A055577 -id:A055577 - cp's OEIS frontend

A152147 Irregular triangle in which row n lists k > 0 such that the sum of digits of k^n equals k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 9, 1, 8, 17, 18, 26, 27, 1, 7, 22, 25, 28, 36, 1, 28, 35, 36, 46, 1, 18, 45, 54, 64, 1, 18, 27, 31, 34, 43, 53, 58, 68, 1, 46, 54, 63, 1, 54, 71, 81, 1, 82, 85, 94, 97, 106, 117, 1, 98, 107, 108, 1, 108, 1, 20, 40, 86, 103, 104, 106, 107, 126, 134, 135

Offset: 1

T. D. Noe, Nov 26 2008

Each row begins with 1 and has length A046019(n).

			1, 2, 3, 4, 5, 6, 7, 8, 9;
1, 9;
1, 8, 17, 18, 26, 27;              (A046459, with 0)
1, 7, 22, 25, 28, 36;              (A055575    "   )
1, 28, 35, 36, 46;                 (A055576    "   )
1, 18, 45, 54, 64;                 (A055577    "   )
1, 18, 27, 31, 34, 43, 53, 58, 68; (A226971    "   )
1, 46, 54, 63;
1, 54, 71, 81,
1, 82, 85, 94, 97, 106, 117,
1, 98, 107, 108, etc.

T. D. Noe, Rows n = 1..1000 of triangle, flattened

Cf. A046000, A046017, A046471, A133509.

def ok(k, r): return sum(map(int, str(k**r))) == k
def agen(rows, startrow=1, withzero=0):
  for r in range(startrow, rows + startrow):
    d, lim = 1, 1
    while lim < r*9*d: d, lim = d+1, lim*10
    yield from [k for k in range(1-withzero, lim+1) if ok(k, r)]
print([an for an in agen(13)]) # Michael S. Branicky, May 23 2021

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

Original entry on oeis.org

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

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

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

A281917 6th power analog of Keith numbers.

Original entry on oeis.org

1, 18, 45, 54, 64, 125, 218, 246, 935, 1125, 6021, 6866, 7887, 40210, 89330, 457625, 577655, 613385, 640118, 5200210, 6809148, 7293243, 10013591, 50980917, 216864574, 885859983, 4556794863, 4939169289, 8580755055, 8672110451, 18562634876, 18992278338, 36013476739

Offset: 1

Paolo P. Lava, Feb 02 2017

Like Keith numbers but starting from n^6 digits to reach n.

Consider the digits of n^6. Take their sum and repeat the process deleting the first addend and adding the previous sum. The sequence lists the numbers that after some number of iterations reach a sum equal to n.

			125^6 = 3814697265625:
3 + 8 + 1 + 4 + 6 + 9 + 7 + 2 + 6 + 5 + 6 + 2 + 5 = 64;
8 + 1 + 4 + 6 + 9 + 7 + 2 + 6 + 5 + 6 + 2 + 5 + 64 = 125.

Cf. A055577, A007629, A246544, A263534.

Cf. A274769, A274770, A281915, A281916, A281918, A281919, A281920, A281921.

with(numtheory): P:=proc(q, h,w) local a, b, k, t, v; global n; v:=array(1..h);
for n from 1 to q do b:=n^w; a:=[];
for k from 1 to ilog10(b)+1 do a:=[(b mod 10), op(a)]; b:=trunc(b/10); od;
for k from 1 to nops(a) do v[k]:=a[k]; od; b:=ilog10(n^w)+1;
t:=nops(a)+1; v[t]:=add(v[k], k=1..b); while v[t]

(* function keithQ[n_, e_] is defined in A007629 *)
a281917[n_] := Join[{1}, Select[Range[10, n], keithQ[#, 6]&]]
a281917[10^4] (* Hartmut F. W. Hoft, Jun 03 2021 *)

a(24) from Jinyuan Wang, Jan 31 2020

a(25)-a(33) from Giovanni Resta, Jan 31 2020

A055572 Sum of digits of a(n)^6 is greater than or equal to a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 42, 44, 45, 46, 51, 52, 54, 64

Offset: 0

Henry Bottomley, May 26 2000

			a(2) = 2 because 2^6 = 64 and 6+4 = 10>= 2

Cf. A055567, A055577, A055568, A055569, A055570, A055571.

A055567 Sum of digits of n^6.

Original entry on oeis.org

0, 1, 10, 18, 19, 19, 27, 28, 19, 18, 1, 28, 45, 37, 37, 27, 37, 37, 18, 37, 10, 36, 37, 46, 36, 28, 46, 45, 37, 37, 18, 46, 37, 54, 37, 46, 45, 46, 37, 45, 19, 28, 45, 37, 46, 45, 64, 46, 36, 37, 19, 54, 55, 37, 54, 46, 55, 54, 55, 37, 27, 37, 46, 36, 64, 55, 45, 55, 64, 45

Offset: 0

Henry Bottomley, May 26 2000

			a(2) = 10 because 2^6 = 64 and 6+4 = 10.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000012, A004159, A004164, A055565, A055566, A055572, A055577, A066588.

DigitSum[Range[0, 100]^6] (* Paolo Xausa, Jul 03 2024 *)

a(n) = sumdigits(n^6); \\ Seiichi Manyama, Nov 16 2021

A226971 Numbers k such that the sum of digits of k^7 is equal to k.

Original entry on oeis.org

0, 1, 18, 27, 31, 34, 43, 53, 58, 68

Offset: 1

Michel Lagneau, Jun 24 2013

Only the ten integers listed have this property.

			a(3) = 18 because 18^7 = 612220032 and 6+1+2+2+2+0+0+3+2 = 18.

Cf. A046459, A055575, A055576, A055577.

Cf. A152147.

[n: n in [0..80] | &+Intseq(n^7) eq n]; // Vincenzo Librandi, Feb 23 2015

Select[Range[0, 100], #==Total[IntegerDigits[#^7]]&]

isok(k)=sumdigits(k^7)==k \\ Patrick De Geest, Dec 13 2024

[n for n in (0..70) if sum((n^7).digits()) == n] # Bruno Berselli, 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

cp's OEIS Frontend

A152147 Irregular triangle in which row n lists k > 0 such that the sum of digits of k^n equals k.

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

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

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

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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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A281917 6th power analog of Keith numbers.

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Maple

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A055572 Sum of digits of a(n)^6 is greater than or equal to a(n).

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A055567 Sum of digits of n^6.