A003321 -id:A003321 - cp's OEIS frontend

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

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

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

A134703 Powerful numbers (2b): a sum of nonnegative powers of its digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 254, 258, 262, 263, 264, 267, 283, 308, 332, 333, 334, 347, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 472, 518, 538, 598, 629, 635, 653, 675, 730, 731, 732, 733, 734

Offset: 1

David W. Wilson, Sep 05 2009

Here 0 digits may be used, with the convention that 0^0 = 1. Of course 0^1 = 0, so one is free to use the 0 digit to get an extra 1, or not.

			43 = 4^2 + 3^3; 254 = 2^7 + 5^3 + 4^0 = 128 + 125 + 1.
209 = 2^7 + 0^1 + 9^2.
732 = 7^0 + 3^6 + 2^1.

David Wilson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074.

Different from A007532 and A061862, which are variations.

If n = d_1 d_2 ... d_k in decimal then there are integers m_1 m_2 ... m_k >= 0 such that n = d_1^m_1 + ... + d_k^m_k.

A061862 Powerful numbers (2a): a sum of nonnegative powers of its digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 254, 258, 262, 263, 264, 267, 283, 332, 333, 334, 347, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 472, 518, 538, 598, 629, 635, 653, 675, 730, 731, 732

Offset: 1

Erich Friedman, Jun 23 2001

Zero digits cannot be used in the sum. - N. J. A. Sloane, Aug 31 2009

More precisely, digits 0 do not contribute to the sum, in contrast to A134703 where it is allowed to use 0^0 = 1. - M. F. Hasler, Nov 21 2019

			43 = 4^2 + 3^3; 254 = 2^7 + 5^3 + 4^0 = 128 + 125 + 1.
209 = 2^7 + 9^2.
732 = 7^0 + 3^6 + 2^1.

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074.

Different from A007532 and A134703, which are variations.

a061862 n = a061862_list !! (n-1)
a061862_list = filter f [0..] where
   f x = g x 0 where
     g 0 v = v == x
     g u v = if d <= 1 then g u' (v + d) else v <= x && h 1
             where h p = p <= x && (g u' (v + p) || h (p * d))
                   (u', d) = divMod u 10
-- Reinhard Zumkeller, Jun 02 2013

f[ n_ ] := Module[ {}, a=IntegerDigits[ n ]; e=g[ Length[ a ] ]; MemberQ[ Map[ Apply[ Plus, a^# ] &, e ], n ] ] g[ n_ ] := Map[ Take[ Table[ 0, {n} ]~Join~#, -n ] &, IntegerDigits[ Range[ 10^n ], 10 ] ] For[ n=0, n >= 0, n++, If[ f[ n ], Print[ n ] ] ]

If n = d_1 d_2 ... d_k in decimal then there are integers m_1 m_2 ... m_k >= 0 such that n = d_1^m_1 + ... + d_k^m_k.

A236067 a(n) is the least number m such that m = n^d_1 + n^d_2 + ... + n^d_k where d_k represents the k-th digit in the decimal expansion of m, or 0 if no such number exists.

Original entry on oeis.org

1, 0, 12, 4624, 3909511, 0, 13177388, 1033, 10, 0, 0, 0, 0, 0, 2758053616, 1053202, 7413245658, 419370838921, 52135640, 1347536041, 833904227332, 5117557126, 3606012949057, 5398293152472, 31301, 0, 15554976231978, 405287637330, 35751665247, 19705624111111

Offset: 1

Derek Orr, Jan 19 2014

The 0's in the sequence are definite. There exists both a maximum and a minimum number that a(n) can be based on n. They are given in the programs below as Max(n) and Min(n), respectively.
It is known that a(22) = 5117557126, a(25) = 31301, a(29) = 35751665247, a(32) = 2112, a(33) = 1224103, a(37) = 111, a(40) = 102531321, a(48) = 25236435456, a(50) = 101, a(66) = 2524232305, a(78) = 453362316342, a(98) = 100, and a(100) = 20102.
There are an infinite number of nonzero entries. First, note if a(n) is nonzero, a(n) >= n. Further, a(9) = 10, a(98) = 100, a(997) = 1000, ..., a(10^k-k) = 10^k for all k >= 0.
For n = 21, 23, and 24, a(n) > 10^10.
For n in {26, 27, 28, 30, 31, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49}, a(n) > 5*10^10.
For n in {51, 52, 53, ..., 64, 65} and {67, 68, 69, ..., 73, 74}, a(n) > 10^11.
For n in {75, 76, 77} and {79, 80, 81, ..., 96, 97, 99}, a(n) > 5*10^11.
A few nonzero terms were added by math4pad.net @PascalCardin
a(1000) = 1000000000000002002017, a(10000) = 0, a(1000000) = 1000002000010, a(10000000) = 200000020000011. It looks like a(10^k) in decimal consists of mostly the digits 0, 1 and 2. - Chai Wah Wu, Dec 07 2017

			12 is the smallest number such that 3^1 + 3^2 = 12 so a(3) = 12.
4624 is the smallest number such that 4^4 + 4^6 + 4^2 + 4^4 = 4624 so a(4) = 4624.
1033 is the smallest number such that 8^1 + 8^0 + 8^3 + 8^3 = 1033 so a(8) = 1033.

Chai Wah Wu, Table of n, a(n) for n = 1..500 (n = 1..100 from Hiroaki Yamanouchi)
John D. Cook, Monday morning math puzzle (2012).
Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
Dean Morrow, Cycles of a family of digit functions

Cf. A139410 (for 4th term), A003321, A296138, A296139.

Min(n)=for(k=0,oo,if(n+k<=10^k,return(10^k)))
Max(n)=for(k=1,oo,if(k*n^9<=10^k-1,return(10^(k-1))))
a(n)={for(k=Min(n), Max(n), my(d=digits(k)); if(sum(i=1,#d,n^d[i])==k, return(k))); 0}
{ for(n=1, 100,print1(a(n), ", ")) } \\ Derek Orr, Aug 01 2014; corrected by Jason Yuen, Feb 25 2025

More terms and edited extensively by Derek Orr, Aug 26 2014

a(21)-a(30) from Hiroaki Yamanouchi, Sep 27 2014

A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 14 2015

Row lengths of the table A252648.

For a number with d digits, the sum of n-th powers cannot exceed d*9^n, but the number is not less than 10^(d-1). 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).

			a(0)=1 because 1 is the only number equal to the sum of 0th powers of its digits.
a(1)=10 because { 0, 1, ... 9 } are the only numbers equal to the sum of their digits (taken to the power 1).
a(2)=2 because 0 and 1 are the only numbers equal to the sum of the squares of their digits.
a(3)=6 because { 0, 1, 153, 370, 371, 407 } is the set of all numbers equal to the sum of the 3rd powers of their digits, cf. A046197.
For more examples, see the table A252648.

Don Knuth, Table of n, a(n) for n = 0..172
Table of a(n) for n=0..172 [From _Don Knuth_, Sep 09 2015]

Cf. A252648, A003321, A046197, A052455, A052464, A124068, A124069, A226970.

Reap@ For[n = 0, n < 6, n++, Sow@ Length@ Select[Range[0, 10^(n + 1)], Plus @@ (IntegerDigits[#]^n) == # &]] // Flatten // Rest (* Michael De Vlieger, Apr 14 2015 *)

a(n) >= 2 for all n > 0, since 0 and 1 are digital invariants for any power n > 0.

a(10)-a(90) from Don Knuth, Sep 09 2015

A243025 Fixed points of the transform n = d_(k)10^(k-1) + d_(k-1)10^(k-2) + ... + d_(2)*10 + d_(1) -> Sum_{i=1..k-1}{d_(i)^d(i+1)}+d(k)^d(1) (A243023).

Original entry on oeis.org

1, 4155, 4355, 1953504, 1954329, 522169982

Offset: 1

Paolo P. Lava, May 29 2014

Subset of A243023.

This sequence is finite by using the same argument that Armstrong numbers (A005188) are finite. - Robert G. Wilson v, Jun 01 2014

			1^1 = 1.
5^5 + 5^1 + 1^4 + 4^5 = 4155.
5^5 + 5^3 + 3^4 + 4^5 = 4355.
4^0 + 0^5 + 5^3 + 3^5 + 5^9 + 9^1 + 1^4 = 1953504.
9^2 + 2^3 + 3^4 + 4^5 + 5^9 + 9^1 + 1^9 = 1954329.

Cf. A243023, A243024.

Cf. A005188, A003321.

with(numtheory): P:=proc(q) local a,b,k,ok,n; for n from 10 to q do a:=[]; b:=n;
while b>0 do a:=[op(a),b mod 10]; b:=trunc(b/10); od; b:=0; ok:=1; for k from 2 to nops(a)
do if a[k-1]=0 and a[k]=0 then ok:=0; break; else b:=b+a[k-1]^a[k]; fi; od;
if ok=1 then if n=(b+a[nops(a)]^a[nops(1)]) then print(n);
fi; fi; od; end: P(10^10);

fQ[n_] := Block[{r = Reverse@ IntegerDigits@ n}, n == Plus @@ (r^RotateLeft@ r)]; k = 1; lst = {}; While[k < 1000000001, If[ fQ@ k, AppendTo[ lst, k]; Print@ k]; k++] (* Robert G. Wilson v, Jun 01 2014 *)

Added a(1) as 1 and a(6) by Robert G. Wilson v, Jun 01 2014

A192636 Powerful sums of two powerful numbers.

Original entry on oeis.org

8, 9, 16, 25, 32, 36, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Charles R Greathouse IV, Jul 06 2011

nonn

Browning & Valckenborgh conjecture that a(n) ~ kn^2 with k approximately 0.139485255. See their Conjecture 1 and equation (14). Their Theorems 1 and 2 establish upper and lower asymptotic bounds.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Charles R Greathouse IV)
Tim D. Browning and K. Van Valckenborgh, Sums of three squareful numbers, Experimental Mathematics, Vol. 21, No. 2 (2012), pp. 204-211; arXiv preprint, arXiv:1106.4472 [math.NT], 2011.

Subsequence of A001694 and of A076871.

Cf. A001694, A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871, A143813. - Jonathan Vos Post, Jul 10 2011

With[{m = 1225}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Intersection[pow, Plus @@@ Tuples[pow, {2}]]] (* Amiram Eldar, Feb 12 2023 *)

isPowerful(n)=if(n>3,vecmin(factor(n)[,2])>1,n==1)
sumset(a,b)={
  my(c=vectorsmall(#a*#b));
  for(i=1,#a,
    for(j=1,#b,
      c[(i-1)*#b+j]=a[i]+b[j]
    )
  );
  vecsort(c,,8)
}; selfsum(a)={
  my(c=vectorsmall(binomial(#a+1,2)),k);
  for(i=1,#a,
    for(j=i,#a,
      c[k++]=a[i]+a[j]
    )
  );
  vecsort(c,,8)
};
list(lim)={
  my(v=select(isPowerful, vector(floor(lim),i,i)));
  select(n->n<=lim && isPowerful(n), Vec(selfsum(v)))
};

Numbers k such that there exists some a, b, c with A001694(a) + A001694(b) = k = A001694(c).

Corrected (on the advice of Donovan Johnson) by Charles R Greathouse IV, Sep 25 2012

A262094 Orders m for which the only perfect digital invariants of order m are 0 and 1.

Original entry on oeis.org

1, 2, 12, 15, 18, 22, 26, 28, 30, 40, 41, 48, 50, 52, 58, 80, 82, 88, 98, 103, 106, 130, 135, 136, 138, 139, 140, 142, 150, 152, 159, 164, 165, 166

Offset: 1

Don Knuth, Sep 10 2015

a(1), a(2), ... are the indices m for which A255668(m)=2, also A003321(m)=0, also A046761(m)=1.

The values seem to appear in clusters.

			a(3)=12 because there's no solution to x=(x_{12}...x_1x_0)_{10} = x_{12}^{12}+...+x_1^{12}+x_0^{12} except x=0 and x=1.

Cf. A003321, A046761, A255668.

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

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

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A134703 Powerful numbers (2b): a sum of nonnegative powers of its digits.

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A061862 Powerful numbers (2a): a sum of nonnegative powers of its digits.

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A236067 a(n) is the least number m such that m = n^d_1 + n^d_2 + ... + n^d_k where d_k represents the k-th digit in the decimal expansion of m, or 0 if no such number exists.

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A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

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A243025 Fixed points of the transform n = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + ... + d_(2)*10 + d_(1) -> Sum_{i=1..k-1}{d_(i)^d(i+1)}+d(k)^d(1) (A243023).

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A192636 Powerful sums of two powerful numbers.

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A243025 Fixed points of the transform n = d_(k)10^(k-1) + d_(k-1)10^(k-2) + ... + d_(2)*10 + d_(1) -> Sum_{i=1..k-1}{d_(i)^d(i+1)}+d(k)^d(1) (A243023).