A152147 -id:A152147 - cp's OEIS frontend

A046459 Dudeney numbers: integers equal to the sum of the digits of their cubes.

0, 1, 8, 17, 18, 26, 27

Offset: 1

Patrick De Geest, Aug 15 1998

This sequence was first found by the French mathematician Claude (Séraphin) Moret-Blanc in 1879. See Le Lionnais page 27 for the last term of this sequence: 27. - Bernard Schott, Dec 07 2012

The name "Dudeney numbers" appears in the October 2018 issue of Mathematics Teacher (see link). - N. J. A. Sloane, Oct 10 2018

			a(3) = 8 because 8^3 = 512 and 5 + 1 + 2 = 8.
a(7) = 27 because 27^3 = 19683 and 1 + 9 + 6 + 8 + 3 = 27.

H. E. Dudeney, 536 Puzzles & Curious Problems, reprinted by Souvenir Press, London, 1968, p. 36, #120.
Italo Ghersi, Matematica dilettevole e curiosa, p. 115, Hoepli, Milano, 1967. [From Vincenzo Librandi, Jan 02 2009]
F. Le Lionnais, Les nombres remarquables, Hermann, 1983.
J. Roberts, Lure of the Integers, The Mathematical Association of America, 1992, p. 172.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 96.

H. E. Dudeney, 536 Puzzles & Curious Problems.
The Mathematics Teacher, October 2018 Calendar and Solutions, Volume 112, Number 2, October 2018, pages 120 and 122.
ProofWiki, Sequence of Dudeney Numbers.
Bernard Schott and Norbert Verdier, QDL 19: Quels beaux cubes! (French mathematical forum les-mathematiques.net).
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A004164, A055569, A055575, A055576, A055577.

Cf. A152147.

[n: n in [0..100] | &+Intseq(n^3) eq n ]; // Vincenzo Librandi, Sep 16 2015

Select[Range[0,30],#==Total[IntegerDigits[#^3]]&] (* Harvey P. Dale, Dec 21 2014 *)

isok(k)=sumdigits(k^3)==k \\ Patrick De Geest, Dec 10 2024

a = [n for n in range(100) if sum(map(int, str(n ** 3))) == n] # David Radcliffe, Aug 18 2022

Offset corrected by Arkadiusz Wesolowski, Aug 09 2013

A046019 a(n) gives the number of different powers m^n for which the sum of the digits is equal to m.

Original entry on oeis.org

1, 9, 2, 6, 6, 5, 5, 9, 4, 4, 7, 4, 2, 12, 6, 8, 7, 5, 3, 10, 4, 4, 8, 4, 4, 14, 5, 3, 7, 6, 2, 11, 2, 8, 4, 6, 3, 9, 3, 3, 7, 2, 5, 10, 6, 4, 9, 9, 5, 12, 2, 4, 5, 5, 6, 3, 2, 7, 4, 5, 5, 6, 3, 4, 5, 5, 4, 9, 2, 6, 4, 3, 3, 6, 5, 6, 4, 4, 5, 9, 5, 3, 5, 5, 2, 6, 3, 7, 7, 4, 3, 8, 4, 4, 9, 6, 2, 8, 2, 5, 6, 3

Offset: 0

David W. Wilson

Number of m >= 1 with m = sum of digits of m^n.

			a(7)=9 because:
1^7=1
18^7= 612220032 and 6+1+2+2+2+3+2=18
27^7= 10460353203 and 1+4+6+3+5+3+2+3=27
31^7= 27512614111 and 2+7+5+1+2+6+1+4+1+1+1=31
34^7= 52523350144 and 5+2+5+2+3+3+5+1+4+4=34
43^7= 271818611107 and 2+7+1+8+1+8+6+1+1+1+7=43
53^7= 1174711139837 and 1+1+7+4+7+1+1+1+3+9+8+3+7=53
58^7= 2207984167552 and 2+2+7+9+8+4+1+6+7+5+5+2=58
68^7= 6722988818432 and 6+7+2+2+9+8+8+8+1+8+4+3+2=68
a(9)=4 because:
1^9=1
54^9=3904305912313344 and 3+9+4+3+5+9+1+2+3+1+3+3+4+4=54
71^9=45848500718449031 and 4+5+8+4+8+5+7+1+8+4+4+9+3+1=71
81^9=150094635296999121 and 1+5+9+4+6+3+5+2+9+6+9+9+9+1+2+1=81

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A124359, A152147 (table of m such that the sum of digits of m^n equals m)

a(n) = 1 + A046471(n). - T. D. Noe, Nov 26 2008

Examples provided by Paolo P. Lava, Oct 30 2006

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

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

A055575 Sum of digits of n^4 is equal to n.

Original entry on oeis.org

0, 1, 7, 22, 25, 28, 36

Offset: 1

Henry Bottomley, May 26 2000

			7 is a member because 7^4 = 2401 and 2+4+0+1 = 7.

Cf. A055565, A055570, A046459, A055576, A055577.

Cf. A152147.

[n: n in [0..50] | &+Intseq(n^4) eq n ]; // Vincenzo Librandi, Feb 23 2015

Select[Range[0, 50], #==Total[IntegerDigits[#^4]] &] (* Vincenzo Librandi, Feb 23 2015 *)

isok(k)=sumdigits(k^4)==k \\ Patrick De Geest, Dec 10 2024

[n for n in (0..50) if sum((n^4).digits()) == n] # Bruno Berselli, Feb 23 2015

A055576 Sum of digits of a(n)^5 is equal to a(n).

Original entry on oeis.org

0, 1, 28, 35, 36, 46

Offset: 1

Henry Bottomley, May 26 2000

			a(2) = 28 because 28^5 = 17210368 and 1+7+2+1+0+3+6+8 = 28

Cf. A055566, A055571, A046459, A055575, A055577.

Cf. A152147.

[n: n in [0..50] | &+Intseq(n^5) eq n ]; // Vincenzo Librandi, Feb 23 2015

Select[Range[0, 60], #==Total[IntegerDigits[#^5]] &] (* Vincenzo Librandi, Feb 23 2015 *)

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

Offset changed to 1 by Michel Marcus, Feb 23 2015

A055577 Numbers k such that the sum of digits of k^6 is equal to k.

Original entry on oeis.org

0, 1, 18, 45, 54, 64

Offset: 1

Henry Bottomley, May 26 2000

			a(2) = 18 because 18^6 = 34012224 and 3+4+0+1+2+2+2+4 = 18

Cf. A055567, A055572, A046459, A055575, A055576, A226971.

Cf. A152147.

[n: n in [0..100] | &+Intseq(n^6) eq n ]; // Vincenzo Librandi, Feb 23 2015

Select[Range[0,100],#==Total[IntegerDigits[#^6]]&] (* Harvey P. Dale, Oct 26 2011 *)

isok(k)=sumdigits(k^6)==k \\ Patrick De Geest, Dec 13 2024

[n for n in (0..70) if sum((n^6).digits()) == n] # Bruno Berselli, Feb 23 2015

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

A046471 Number of numbers k>1 such that k equals the sum of digits in k^n.

Original entry on oeis.org

8, 1, 5, 5, 4, 4, 8, 3, 3, 6, 3, 1, 11, 5, 7, 6, 4, 2, 9, 3, 3, 7, 3, 3, 13, 4, 2, 6, 5, 1, 10, 1, 7, 3, 5, 2, 8, 2, 2, 6, 1, 4, 9, 5, 3, 8, 8, 4, 11, 1, 3, 4, 4, 5, 2, 1, 6, 3, 4, 4, 5, 2, 3, 4, 4, 3, 8, 1, 5, 3, 2, 2, 5, 4, 5, 3, 3, 4, 8, 4, 2, 4, 4, 1, 5, 2, 6, 6, 3, 2, 7, 3, 3, 8, 5, 1, 7, 1, 4, 5, 2, 3, 9

Offset: 1

Patrick De Geest, Aug 15 1998

The number of digits in k^n is at most 1+n*log(k). Hence the maximum sum of digits of k^n is 9(1+n*log(k)). By solving k=9(1+n*log(k)), we can compute an upper bound on k for each n. Sequence A133509 lists the n for which a(n)=0.

			a(17)=4 -> sum-of-digits{x^17}=x for x=80,143,171 and 216 (x>1).

Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 172.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A046459, A046469, A046000.
a(n) = A046019(n) - 1.
Cf. A152147 (table of k such that the sum of digits of k^n equals k)

Edited by T. D. Noe, Nov 25 2008

A247889 Least number k > 0 such that digsum(n^k) = n, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 13, 0, 4, 0, 0, 4, 3, 3, 4, 0, 0, 7, 0, 0, 7, 5, 4, 0, 0, 0, 13, 0, 0, 7, 0, 6, 5, 0, 0, 0, 0, 0, 0, 7, 6, 0, 0, 0, 7, 0, 0, 0, 0, 8, 6, 0, 0, 0, 7, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 17, 9, 10, 0, 0, 10, 13, 0, 0, 0, 19, 14, 0, 0, 10, 0, 0, 10, 11, 0, 0

Offset: 0

Derek Orr, Sep 25 2014

a(10^n) = 0 for all n > 0.
a(n) = 0 if and only if n is in A124367, complement of A124359.
The PARI code uses that, if sumdigit(n^k) <> n until sumdigits(n^k) > 2n for the first time, then sumdigits(n^k) <> n for all larger k. This might not be true for all n, although statistically sumdigits(n^k) ~ 4.5*log_10(n')*k within a few % (n' = n with trailing 0's removed), when k is not too small. - M. F. Hasler, May 18 2017

Hans Havermann, Table of n, a(n) for n = 0..10000

Cf. A124359 (a(n) > 0), A124367 (a(n) = 0), A007953, A152147.

A247889(n,L=2*n)= if(n<10||vecmin(digits(n-1))==9,return(n<10));k=1;while(L >= s=sumdigits(n^k),if(s,return(k));k++) \\ Renamed to A247889 for use in A124359, A124367 (and elsewhere?), and minor edits by M. F. Hasler, May 18 2017
apply(A247889,[0..100])

Data corrected (initial 1 removed, terminal 0 added) by Hans Havermann, Jul 13 2018

cp's OEIS Frontend

A046459 Dudeney numbers: integers equal to the sum of the digits of their cubes.

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A046019 a(n) gives the number of different powers m^n for which the sum of the digits is equal to m.

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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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A055575 Sum of digits of n^4 is equal to n.

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Sage

A055576 Sum of digits of a(n)^5 is equal to a(n).

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Magma

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PARI

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A055577 Numbers k such that the sum of digits of k^6 is equal to k.

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Magma

Mathematica

PARI

Sage

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

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