A061210 -id:A061210 - cp's OEIS frontend

A055013 Sum of 4th powers of digits of n.

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

A061211 Largest number m such that m is the n-th power of the sum of its digits.

Original entry on oeis.org

9, 81, 19683, 1679616, 205962976, 68719476736, 6722988818432, 248155780267521, 150094635296999121, 480682838924478847449, 23316389970546096340992, 2518170116818978404827136, 13695791164569918553628942336, 4219782742781494680756610809856

Offset: 1

Amarnath Murthy, Apr 21 2001

Clearly m = 1 always works, so a(n) exists for all n. - Farideh Firoozbakht, Nov 23 2007

105 is the smallest number n such that a(n)=1. This means that if n<105 there exists at least one number m greater than 1 such that m is the n-th power of the sum of its digits while 1 is the only number m such that m is the 105th power of the sum of its digits. A133509 gives n such that a(n) = 1. - Farideh Firoozbakht, Nov 23 2007

			a(3) = 19683 = 27^3 and no bigger number can have this property. (This has been established in the Murthy reference.)
a(4) = 1679616 = (1+6+7+9+6+1+6)^4 = 36^4.

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.
Amarnath Murthy, e-book, "Ideas on Smarandache Notions", manuscript.

T. D. Noe, Table of n, a(n) for n = 1..105

Cf. A061209, A061210, A046000, A076090, A046017.

meanDigit = 9/2; translate = 900; upperm[1] = translate;
upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
a[n_] := (For[max = m = 1, m <= upperm[n], m++, If[m == Total[ IntegerDigits[ m^n ] ], max = m]]; max^n);
Array[a, 14] (* Jean-François Alcover, Jan 09 2018 *)

More terms from Ulrich Schimke, Feb 11 2002

Edited by N. J. A. Sloane at the suggestion of Farideh Firoozbakht, Dec 04 2007

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

A254000 Numbers equal to the fifth powers of the sums of their digits.

Original entry on oeis.org

0, 1, 17210368, 52521875, 60466176, 205962976

Offset: 1

Michal Paulovic, Jan 21 2015

			205962976 = 46^5 = (2 + 0 + 5 + 9 + 6 + 2 + 9 + 7 + 6)^5.

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

Cf. A061209 (with cubes), A061210 (with 4th powers), A061211.

Select[Range@ 210000000, Plus @@ IntegerDigits@ # ^ 5 == # &] (* Michael De Vlieger, Feb 25 2015 *)

lista(nn) = {for (n=0, nn, if (n^5 == sumdigits(n^5)^5, print1(n^5, ", ")););} \\ Michel Marcus, Feb 23 2015

a(n) = A055576(n)^5. - Michel Marcus, Feb 23 2015

A366507 Numbers k such that the sum of the digits of k times the square of the sum of the digits cubed of k equals k.

Original entry on oeis.org

1, 4147200, 12743163, 21147075, 39143552, 52921472, 156754936, 205889445, 233935967

Offset: 1

René-Louis Clerc, Oct 11 2023

There are exactly 9 such numbers (Property 13 of Clerc).

			4147200 = (4+1+4+7+2)*(4^3+1+4^3+7^3+2^3)^2 = 18*230400.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, pp. 1-17, hal-04235744, 2023.
René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Cf. A115518, A257766, A061209, A061210, A254000, A130680.

niven12()={for(a=0,9,for(b=0,9,for(c=0,9,for(d=0,9,for(e=0,9,for(f=0,9,for(g=0,9,for(h=0,9,for(i=0,9,for(j=0,9,if((a+b+c+d+e+f+g+h+i+j)*(a^3+b^3+c^3+d^3+e^3+f^3+g^3+h^3+i^3+j^3)^2==1000000000*a+100000000*b+10000000*c+1000000*d+100000*e+10000*f+1000*g+100*h+10*i+j,print1(1000000000*a+100000000*b+10000000*c+1000000*d+100000*e+10000*f+1000*g+100*h+10*i+j,";"))))))))))))}

isok(k) = my(d=digits(k)); vecsum(d)*sum(i=1, #d, d[i]^3)^2 == k; \\ Michel Marcus, Oct 12 2023

A366512 Numbers k such that the square of the sum of the digits times the sum of the cubes of the digits equals k.

Original entry on oeis.org

1, 32144, 37000, 111616, 382360

Offset: 1

René-Louis Clerc, Oct 11 2023

There are exactly 5 such numbers (Property 14 of Clerc).

			32144 = ((3+2+1+4+4)^2)*(3^3 + 2^3 + 1^3 + 4^3 + 4^3) = 196*164.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, pp. 1-17, hal-04235744, 2023.
René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Cf. A115518, A257766, A061209, A061210, A254000, A130680.

Select[Range[10^6], #1 == Total[#2]^2*Total[#2^3] & @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Mar 25 2024 *)

niven23()={for(a=0,9,for(b=0,9,for(c=0,9,for(d=0,9,for(e=0,9,for(f=0,9,for(g=0,9,for(h=0,9,if((a+b+c+d+e+f+g+h)^2*(a^3+b^3+c^3+d^3+e^3+f^3+g^3+h^3)==10000000*a+1000000*b+100000*c+10000*d+1000*e+100*f+10*g+h,print1(10000000*a+1000000*b+100000*c+10000*d+1000*e+100*f+10*g+h,", "))))))))))}

isok(k) = my(d=digits(k)); vecsum(d)^2*sum(i=1, #d, d[i]^3) == k; \\ Michel Marcus, Oct 12 2023

A259313 Numbers m for which there exists a k>=2 such that m equals the average of digitsum(m^p) for p from 1 to k.

Original entry on oeis.org

1, 9, 12, 13, 16, 19, 21, 49, 61, 67, 84, 106, 160, 191, 207, 250, 268, 373, 436, 783, 2321, 3133, 3786, 3805, 4842, 5128, 8167, 13599, 29431, 35308

Offset: 1

Pieter Post, Jun 24 2015

Digitsum = (A007953).

The 'k's are 2, 2, 4, 3, 4, 5, 7, 12, 15, 16, 19, 21, 57, 37, 38, 79, 48, 63, 72, 119, 306, 397, 469, 472, 582, 613, 927, 1461, 2926, 3449, ..., . - Robert G. Wilson v, Jul 30 2015

			Digitsum(9) is 9, digitsum(9^2) is 9. (9+9)/2 = 9. So 9 is in this sequence.
12^1 = 12, 12^2 = 144, 12^3 = 1728 and 12^4 = 20736. Digitsum(12) = 3, digitsum(144) = 9, digitsum(1728) = 18, digitsum(20736) = 18, (3+9+18+18)/4 = 12. So 12 is in this sequence.

Cf. A007953, A061910, A061209, A061210.

fQ[n_] := If[ IntegerQ@ Log10@ n, False, Block[{pwr = 2, s = Plus @@ IntegerDigits@ n}, While[s = s + Plus @@ IntegerDigits[n^pwr]; s < n*pwr, pwr++]; If[s == n*pwr, True, False]]]; k = 1; lst = {1}; While[k < 100001, If[fQ@ k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Jul 30 2015 *)

def sod(n):
    kk = 0
    while n > 0:
        kk= kk+(n%10)
        n =int(n//10)
    return kk
for c in range (2, 10**3):
    bb=0
    for a in range(1,200):
        bb=bb+sod(c**a)
        if bb==c*a:
            print (c,a)

a(21)-a(28) from Giovanni Resta, Jun 24 2015
a(1)-a(28) checked by Robert G. Wilson v, Jul 30 2015
a(29)-a(30) from Robert G. Wilson v, Jul 30 2015

A368939 Numbers k such that the sum of the digits times the sum of the fourth powers of the digits equals k.

Original entry on oeis.org

0, 1, 182380, 444992

Offset: 1

René-Louis Clerc, Jan 10 2024

There are exactly 4 such numbers (Property 16 of Clerc).

			182380 = (1+8+2+3+8)*(1^4 + 8^4 + 2^4 + 3^4 + 8^4) = 22*8290.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.

Cf. A115518, A257766, A061209, A061210, A254000, A130680, A366507, A366512.

Select[Range[0,10^7],#==Total[IntegerDigits[#]]*Total[IntegerDigits[#]^4]&] (* James C. McMahon, Jan 11 2024 *)

niven14(k) = my(d=digits(k)); vecsum(d)*sum(i=1, #d, d[i]^4) == k;
for(k=1,10^7,if(niven14(k)==1,print1(k,", ")))

A375343 Numbers which are the sixth powers of their digit sum.

Original entry on oeis.org

0, 1, 34012224, 8303765625, 24794911296, 68719476736

Offset: 1

René-Louis Clerc, Aug 12 2024

Solutions can have no more than 13 digits, since (13*9)^6 < 10^13.

			68719476736 = (6+8+7+1+9+4+7+6+7+3+6)^6 = 64^6.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.

Cf. A001014, A007953, A061209, A061210, A254000, A061211.

Cf. A055577, A152147.

for (k=0, sqrtnint(10^13,6), if (k^6 == sumdigits(k^6)^6, print1(k^6, ", ")); )

{ k : k = A007953(k)^6}.

a(n) = A055577(n)^6. - Alois P. Heinz, Aug 24 2024

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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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A061211 Largest number m such that m is the n-th power of the sum of its digits.

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

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A254000 Numbers equal to the fifth powers of the sums of their digits.

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A366507 Numbers k such that the sum of the digits of k times the square of the sum of the digits cubed of k equals k.

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A366512 Numbers k such that the square of the sum of the digits times the sum of the cubes of the digits equals k.

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