

Offset: 1

Paolo P. Lava, Aug 20 2015

The terms of the sequence could be called "Shy n n-digit numbers" as suggested by Geoffrey Campbell, Long Term Visitor (Visiting Fellow), Mathematical Sciences Institute, Australian National University, cf Links.
In base 10, x is a "Shy k n-digit number" if it is an n-digit (d_i) number such that x = Sum_{i=1..n}{(10-d_i)^k}. For instance, 2240 is a "Shy 3 4-digit number": (10 - 2)^3 + (10 - 2)^3 + (10 - 4)^3 + (10 - 0)^3 = 512 + 512 + 216 + 1000 = 2240. Again, 2149042 is a "Shy 6 7-digit number": (10 - 2)^6 + (10 - 1)^6 + (10 - 4)^6 + (10 - 9)^6 + (10 - 0)^6 + (10 - 4)^6 + (10 - 2)^6 =  262144 + 531441 + 46656 + 1 + 1000000 + 46656 + 262144 = 2149042.
It is not known if the sequence is finite. At least there are no other terms up to 18-digit numbers (as tested by Marco Cecchi at LinkedIn link).
If there are further terms, they are greater than 10^33. - Giovanni Resta, Aug 20 2015
Subsequence of A052382. Sequence is finite and complete as verified by exhaustive search since all terms have 60 or fewer digits. Since all terms are zeroless, they are less than k*9^k which would be less than 10^(k-1) (i.e., have fewer than k digits) if k > 60. - Chai Wah Wu, Apr 07 2018

			(10 - 5)^1 = 5,
(10 - 3)^3 + (10 - 7)^3 + (10 - 8)^3 = 343 + 27 + 8 = 378,
(10 - 9)^5 + (10 - 1)^5 + (10 - 8)^5 + (10 - 8)^5 + (10 - 2)^5 = 1 + 59049 + 32 + 32 + 32768 = 91882, etc.

Geoffrey Campbell, Related to Narcissistic numbers: the Shy numbers, Number Theory group on LinkedIn.com
Marco Cecchi, Python program based on partitions.

Cf. A005188, A052382.

with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do a:=ilog10(n)+1; b:=0; c:=n;
for k from 1 to a do b:=b+(10-(c mod 10))^a; c:=trunc(c/10); od;
if b=n then print(n); fi; od; end: P(10^9);

Select[Range[10^5], # == Total[(10 - IntegerDigits@ #)^ IntegerLength[#]] &] (* Giovanni Resta, Aug 20 2015 *)

isok(n) = (d = digits(n)) && (sum(k=1, #d, (10-d[k])^#d) == n); \\ Michel Marcus, Aug 24 2015

from itertools import combinations_with_replacement
A261433_list = []
for k in range(1,10):
    a, k10 = tuple([i**k for i in range(10,0,-1)]), 10**k
    for b in combinations_with_replacement(range(1,10),k):
        x = sum(list(map(lambda y:a[y],b)))
        if x < k10 and tuple(int(d) for d in sorted(str(x))) == b:
            A261433_list.append(x)
A261433_list = sorted(A261433_list) # Chai Wah Wu, Aug 25 2015, updated Apr 06, 2018

a(4)-a(6) found by Aleksander Zujev

A281857 Numbers occurring in a curious cubic identity.

Original entry on oeis.org

153, 165033, 166500333, 166650003333, 166665000033333, 166666500000333333, 166666650000003333333, 166666665000000033333333, 166666666500000000333333333, 166666666650000000003333333333, 166666666665000000000033333333333, 166666666666500000000000333333333333

Offset: 1

Wolfdieter Lang, Feb 07 2017

See A246057 for the van der Poorten et al. reference and a comment.

153 is the Armstrong number A005188(10). [Typo corrected by Jeremy Tan, Feb 25 2023]

			1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ...

Colin Barker, Table of n, a(n) for n = 1..333
Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000).

Cf. A002277, A005188, A093143, A246057, A281858, A281859, A281860.

Table[FromDigits@ Join[ReplacePart[ConstantArray[6, n], 1 -> 1], ReplacePart[ConstantArray[0, n], 1 -> 5], ConstantArray[3, n]], {n, 12}] (* Michael De Vlieger, Feb 08 2017 *)

Vec(9*x*(17 - 550*x + 33500*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)) + O(x^15)) \\ Colin Barker, Feb 08 2017

a(n) = (((10^n - 4)/6)^3) + ((10^n/2)^3) + (((10^n - 1)/3)^3) \\ Jean-Jacques Vaudroz, Aug 11 2024

a(n) = A246057(n-1)^3 + A093143(n)^3 + A002277(n)^3, n >= 1.
From Colin Barker, Feb 08 2017: (Start)
G.f.: 9*x*(17 - 550*x + 33500*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).
a(n) = (-2 + 2^(1+n)*5^n - 100^n + 1000^n) / 6.
a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4) for n>4. (End)

A056733 Each number is the sum of the cubes of its 3 sections.

Original entry on oeis.org

153, 370, 371, 407, 165033, 221859, 336700, 336701, 340067, 341067, 407000, 407001, 444664, 487215, 982827, 983221, 166500333, 296584415, 333667000, 333667001, 334000667, 710656413, 828538472, 142051701000, 166650003333, 262662141664, 333366670000

Offset: 1

Carlos Rivera, Aug 13 2000

The first four terms are also called Narcissistic or Armstrong numbers. The first 16 terms are found in Spencer's book, pages 65 and 101.
The sequence contains several infinite subsequences such as 153, 165033, 166500333, 166650003333, ...; 370, 336700, 333667000, 333366670000, ... or 371, 336701, 333667001, 333366670001, ... - Ulrich Schimke (ulrschimke(AT)aol.com), Jun 08 2001
From Daniel Forgues, Jan 30 2015: (Start)
The subsequence {153, 165033, 166500333, ...} consists of numbers of the form
  [(10^n - 4) / 6] * (10^n)^2 + [(10^n) / 2] * (10^n)^1 +
  [(10^n - 1) / 3] * (10^n)^0 =
  [(10^n - 4) / 6]^3 + [(10^n) / 2]^3 + [(10^n - 1) / 3]^3, n >= 1,
thus equal to the sum of the cube of their "digits" in base 10^n.
The subsequence {370, 336700, 333667000, ...} consists of numbers of the form
  [(10^n - 1) / 3] * (10^n)^2 + {10^n - [(10^n - 1) / 3]} * (10^n)^1 =
  [(10^n - 1) / 3]^3 + {10^n - [(10^n - 1) / 3]}^3, n >= 1,
thus equal to the sum of their "digits" in base 10^n.
The subsequence {371, 336701, 333667001, ...} is trivially derived from the subsequence {370, 336700, 333667000, ...}, since 1^3 = 1.
The subsequence {407, 340067, 334000667, ...} consists of numbers of the form
  {10^n - 2 * [(10^n - 1) / 3]} * (10^n)^2 +
  {10^n - [(10^n - 1) / 3]} * (10^n)^0 =
  {10^n - 2 * [(10^n - 1) / 3]}^3 + {10^n - [(10^n - 1) / 3]}^3, n >= 1,
thus equal to the sum of their "digits" in base 10^n.
"There are just four numbers (after 1) which are the sums of the cubes of their digits, viz. 153 = 1^3 + 5^3 + 3^3, 370 = 3^3 + 7^3 + 0^3, 371 = 3^3 + 7^3 + 1^3, and 407 = 4^3 + 0^3 + 7^3. This is an odd fact, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in it which appeals much to a mathematician. The proof is neither difficult nor interesting--merely a little tiresome. The theorem is not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciation and the proof, which is not capable of any significant generalization." -- G. H. Hardy, "A Mathematician’s Apology" (End)
From Daniel Forgues, Feb 04 2015: (Start)
The subsequence {341067, 333401006667, 333334001000666667, ...} is trivially derived from the even-indexed terms 2n, n >= 1, of the subsequence {407, 340067, 334000667, 333400006667, ...}, since (10^n)^3 = 10^n * 10^(2n). These numbers are equal to the sum of the cube of their "digits" in base 10^(2n), n >= 1.
The number 407000 is trivially derived from 407, since 40^3 + 70^3 =
(4 * 10)^3 + (7 * 10)^3 = (4^3 + 7^3) * 10^3 = 407 * 1000 = 407000.
The number 407001 is trivially derived from 407000, since 1^3 = 1. (End)
From Jose M. Arenas, Mar 08 2017: (Start)
The subsequence {340067000000, 334000667000000000, 333400006667000000000000, ...} consists of numbers of the form
  (4 * 10^(n + 2) + ((10^(n + 1) - 1) / 3) * 10^(n + 3)) * 10^(4 * n + 8) +
  (7 * 10^(n + 2) + (2 * (10^(n + 1) - 1) / 3) * 10^(n + 3)) * 10^(2 * n + 4) =
  (4 * 10^(n + 2) + ((10^(n + 1) - 1) / 3) * 10^(n + 3))^3 +
  (7 * 10^(n + 2) + (2 * (10^(n + 1) - 1) / 3) * 10^(n + 3))^3, n >= 0,
thus equal to the sum of their 3 sections, each section of (2 * n + 4) digits.
The subsequence {340067000001, 334000667000000001, 333400006667000000000001, ...} is trivially derived from the subsequence {340067000000, 334000667000000000, 333400006667000000000000, ...}, since 1^3 = 1. (End)

			333667001 = 333^3 + 667^3 + 001^3, so 333667001 is a term.

J. S. Madachy, Madachy's Mathematical Recreations, pp. 166, Dover, NY, 1979.
Donald D. Spencer, "Exploring number theory with microcomputers", pp. 65 and 101, Camelot Publishing Co.

Jose M. Arenas and Giovanni Resta, Table of n, a(n) for n = 1..69 (terms < 10^18, first 49 terms from Jose M. Arenas)
Jose M. Arenas, Python code.

Cf. A005188.

See A271730 for a related sequence.

f[n_] := Block[{len = IntegerLength@ n}, If[IntegerQ[len/3], n == Plus @@ Flatten[(FromDigits /@ Partition[IntegerDigits@ n, len/3])^3], False]]; Select[Range[10^6], f] (* Michael De Vlieger, Jan 31 2015 *)

def a():
  n = 1
  while n < 10**9:
    st = str(n)
    if len(st) % 3 == 0:
      s1 = st[:int(len(st)/3)]
      s2 = st[int(len(st)/3):int(2*len(st)/3)]
      s3 = st[int(2*len(st)/3):int(len(st))]
      if int(s1)**3+int(s2)**3+int(s3)**3 == int(st):
        print(n, end=', ')
        n += 1
      else:
        n += 1
    else:
      n = 10*n
a()
# Derek Orr, Jul 03 2014

def a():
  for i in range(1,10):
    for j in range(10):
      for k in range(10):
        if i**3 + j**3 + k**3 == i*100 + j*10 + k:
          print(i*100 + j*10 + k)
  for i in range(10,100):
    for j in range(100):
      for k in range(100):
        if i**3 + j**3 + k**3 == i*10000 + j*100 + k:
          print(i*10000 + j*100 + k)
  for i in range(100,1000):
    for j in range(1000):
      for k in range(1000):
        if i**3 + j**3 + k**3 == i*1000000 + j*1000 + k:
          print(i*1000000 + j*1000 + k)
a()
# Denys Contant, Feb 23 2017

Offset changed to 1 by N. J. A. Sloane, Jul 07 2014

a(24)-a(27) from Jose M. Arenas, Mar 08 2017

cp's OEIS Frontend

A256363 Numbers that are multiple-digit narcissistic numbers in exactly four bases.

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PARI

A256364 Numbers that are multiple-digit narcissistic numbers in exactly five bases.

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PARI

A256365 Numbers that are multiple-digit narcissistic numbers in exactly six bases.

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PARI

A256459 Least number that is a multiple-digit narcissistic number in exactly n bases.

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PARI

A134703 Powerful numbers (2b): a sum of nonnegative powers of its digits.

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A281858 Curious cubic identities based on the Armstrong number 370.

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Mathematica

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A157714 Base-10 pseudo-altruistic numbers.

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A261433 k-digit integers equal to the sum of the k-th powers of the tens' complements of their digits.

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