A124068 -id:A124068 - cp's OEIS frontend

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

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

A046197 Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits.

Original entry on oeis.org

0, 1, 153, 370, 371, 407

Offset: 1

Richard C. Schroeppel

Suppose n has d digits; then the sum of the cubes of its digits is <= 729d and n >= 10^(d-1). So d <= 5. It is now easy to check that the numbers shown are the only solutions. [Corrected by M. F. Hasler, Apr 12 2015]

This is row n=3 of A252648. - M. F. Hasler, Apr 12 2015

			1^3 + 5^3 + 3^3 = 153. 3^3+7^3 +0^3 = 370.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 153, p. 50, Ellipses, Paris 2008.
G. H. Hardy, A Mathematician's Apology, Cambridge, 1967.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 60-62.
J. Shallit, Number theory and formal languages, in Emerging applications of number theory (Minneapolis, MN, 1996), 547-570, IMA Vol. Math. Appl., 109, Springer, New York, 1999.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 140.

H. Lehning, La migration des nombres vers le bonheur, Tangente: L'aventure mathématique, pp. 27 No. 108 Jan-Feb 2006 Pole Paris.

Cf. A005188, A023052, A046156.
Cf. A165330, A035504, A008585, A165333, A165334, A165335.
Cf. A052455, A052464, A124068, A124069, A226970, A003321.

Select[Range[0,407],Total[IntegerDigits[#]^3]==# &] (* Stefano Spezia, Sep 08 2024 *)

for(n=0,10^5,A055012(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

A055012(a(n))=a(n); A165331(a(n))=0; subset of A031179. - Reinhard Zumkeller, Sep 17 2009

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

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

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

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

A123253 Sum of 7th powers of digits of n.

Original entry on oeis.org

0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 1, 2, 129, 2188, 16385, 78126, 279937, 823544, 2097153, 4782970, 128, 129, 256, 2315, 16512, 78253, 280064, 823671, 2097280, 4783097, 2187, 2188, 2315, 4374, 18571, 80312, 282123

Offset: 0

Zerinvary Lajos, Nov 06 2006

Fixed points are listed in A124068 = row n=7 of A252648. - M. F. Hasler, Apr 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A003132, A055012, A055013, A055014, A055015, A210840.

[0] cat [&+[d^7: d in Intseq(n)]: n in [1..40]]; // Bruno Berselli, Feb 01 2013

A123253 := proc(n)
        add(d^7,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Jan 16 2013

Table[Sum[DigitCount[n][[i]] i^7, {i, 9}], {n, 0, 40}] (* Bruno Berselli, Feb 01 2013 *)

A123253(n)=sum(i=1,#n=digits(n),n[i]^7) \\ M. F. Hasler, Apr 12 2015

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

cp's OEIS Frontend

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

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A046197 Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its 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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A052464 Fixed points for operation of repeatedly replacing a number with the sum of the fifth power of its digits.

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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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A123253 Sum of 7th powers of digits of n.