A061211 -id:A061211 - cp's OEIS frontend

A076088 Duplicate of A061211.

Original entry on oeis.org

9, 81, 19683, 1679616, 205962976, 68719476736, 6722988818432, 248155780267521

Offset: 1

dead

A061209 Numbers which are the cubes of their digit sum.

Original entry on oeis.org

0, 1, 512, 4913, 5832, 17576, 19683

Offset: 1

Amarnath Murthy, Apr 21 2001

It can be shown that 19683 = (1 + 9 + 6 + 8 + 3)^3 = 27^3 is the largest such number.
Numbers of Dudeney. - Philippe Deléham, May 11 2013
If a number n has d digits, 10^(d-1) <= n < 10^d, the cube of the digit sum is at most (d*9)^3 = 729*d^3; if d > 6 this is strictly smaller than 10^(d-1) and cannot be equal to n. See also A061211. - M. F. Hasler, Apr 12 2015

			4913 = (4 + 9 + 1 + 3)^3.

H. E. Dudeney, 536 Puzzles & Curious Problems, Souvenir Press, London, 1966, p. 36, #120.
Amarnath Murthy, The largest and the smallest m-th power whose digit sum is the m-th root. (To be published)
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Joseph S. Madachy, Recreational Mathematics, Fibonacci Quarterly 6:1 (1968), pp. 60-68.
Wikipedia, Dudeney number

Cf. A007953, A061210, A061211, A252648.

Select[Range[20000],Total[IntegerDigits[#]]^3==#&] (* Harvey P. Dale, Apr 11 2015 *)

for(n=0,999999,sumdigits(n)^3==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

a(n) = A007953(a(n))^3. - M. F. Hasler, Apr 12 2015

Initial term 0 added by M. F. Hasler, Apr 12 2015

A061210 Numbers which are the fourth powers of their digit sum.

Original entry on oeis.org

0, 1, 2401, 234256, 390625, 614656, 1679616

Offset: 1

Amarnath Murthy, Apr 21 2001

It can be shown that 1679616 = 36^4 is the largest such number.

			614656 = ( 6+1+4+6+5+6)^4 =28^4.

Amarnath Murthy, The largest and the smallest m-th power whose digit sum is the m-th root. (To be published)
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

Cf. A061209 (with cubes), A061211.

Cf. A046000, A076090, A046017; A252648 and references there.

Select[Range[0,17*10^5],#==Total[IntegerDigits[#]]^4&] (* Harvey P. Dale, Sep 22 2019 *)

isok(n) = n == sumdigits(n)^4; \\ Michel Marcus, Jan 22 2015

Corrected by Ulrich Schimke, Feb 11 2002

Initial 0 added by M. F. Hasler, Apr 12 2015

A046017 Least k > 1 with k = sum of digits of k^n, or 0 if no such k exists.

Original entry on oeis.org

2, 9, 8, 7, 28, 18, 18, 46, 54, 82, 98, 108, 20, 91, 107, 133, 80, 172, 80, 90, 90, 90, 234, 252, 140, 306, 305, 90, 305, 396, 170, 388, 170, 387, 378, 388, 414, 468, 449, 250, 432, 280, 461, 280, 360, 360, 350, 370, 270, 685, 360, 625, 648, 370, 677, 684, 370, 667, 370, 694, 440, 855, 827, 430, 818

Offset: 1

David W. Wilson

First non-occurrence happens with exponent 105. There is no x such that sum-of-digits{x^105}=x (x>1). - Patrick De Geest, Aug 15 1998

			a(3) = 8 since 8^3 = 512 and 5+1+2 = 8; a(5) = 28 because 28 is least number > 1 with 28^5 = 17210368, 1+7+2+1+0+3+6+8 = 28. 53^7 = 1174711139837 -> 1+1+7+4+7+1+1+1+3+9+8+3+7 = 53.
a(10) = 82 because 82^10 = 13744803133596058624 and 1 + 3 + 7 + 4 + 4 + 8 + 0 + 3 + 1 + 3 + 3 + 5 + 9 + 6 + 0 + 5 + 8 + 6 + 2 + 4 = 82.
a(13) = 20: 20^13=81920000000000000, 8+1+9+2=20.
a(17) = 80: 80^17=225179981368524800000000000000000, 2+2+5+1+7+9+9+8+1+3+6+8+5+2+4+8 = 80.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 208-210.
Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 172.

Carole Dubois, Table of n, a(n) for n = 1..4522 (terms 1..1000 from T. D. Noe).
Carole Dubois, Scatterplot of A046017

Cf. A046459, A046000, A046471, A061211.

Cf. A133509 (n for which a(n)=0), A152147 (table of k for each n).

a[n_] := For[k = 2, k <= 20*n, k++, Which[k == Total[IntegerDigits[k^n]], Return[k], k == 20*n, Return[0]]]; Table[a[n] , {n, 1, 105}] (* Jean-François Alcover, May 23 2012 *)
sdk[n_]:=Module[{k=2},While[k!=Total[IntegerDigits[k^n]],k++];k]; Array[sdk,70] (* Harvey P. Dale, Jan 07 2024 *)

from itertools import chain
def c(k, n): return sum(map(int, str(k**n))) == k
def a(n):
    if n == 0: return False
    d, lim = 1, 1
    while lim < n*9*d: d, lim = d+1, lim*10
    m = next(k for k in chain(range(2, lim+1), (0,)) if c(k, n))
    return m
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Jul 06 2022

More terms from Asher Auel, Jun 01 2000

A046000 a(n) is the largest number m equal to the sum of digits of m^n.

Original entry on oeis.org

1, 9, 9, 27, 36, 46, 64, 68, 63, 81, 117, 108, 108, 146, 154, 199, 187, 216, 181, 207, 207, 225, 256, 271, 288, 337, 324, 307, 328, 341, 396, 443, 388, 423, 463, 477, 424, 495, 469, 523, 502, 432, 531, 572, 603, 523, 592, 666, 667, 695, 685, 685, 739, 746, 739, 683, 684, 802, 754, 845, 793, 833, 865

Offset: 0

David W. Wilson and Patrick De Geest

Cases a(n) = 1 begin: 0, 105, 164, 186, 194, 206, 216, 231, 254, 282, 285, ... Cf. A133509. - Jean-François Alcover, Jan 09 2018

			a(3) = 27 because 27 is the largest number with 27^3 = 19683 and 1+9+6+8+3 = 27.
a(5) = 46 because 46 is the largest number with 46^5 = 205962976 and 2+0+5+9+6+2+9+7+6 = 46.

Amarnath Murthy, The largest and the smallest m-th power whose digits sum /product is its m-th root. To appear in Smarandache Notions Journal, 2003.
Amarnath Murthy, e-book, "Ideas on Smarandache Notions" MS.LIT
Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 172.

T. D. Noe, Table of n, a(n) for n = 0..1000 (a(105), a(164), a(186), ... corrected by Mohammed Yaseen)

Cf. A046459, A055575, A055576, A055577.
Cf. A046017, A046019, A046471, A133509, A152147.
Cf. A061211, A076090.

meanDigit = 9/2; translate = 900; upperm[1] = translate;
upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
(* assuming that upper bound of m fits the implicit curve m = Log[10, m^n]*9/2 *)
a[0] = 1; a[n_] := (For[max = m = 0, m <= upperm[n], m++, If[m == Total[IntegerDigits[m^n]], max = m]]; max);
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Jan 09 2018, updated Jul 07 2022 *)

def ok(k, n): return sum(map(int, str(k**n))) == k
def a(n):
    d, lim = 1, 1
    while lim < n*9*d: d, lim = d+1, lim*10
    return next(k for k in range(lim, 0, -1) if ok(k, n))
print([a(n) for n in range(63)]) # Michael S. Branicky, Jul 06 2022

a(n) = A061211(n)^(1/n), for n > 0.

More terms from Asher Auel, Jun 01 2000
More terms from Franklin T. Adams-Watters, Sep 01 2006
Edited by N. J. A. Sloane at the suggestion of David Wasserman, Dec 12 2007

A133509 Numbers k such that m=1 is the only number for which the sum of digits of m^k equals m.

Original entry on oeis.org

0, 105, 164, 186, 194, 206, 216, 231, 254, 282, 285, 302, 314, 324, 374, 386, 402, 416, 456, 468, 491, 504, 521, 552, 588, 606, 610, 615, 629, 651, 656, 657, 696, 759, 794, 830, 842, 854, 870, 903, 906, 954, 956, 981, 998, 1029, 1064, 1079, 1082, 1109, 1112, 1131

Offset: 1

Farideh Firoozbakht, Dec 04 2007

Carole Dubois, Table of n, a(n) for n = 1..226 (terms 1..100 from Michael S. Branicky)
Carole Dubois, Pin plot of A133509

Cf. A046000, A046017, A046019, A046471, A061211, A152147.

def ok(n):
    d, lim = 1, 1
    while lim < n*9*d: d, lim = d+1, lim*10
    return not any(sum(map(int, str(k**n))) == k for k in range(2, lim+1))
for k in range(195):
    if ok(k): print(k, end=", ") # Michael S. Branicky, Jul 06 2022

If t is a term, A046000(t)=1, A046017(t)=0, A046019(t)=1, A046471(t)=0 and A061211(t)=1. - Mohammed Yaseen, Jun 29 2022

Description improved by T. D. Noe, Nov 26 2008
Extension by T. D. Noe, Nov 26 2008
Edited by Charles R Greathouse IV, Aug 02 2010
a(1) = 0 and a(46) and beyond from Michael S. Branicky, Jul 06 2022

A023106 a(n) is a power of the sum of its digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 512, 2401, 4913, 5832, 17576, 19683, 234256, 390625, 614656, 1679616, 17210368, 34012224, 52521875, 60466176, 205962976, 612220032, 8303765625, 10460353203, 24794911296, 27512614111, 52523350144, 68719476736

Offset: 0

David W. Wilson

Base-10 Reacher numbers: named for the character Jack Reacher in the series of books by Lee Child. Reacher likes the number 81 because it is the square of the sum of its base-10 digits. - Jeffrey Shallit, Apr 03 2015

Contains A061209 and A061210 and all terms of A061211. See A252648 for numbers which are the sum of some power of their digits. - M. F. Hasler, Apr 13 2015

			2401 is an element because 2401 = 7^4 is a power of its digit sum 7.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

David W. Wilson, Table of n, a(n) for n = 0..1137
Jeffrey Shallit, Mathematics in a Jack Reacher Novel, blog post, September 8 2007.

Cf. A061209, A061210, A061211, A252648.

fQ[n_] := Block[{b = Plus @@ IntegerDigits[n]}, If[b > 1, IntegerQ[ Log[b, n]] ]]; Take[ Select[ Union[ Flatten[ Table[n^m, {n, 55}, {m, 9}]]], fQ[ # ] &], 31] (* Robert G. Wilson v, Jan 28 2005 *)
Join[{0,1},Select[Range[0,1700000],IntegerQ[Log[Total[IntegerDigits[#]],#]]&]//Quiet] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Mar 30 2024 *)

is(n)={n<10||(!(n%s=sumdigits(n))&&s>1&&n==s^round(log(n)/log(s)))} \\ M. F. Hasler, Apr 13 2015

import math
def is_valid(n): dsum = sum(map(int, str(n))); return dsum ** int(round(math.log(n, dsum))) == n if dsum > 1 else n < 2
# Victor Dumbrava, May 02 2018

A254000 Numbers equal to the fifth powers of the sums of their digits.

Original entry on oeis.org

0, 1, 17210368, 52521875, 60466176, 205962976

Offset: 1

Michal Paulovic, Jan 21 2015

			205962976 = 46^5 = (2 + 0 + 5 + 9 + 6 + 2 + 9 + 7 + 6)^5.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

Cf. A061209 (with cubes), A061210 (with 4th powers), A061211.

Select[Range@ 210000000, Plus @@ IntegerDigits@ # ^ 5 == # &] (* Michael De Vlieger, Feb 25 2015 *)

lista(nn) = {for (n=0, nn, if (n^5 == sumdigits(n^5)^5, print1(n^5, ", ")););} \\ Michel Marcus, Feb 23 2015

a(n) = A055576(n)^5. - Michel Marcus, Feb 23 2015

A375343 Numbers which are the sixth powers of their digit sum.

Original entry on oeis.org

0, 1, 34012224, 8303765625, 24794911296, 68719476736

Offset: 1

René-Louis Clerc, Aug 12 2024

Solutions can have no more than 13 digits, since (13*9)^6 < 10^13.

			68719476736 = (6+8+7+1+9+4+7+6+7+3+6)^6 = 64^6.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.

Cf. A001014, A007953, A061209, A061210, A254000, A061211.

Cf. A055577, A152147.

for (k=0, sqrtnint(10^13,6), if (k^6 == sumdigits(k^6)^6, print1(k^6, ", ")); )

{ k : k = A007953(k)^6}.

a(n) = A055577(n)^6. - Alois P. Heinz, Aug 24 2024

A379767 Triangle read by rows: row n lists numbers which are the n-th powers of their digit sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 81, 0, 1, 512, 4913, 5832, 17576, 19683, 0, 1, 2401, 234256, 390625, 614656, 1679616, 0, 1, 17210368, 52521875, 60466176, 205962976, 0, 1, 34012224, 8303765625, 24794911296, 68719476736, 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432

Offset: 1

René-Louis Clerc, Jan 02 2025

Each row begins with 0, 1. Solutions can have no more than R(n) digits, since (R(n)*9)^n < 10^R(n), hence, for each n, there are a finite number of solutions (Property 1 and table 1 of Clerc).

			Triangle begins:
  1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
  2 | 0, 1, 81;
  3 | 0, 1, 512, 4913, 5832, 17576, 19683;
  4 | 0, 1, 2401, 234256, 390625, 614656, 1679616;
  5 | 0, 1, 17210368, 52521875, 60466176, 205962976;
  6 | 0, 1, 34012224, 8303765625, 24794911296, 68719476736;
  7 | 0, 1, 612220032, 10460353203, 27512614111, 52523350144, 271818611107, 1174711139837, 2207984167552, 6722988818432;
  8 | 0, 1, 20047612231936, 72301961339136, 248155780267521;
  9 | 0, 1, 3904305912313344, 45848500718449031, 150094635296999121;
  ...

René-Louis Clerc, Rows n=1...270 of triangle, flattened
René-Louis Clerc, Nombres de Niven-Harshad égaux à une puissance de la somme de leurs chiffres, 2024.

Rows 3..6 are A061209, A061210, A254000, A375343.
Row lengths are 1 + A046019(n).
Cf. A001014, A007953, A061211 (largest terms), A133509.
Cf. A152147.

R(n) = for(j=2,oo, if((j*9)^n <10^j, return(j)));
row(n) = my(L=List()); for (k=0, sqrtnint(10^R(n),n), if (k^n == sumdigits(k^n)^n, listput(L, k^n))); Vec(L)

cp's OEIS Frontend

A076088 Duplicate of A061211.

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A061209 Numbers which are the cubes of their digit sum.

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A061210 Numbers which are the fourth powers of their digit sum.

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A046017 Least k > 1 with k = sum of digits of k^n, or 0 if no such k exists.

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A046000 a(n) is the largest number m equal to the sum of digits of m^n.

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A133509 Numbers k such that m=1 is the only number for which the sum of digits of m^k equals m.

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A023106 a(n) is a power of the sum of its digits.

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