A046459 -id:A046459 - cp's OEIS frontend

A055012 Sum of cubes of the digits of n written in base 10.

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 2, 9, 28, 65, 126, 217, 344, 513, 730, 8, 9, 16, 35, 72, 133, 224, 351, 520, 737, 27, 28, 35, 54, 91, 152, 243, 370, 539, 756, 64, 65, 72, 91, 128, 189, 280, 407, 576, 793, 125, 126, 133, 152, 189, 250, 341, 468, 637, 854

Offset: 0

Henry Bottomley, May 31 2000

For n > 1999, a(n) < n. The iteration of this map on n either stops at a fixed point (A046197) or has a period of 2 or 3: {55,250,133}, {136,244}, {160,217,352}, or {919,1459}. - T. D. Noe, Jul 17 2007

A165330 and A165331 give the final value and the number of steps when iterating until a fixed point or cycle is reached. - Reinhard Zumkeller, Sep 17 2009

T. D. Noe, Table of n, a(n) for n = 0..11000
K. Iséki, A problem of number theory, Proc. Japan Academy 36 (1960), 578-583.
B. M. Stewart, Sums of functions of digits, Canad. J. Math., 12 (1960), 374-389.
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A031179.
Cf. A046197 Fixed points; A046459: integers equal to the sum of the digits of their cubes; A072884: 3rd-order digital invariants: the sum of the cubes of the digits of n equals some number k and the sum of the cubes of the digits of k equals n; A164883: cubes with the property that the sum of the cubes of the digits is also a cube.
Cf. A003132, A007953, A055017, A076313, A076314, A165340.

[0] cat [&+[d^3: d in Intseq(n)]: n in [1..60]]; // Bruno Berselli, Feb 01 2013

A055012 := proc(n)
        add(d^3,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Dec 15 2011

Total/@((IntegerDigits/@Range[0,60])^3) (* Harvey P. Dale, Jan 27 2012 *)
Table[Sum[DigitCount[n][[i]] i^3, {i, 9}], {n, 0, 60}] (* Bruno Berselli, Feb 01 2013 *)

A055012(n)=sum(i=1,#n=digits(n),n[i]^3) \\ Charles R Greathouse IV, Jul 01 2013

def a(n): return sum(map(lambda x: x*x*x, map(int, str(n))))
print([a(n) for n in range(60)]) # Michael S. Branicky, Jul 13 2022

a(n) = Sum_{k>=1} (floor(n/10^k) - 10*floor(n/10^(k+1)))^3. - Hieronymus Fischer, Jun 25 2007
a(10n+k) = a(n) + k^3, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007
From Reinhard Zumkeller, Sep 17 2009: (Start)
a(n) <= 729*A055642(n);
a(A165370(n)) = n and a(m) <> n for m < A165370(n);
a(A031179(n)) = A031179(n);
a(a(A165336(n))) = A165336(n) or a(a(a(A165336(n)))) = A165336(n). (End)
G.f. g(x) = Sum_{k>=0} (1-x^(10^k))*(x^(10^k)+8*x^(2*10^k)+27*x^(3*10^k)+64*x^(4*10^k)+125*x^(5*10^k)+216*x^(6*10^k)+343*x^(7*10^k)+512*x^(8*10^k)+729*x^(9*10^k))/((1-x)*(1-x^(10^(k+1))))
satisfies
g(x) = (x+8*x^2+27*x^3+64*x^4+125*x^5+216*x^6+343*x^7+512*x^8+729*x^9)/(1-x^10) + (1-x^10)*g(x^10)/(1-x). - Robert Israel, Jan 26 2017

Edited by M. F. Hasler, Apr 12 2015

Iséki and Stewart links added by Don Knuth, Sep 07 2015

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

Original entry on oeis.org

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

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

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

A055569 Sum of digits of a(n)^3 is greater than or equal to a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 26, 27

Offset: 0

Henry Bottomley, May 26 2000

			a(4) = 4 because 4^3 = 64 and 6+4 = 10>= 4

Cf. A004164, A046459, A055568, A055570, A055571, A055572.

Select[Range[300],Total[IntegerDigits[#^3]]>=#&] (* Harvey P. Dale, Aug 27 2013 *)

A225534 Numbers whose sum of cubed digits is prime.

Original entry on oeis.org

11, 101, 110, 111, 113, 115, 122, 124, 128, 131, 139, 142, 146, 148, 151, 155, 164, 166, 182, 184, 193, 199, 212, 214, 218, 221, 223, 227, 232, 236, 238, 241, 245, 254, 256, 263, 265, 269, 272, 278, 281, 283, 287, 289, 296, 298, 311, 319, 322, 326, 328, 335

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

Note that 11 is the only two-digit number in the sequence.

a(n) ~ n. For 414 < n < 10000, 6.38*n - 528 provides an estimate of a(n) to within 6%.

			139 is in the sequence because 1^3 + 3^3 + 9^3 = 757, which is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A197039, A055012, A046459, A225535.
Cf. A165330, A046197.
Cf. A046704, A023194.

Select[Range[350],PrimeQ[Total[IntegerDigits[#]^3]]&] (* Harvey P. Dale, Mar 16 2016 *)

digcubesum<-function(x) sum(as.numeric(strsplit(as.character(x),split="")[[1]])^3); library(gmp);
which(sapply(1:1000,function(x) isprime(digcubesum(x))>0))

cp's OEIS Frontend

A055012 Sum of cubes of the digits of n written in base 10.

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

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

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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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