A046197 -id:A046197 - 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

A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, 92727, 93084, 194979, 548834, 1741725, 4210818, 9800817, 9926315, 14459929, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153

Offset: 1

David W. Wilson

The old name was "Powerful numbers, definition (3)". Cf. A001694, A007532. - N. J. A. Sloane, Jan 16 2022.
Randle has suggested that these numbers be called "powerful", but this usually refers to a distinct property related to prime factorization, cf. A001694, A036966, A005934.
Numbers m such that m = Sum_{i=1..k} d(i)^s for some s, where d(1..k) are the decimal digits of m.
Superset of A005188 (Plusperfect, narcissistic or Armstrong numbers: s=k), A046197 (s=3), A052455 (s=4), A052464 (s=5), A124068 (s=6, 7), A124069 (s=8). - R. J. Mathar, Jun 15 2009, Jun 22 2009

			153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

Jerome Raulin, Table of n, a(n) for n = 1..345 (terms 1..255 from Joseph Myers)
Encyclopaedia Britannica, Perfect digital invariant, article "Number patterns and curiosities" online since July 26, 1999, revised Aug 25, 2000.
Hans Havermann, Extended table of values for A023052 and A046074
Donald E. Knuth, The Art of Computer Programming, Volume 4, Pre-Fascicle 9B A Potpourri of Puzzles
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383.
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Narcissistic Number
Index entries for sequences related to powerful numbers

Cf. A001694 (powerful numbers: p|n => p^2|n), A005934 (highly powerful numbers).
Cf. A003321, A007532, A014576, A046074, A046761, A053540, A161752.
Cf. A005188 (here the power must be equal to the number of digits).
In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9).

Select[Range[0, 10^5], Function[m, AnyTrue[Function[k, Total@ Map[Power[#, k] &, IntegerDigits@ m]] /@ Range@ 10, # == m &]]] (* Michael De Vlieger, Feb 08 2016, Version 10 *)

is(n)=if(n<10, return(1)); my(d=digits(n),m=vecmax(d)); if(m<2, return(0)); for(k=3,logint(n,m), if(sum(i=1,#d,d[i]^k)==n, return(1))); 0 \\ Charles R Greathouse IV, Feb 06 2017

select( is_A023052(n,b=10)={nn|| return(t==n))}, [0..10^5]) \\  M. F. Hasler, Nov 21 2019

Computed to 10^50 by G. N. Gusev (GGN(AT)rm.yaroslavl.ru)
Computed to 10^74 by Xiaoqing Tang
A-number typo corrected by R. J. Mathar, Jun 22 2009
Computed to 10^105 by Joseph Myers
Cross-references edited by Joseph Myers, Jun 28 2009
Edited by M. F. Hasler, Nov 21 2019

A252648 Irregular table of perfect digital invariants for n > 1, i.e., numbers equal to the sum of n-th powers of their digits, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 0, 1, 153, 370, 371, 407, 0, 1, 1634, 8208, 9474, 0, 1, 4150, 4151, 54748, 92727, 93084, 194979, 0, 1, 548834, 0, 1, 1741725, 4210818, 9800817, 9926315, 14459929, 0, 1, 24678050, 24678051, 88593477, 0, 1, 146511208, 472335975, 534494836, 912985153, 0, 1, 4679307774

Offset: 0

Derek Orr, Dec 19 2014

The third column is listed in A003321. - M. F. Hasler, Feb 16 2015

For a number x >= 10^(d-1) with d digits, the sum of n-th powers of these digits cannot exceed d*9^n. Therefore there is only a finite number of possible perfect digital invariants for any n, the largest of which has at most d* digits, where d* = 1+(n*log(9)+log d*)/log(10). - M. F. Hasler, Apr 14 2015

			The table starts:
1; (n = 0; 1 = 1^0.)
0, 1, 2, 3, 4, 5, 6, 7, 8, 9; (n = 1)
0, 1; (n = 2)
0, 1, 153, 370, 371, 407; (n = 3, A046197)
0, 1, 1634, 8208, 9474; (n = 4, A052455)
0, 1, 4150, 4151, 54748, 92727, 93084, 194979; (n = 5, A052464)
0, 1, 548834; (n = 6)
0, 1, 1741725, 4210818, 9800817, 9926315, 14459929; (n = 7, A124068)
0, 1, 24678050, 24678051, 88593477; (n = 8, A124069)
0, 1, 146511208, 472335975, 534494836, 912985153; (n = 9, A226970)
The third column corresponds to A003321.
The term 153 is member of the row n=3 because 153 = 1^3 + 5^3 + 3^3. - _M. F. Hasler_, Apr 14 2015

Don Knuth, Table of a(n) for n=0..732
Don Knuth, CWEB program to generate solutions

Cf. A003321, A046197, A052455, A052464, A124068, A124069, A226970.

Cf. A255668 (row lengths).

row(n)={m=1;while(m*9^n>=10^m,m++);for(k=1,10^m,sum(i=1,#d=digits(k),d[i]^n)==k && print1(k,", "))}
for(n=0,7,print1(row(n),", "))

from itertools import combinations_with_replacement
A252648_list = [1]
for m in range(1,21):
    l, L, dm, xlist, q = 1, 1, [d**m for d in range(10)], [0], 9**m
    while l*q >= L:
        for c in combinations_with_replacement(range(1,10),l):
            n = sum(dm[d] for d in c)
            if sorted(int(d) for d in str(n)) == [0]*(len(str(n))-len(c))+list(c):
                xlist.append(n)
        l += 1
        L *= 10
    A252648_list.extend(sorted(xlist)) # Chai Wah Wu, Jan 04 2016

I added two links. - Don Knuth, Sep 10 2015

A052455 Fixed points for operation of repeatedly replacing a number with the sum of the fourth power of its digits.

Original entry on oeis.org

0, 1, 1634, 8208, 9474

Offset: 1

Henry Bottomley, Mar 15 2000

This is row n=4 in A252648. - M. F. Hasler, Apr 12 2015

			a(2)=1634 since 1^4+6^4+3^4+4^4=1+1296+81+256=1634

Cf. A023052, A046074, A046197, A052464, A124068, A124069, A226970, A003321.

for(n=0,10^5,A055013(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

a(n) = A055013(a(n)). - M. F. Hasler, Apr 12 2015

A035504 Numbers that eventually reach 1 under "x -> sum of cubes of digits of x".

Original entry on oeis.org

1, 10, 100, 112, 121, 211, 778, 787, 877, 1000, 1012, 1021, 1102, 1120, 1189, 1198, 1201, 1210, 1234, 1243, 1324, 1342, 1423, 1432, 1579, 1597, 1759, 1795, 1819, 1891, 1918, 1957, 1975, 1981, 2011, 2101, 2110, 2134, 2143, 2314, 2341, 2413, 2431, 2779

Offset: 1

N. J. A. Sloane

Subsequence of A016777; a(n) mod 3 = 1; A165330(a(n))=1. [Reinhard Zumkeller, Sep 17 2009]

R. K. Guy, Unsolved Problems Number Theory, Sect. E34.

R. Zumkeller, Table of n, a(n) for n = 1..1000
R. Styer, Smallest Examples of Strings of Consecutive Happy Numbers, J. Int. Seq. 13 (2010), 10.6.3, Section 4.

Cf. A007770.

Cf. A046197, A008585, A165333, A165334, A165335; subsequence of A031179.

f[n_]:=Plus@@(IntegerDigits[n]^3);Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ, All]];Select[Range[2780],Last[Trajectory[#]]==1 &] (* Ant King, May 24 2013 *)

More terms from Larry Reeves (larryr(AT)acm.org), Mar 31 2000

A052464 Fixed points for operation of repeatedly replacing a number with the sum of the fifth power of its digits.

Original entry on oeis.org

0, 1, 4150, 4151, 54748, 92727, 93084, 194979

Offset: 1

Henry Bottomley, Mar 15 2000

Equivalently, numbers equal to the sum of 5th powers of their decimal digits. Since this sum is <= 9^5*d for a d-digit number n >= 10^(d-1), there cannot be such a number with more than 6 digits. - M. F. Hasler, Apr 12 2015

			a(2) = 4150 since 4^5 + 1^5 + 5^5 + 0^5 = 1024 + 1 + 3125 + 0 = 4150.

G. K. Patil, Ramanujan's Life And His Contributions In The Field Of Mathematics, International Journal of Scientific Research and Engineering Studies (IJSRES), 1(6) (2014), ISSN: 2349-8862.

Cf. A023052, A046074, A046197, A052455, A124068, A124069, A226970, A003321.

for(n=0,10^6,A055014(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

A124068 Fixed points for operation of repeatedly replacing a number with the sum of the seventh power of its digits.

Original entry on oeis.org

0, 1, 1741725, 4210818, 9800817, 9926315, 14459929

Offset: 1

Sébastien Dumortier, Nov 05 2006

The sequence "Fixed points for operation of repeatedly replacing a number by the sum of the sixth power of its digits" has just 3 terms: 0, 1, and 548834.

For a d-digit number n >= 10^(d-1), the sum of 7th powers of its digits is <= 9^7*d, therefore these numbers cannot exceed 41205040. - M. F. Hasler, Apr 12 2015

			1741725 = 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7.

Cf. A046197, A052455, A052464, A124069, A226970, A003321.

isok(n) = my(d = digits(n)); sum(k=1, #d, d[k]^7) == n; \\ Michel Marcus, Feb 21 2015

for(n=0,41205040,A123253(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

A124069 Fixed points for operation of repeatedly replacing a number with the sum of the eighth power of its digits.

Original entry on oeis.org

0, 1, 24678050, 24678051, 88593477

Offset: 1

Sébastien Dumortier, Nov 05 2006

This is row n=8 of A252648. For a d-digit number n >= 10^(d-1), the sum of 8th powers of its digits is <= 9^8*d, therefore n <= 413979400. - M. F. Hasler, Apr 12 2015

			24678050 = 2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 0^8.

Cf. A046197, A052455, A052464, A124068, A226970, A003321, A210840, A252648.

isok(n) = my(d = digits(n)); sum(k=1, #d, d[k]^8) == n; \\ Michel Marcus, Feb 21 2015

for(n=0,413979400,A210840(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

a(n) = A210840(a(n)). - M. F. Hasler, Apr 12 2015

A165331 Number of iterations until a fixed point or cycle is reached when repeatedly replacing a number with the sum of the cubes of its digits.

Original entry on oeis.org

0, 0, 7, 3, 4, 6, 10, 6, 6, 4, 1, 8, 5, 3, 6, 10, 1, 8, 2, 2, 7, 5, 2, 7, 3, 1, 8, 2, 2, 3, 3, 3, 7, 6, 3, 6, 6, 1, 8, 6, 4, 6, 3, 3, 7, 5, 3, 1, 6, 3, 6, 10, 1, 6, 5, 0, 5, 5, 8, 10, 10, 1, 8, 6, 3, 5, 6, 7, 11, 6, 6, 8, 2, 1, 1, 5, 7, 7, 8, 2, 6, 2, 2, 8, 6, 8, 11, 8, 3, 3, 4, 2, 3, 6, 3, 10, 6, 2, 3, 4, 1, 8

Offset: 0

Reinhard Zumkeller, Sep 17 2009

a(A046197(k)) = 0 for k: 1 <= k <= 6;
a(A046156(k)) = 0 for k: 1 <= k <= 16;
a(A165330(n)) = 0;
a(A165340(n,k)) = n - k, 0<=k<=n.
a(A008585(n)) = A003620(n), n>0. [From Reinhard Zumkeller, Nov 21 2009]

R. Zumkeller, Table of n, a(n) for n = 0..20000

A226970 Fixed points for the operation of repeatedly replacing a number with the sum of the ninth powers of its digits.

Original entry on oeis.org

0, 1, 146511208, 472335975, 534494836, 912985153

Offset: 1

Michel Lagneau, Jun 24 2013

The only six integers equal to the sum of the ninth powers of their digits.

This is row n=9 of A252648. For a d-digit number n >= 10^(d-1), the sum of 9th powers of its digits is <= 9^9*d, therefore n <= 4112105981. - M. F. Hasler, Apr 12 2015

			a(3) = A003321(9);
a(4) = 472335975 = 4^9 + 7^9 + 2^9 + 3^9 + 3^9 + 5^9 + 9^9 + 7^9 + 5^9.

Cf. A046197, A052455, A052464, A124068, A124069, A003321.

is_A226970(n)=n==sum(i=1,#n=digits(n),n[i]^9)
for(n=0,4112105981,is_A226970(n)&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

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

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A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

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A252648 Irregular table of perfect digital invariants for n > 1, i.e., numbers equal to the sum of n-th powers of their digits, read by rows.

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A052455 Fixed points for operation of repeatedly replacing a number with the sum of the fourth power of its digits.

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A035504 Numbers that eventually reach 1 under "x -> sum of cubes of digits of x".

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A052464 Fixed points for operation of repeatedly replacing a number with the sum of the fifth power of its digits.

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A124068 Fixed points for operation of repeatedly replacing a number with the sum of the seventh power of its digits.

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