A252648 -id:A252648 - cp's OEIS frontend

A003321 Smallest n-th order perfect digital invariant or PDI: smallest number > 1 equal to sum of n-th powers of its digits, or 0 if no such number exists.

2, 0, 153, 1634, 4150, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 564240140138, 28116440335967, 0, 4338281769391370, 233411150132317, 0, 1517841543307505039, 63105425988599693916

Offset: 1

N. J. A. Sloane

Except for the initial term, this is the third column of A252648. - M. F. Hasler, Feb 16 2015

a(n) = 0 if n>1 and in A262094. - Dmitry Kamenetsky, Jun 05 2020

			1^3 + 5^3 + 3^3 = 153.
1*0^17 + 5*1^17 + 2*2^17 + 4*3^17 + 1*4^17 + 1*5^17 + 1*7^17 = 233411150132317.

M. Gardner, The Magic Numbers of Dr Matrix. Prometheus, Buffalo, NY, 1985, p. 249.
J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons Ltd. 1966, p. 164.
J. S. Madachy, Madachy's Mathematical Recreations, Dover, p. 164.
C. A. Pickover, Keys to Infinity. New York: W. H. Freeman, pp. 169-170, 1995.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joseph Myers, Table of n, a(n) for n=1..109
L. E. Deimel, Narcissistic Numbers
H. Heinz, Narcissistic Numbers
Eric Weisstein's World of Mathematics, Narcissistic Number.

Cf. A001694, A007532, A005934, A005188, A014576, A023052, A046074, A046761.

In other bases: A033835 (base 3), A033836 (base 4), A033837 (base 5), A033838 (base 6), A033839 (base 7), A033840 (base 8), A033841 (base 9).

a(n)=m=1;while(m*9^n>=10^m,m++);for(k=2,10^m,d=digits(k);s=sum(i=1,#d,d[i]^n);if(s==k,return(k)));0
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Dec 19 2014

Additional comments from Lekraj Beedassy, May 23 2001

Extended and cross-references edited by Joseph Myers, Jun 28 2009

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

A055013 Sum of 4th powers of digits of n.

Original entry on oeis.org

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 1, 2, 17, 82, 257, 626, 1297, 2402, 4097, 6562, 16, 17, 32, 97, 272, 641, 1312, 2417, 4112, 6577, 81, 82, 97, 162, 337, 706, 1377, 2482, 4177, 6642, 256, 257, 272, 337, 512, 881, 1552, 2657, 4352, 6817

Offset: 0

Henry Bottomley, May 31 2000

Fixed points are listed in A052455, row 4 of A252648. See also A061210. - M. F. Hasler, Apr 12 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Chikawa, K. Iséki, and T. Kusakabe, On a problem by H. Steinhaus, Acta Arithmetica 7 (1962), 251-252. - _Don Knuth_, Sep 07 2015
Index entries for Colombian or self numbers and related sequences

Cf. A003132, A055012.
Cf. A007953, A055017, A076313, A076314.
Cf. A052455, A252648; A061210.

[0] cat [&+[d^4: d in Intseq(n)]: n in [1..50]]; // Bruno Berselli, Feb 01 2013

A055013 := proc(n)
        add(d^4,d=convert(n,base,10)) ;
end proc:
seq(A055013(n),n=0..20) ; # R. J. Mathar, Nov 07 2011

Table[Sum[DigitCount[n][[i]] i^4, {i, 9}], {n, 0, 50}] (* Bruno Berselli, Feb 01 2013 *)
Table[Total[IntegerDigits[n]^4],{n,0,50}] (* Harvey P. Dale, Jul 28 2019 *)

a(n)=round(normlp(n,4)^4) \\ Quite slow. - M. F. Hasler, Apr 12 2015

A055013(n)=sum(i=1,#n=digits(n),n[i]^4) \\ M. F. Hasler, Apr 12 2015

a(n) = sum{k>0, (floor(n/10^k)-10*floor(n/10^(k+1)))^4}. - Hieronymus Fischer, Jun 25 2007

a(10n+k) = a(n)+k^4, 0<=k<10. - Hieronymus Fischer, Jun 25 2007

A061209 Numbers which are the cubes of their digit sum.

Original entry on oeis.org

0, 1, 512, 4913, 5832, 17576, 19683

Offset: 1

Amarnath Murthy, Apr 21 2001

It can be shown that 19683 = (1 + 9 + 6 + 8 + 3)^3 = 27^3 is the largest such number.
Numbers of Dudeney. - Philippe Deléham, May 11 2013
If a number n has d digits, 10^(d-1) <= n < 10^d, the cube of the digit sum is at most (d*9)^3 = 729*d^3; if d > 6 this is strictly smaller than 10^(d-1) and cannot be equal to n. See also A061211. - M. F. Hasler, Apr 12 2015

			4913 = (4 + 9 + 1 + 3)^3.

H. E. Dudeney, 536 Puzzles & Curious Problems, Souvenir Press, London, 1966, p. 36, #120.
Amarnath Murthy, The largest and the smallest m-th power whose digit sum is the m-th root. (To be published)
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

Henk Koppelaar and Peyman Nasehpour, On Hardy's Apology Numbers, arXiv:2008.08187 [math.NT], 2020.
Joseph S. Madachy, Recreational Mathematics, Fibonacci Quarterly 6:1 (1968), pp. 60-68.
Wikipedia, Dudeney number

Cf. A007953, A061210, A061211, A252648.

Select[Range[20000],Total[IntegerDigits[#]]^3==#&] (* Harvey P. Dale, Apr 11 2015 *)

for(n=0,999999,sumdigits(n)^3==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

a(n) = A007953(a(n))^3. - M. F. Hasler, Apr 12 2015

Initial term 0 added by M. F. Hasler, Apr 12 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

A061210 Numbers which are the fourth powers of their digit sum.

Original entry on oeis.org

0, 1, 2401, 234256, 390625, 614656, 1679616

Offset: 1

Amarnath Murthy, Apr 21 2001

It can be shown that 1679616 = 36^4 is the largest such number.

			614656 = ( 6+1+4+6+5+6)^4 =28^4.

Amarnath Murthy, The largest and the smallest m-th power whose digit sum is the m-th root. (To be published)
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

Cf. A061209 (with cubes), A061211.

Cf. A046000, A076090, A046017; A252648 and references there.

Select[Range[0,17*10^5],#==Total[IntegerDigits[#]]^4&] (* Harvey P. Dale, Sep 22 2019 *)

isok(n) = n == sumdigits(n)^4; \\ Michel Marcus, Jan 22 2015

Corrected by Ulrich Schimke, Feb 11 2002

Initial 0 added by M. F. Hasler, Apr 12 2015

A055015 Sum of 6th powers of digits of n.

Original entry on oeis.org

0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1, 2, 65, 730, 4097, 15626, 46657, 117650, 262145, 531442, 64, 65, 128, 793, 4160, 15689, 46720, 117713, 262208, 531505, 729, 730, 793, 1458, 4825, 16354, 47385, 118378, 262873

Offset: 0

Henry Bottomley, May 31 2000

The only fixed points (n = 0, 1 and 548834) are listed in row 6 of A252648. - M. F. Hasler, Apr 12 2015

Michel Marcus, Table of n, a(n) for n = 0..999
Index entries for Colombian or self numbers and related sequences

Cf. A003132.
Cf. A003132, A055012, A055013, A005514.
Cf. A007953, A055017, A076313, A076314.

[0] cat [&+[d^6: d in Intseq(n)]: n in [1..40]]; // Bruno Berselli, Feb 01 2013

for n from 0 to 3 do seq(n^6+j^6, j=0..9 ); od; # Zerinvary Lajos, Nov 06 2006

Table[Sum[DigitCount[n][[i]] i^6, {i, 9}], {n, 0, 40}] (* Bruno Berselli, Feb 01 2013 *)

A055015(n)=sum(i=1,#n=digits(n),n[i]^6) \\ M. F. Hasler, Apr 12 2015

a(n) = Sum_{k>0} (floor(n/10^k) - 10*floor(n/10^(k+1)))^6. - Hieronymus Fischer, Jun 25 2007

a(10n+k) = a(n) + k^6, 0 <= k < 10. - Hieronymus Fischer, Jun 25 2007

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

A023106 a(n) is a power of the sum of its digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 81, 512, 2401, 4913, 5832, 17576, 19683, 234256, 390625, 614656, 1679616, 17210368, 34012224, 52521875, 60466176, 205962976, 612220032, 8303765625, 10460353203, 24794911296, 27512614111, 52523350144, 68719476736

Offset: 0

David W. Wilson

Base-10 Reacher numbers: named for the character Jack Reacher in the series of books by Lee Child. Reacher likes the number 81 because it is the square of the sum of its base-10 digits. - Jeffrey Shallit, Apr 03 2015

Contains A061209 and A061210 and all terms of A061211. See A252648 for numbers which are the sum of some power of their digits. - M. F. Hasler, Apr 13 2015

			2401 is an element because 2401 = 7^4 is a power of its digit sum 7.

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

David W. Wilson, Table of n, a(n) for n = 0..1137
Jeffrey Shallit, Mathematics in a Jack Reacher Novel, blog post, September 8 2007.

Cf. A061209, A061210, A061211, A252648.

fQ[n_] := Block[{b = Plus @@ IntegerDigits[n]}, If[b > 1, IntegerQ[ Log[b, n]] ]]; Take[ Select[ Union[ Flatten[ Table[n^m, {n, 55}, {m, 9}]]], fQ[ # ] &], 31] (* Robert G. Wilson v, Jan 28 2005 *)
Join[{0,1},Select[Range[0,1700000],IntegerQ[Log[Total[IntegerDigits[#]],#]]&]//Quiet] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Mar 30 2024 *)

is(n)={n<10||(!(n%s=sumdigits(n))&&s>1&&n==s^round(log(n)/log(s)))} \\ M. F. Hasler, Apr 13 2015

import math
def is_valid(n): dsum = sum(map(int, str(n))); return dsum ** int(round(math.log(n, dsum))) == n if dsum > 1 else n < 2
# Victor Dumbrava, May 02 2018

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

cp's OEIS Frontend

A003321 Smallest n-th order perfect digital invariant or PDI: smallest number > 1 equal to sum of n-th powers of its digits, or 0 if no such number exists.

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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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A055013 Sum of 4th powers of digits of n.

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A061209 Numbers which are the cubes of their digit sum.

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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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A061210 Numbers which are the fourth powers of their digit sum.

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A055015 Sum of 6th powers of digits of n.

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