A055565 -id:A055565 - cp's OEIS frontend

A055566 Sum of digits of n^5.

0, 1, 5, 9, 7, 11, 27, 22, 26, 27, 1, 14, 27, 25, 29, 36, 31, 35, 45, 37, 5, 18, 25, 29, 36, 40, 35, 36, 28, 23, 9, 34, 29, 36, 31, 35, 36, 46, 41, 36, 7, 29, 27, 31, 35, 36, 46, 32, 45, 43, 11, 27, 22, 44, 36, 37, 41, 36, 52, 47, 27, 40, 35, 45, 37, 32, 36, 25, 47, 36, 22, 35

Offset: 0

Henry Bottomley, May 26 2000

			a(2) = 5 because 2^4 = 32 and 3+2 = 5.
Trajectories under the map x->a(x):
1 ->1 ->1 ->1 ->1 ->1 ->1 ->1 ->1 ->..
2 ->5 ->11 ->14 ->29 ->23 ->29 ->23 ->29 ->..
3 ->9 ->27 ->36 ->36 ->36 ->36 ->36 ->36 ->..
4 ->7 ->22 ->25 ->40 ->7 ->22 ->25 ->40 ->..
5 ->11 ->14 ->29 ->23 ->29 ->23 ->29 ->23 ->..
6 ->27 ->36 ->36 ->36 ->36 ->36 ->36 ->36 ->..
7 ->22 ->25 ->40 ->7 ->22 ->25 ->40 ->7 ->..

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000012, A004159, A004164, A055565, A055567, A055571, A055576, A066588.

read("transforms") :
A055566 := proc(n)
        digsum(n^5) ;
end proc: # R. J. Mathar, Jul 08 2012

Table[Total[IntegerDigits[n^5]],{n,0,80}] (* Harvey P. Dale, Feb 12 2023 *)

a(n) = sumdigits(n^5); \\ Seiichi Manyama, Nov 16 2021

A055575 Sum of digits of n^4 is equal to n.

Original entry on oeis.org

0, 1, 7, 22, 25, 28, 36

Offset: 1

Henry Bottomley, May 26 2000

			7 is a member because 7^4 = 2401 and 2+4+0+1 = 7.

Cf. A055565, A055570, A046459, A055576, A055577.

Cf. A152147.

[n: n in [0..50] | &+Intseq(n^4) eq n ]; // Vincenzo Librandi, Feb 23 2015

Select[Range[0, 50], #==Total[IntegerDigits[#^4]] &] (* Vincenzo Librandi, Feb 23 2015 *)

isok(k)=sumdigits(k^4)==k \\ Patrick De Geest, Dec 10 2024

[n for n in (0..50) if sum((n^4).digits()) == n] # Bruno Berselli, Feb 23 2015

A373914 a(n) is the largest digit sum of all n-digit fourth powers.

Original entry on oeis.org

1, 9, 13, 19, 25, 37, 43, 52, 55, 70, 76, 79, 85, 99, 103, 108, 118, 127, 135, 142, 144, 153, 171, 166, 178, 181, 189, 198, 205, 211, 220, 232, 234, 243, 252, 261, 265, 274, 279, 283, 297, 298, 313, 316, 325, 334, 337, 346, 358

Offset: 1

Zhining Yang, Jun 22 2024

			a(3) = 13 because 13 is the largest digital sum encountered among all 3-digit fourth powers (attained at both fourth powers: 256, 625).

Kevin Ryde, C Code

Cf. A055565, A018072, A018074, A380111, A380797.

Other powers: A371728, A373727, A374025, A373994.

C
```
/* See links. */
```

Table[Max@Map[Total@IntegerDigits[#^4] &, Range[Ceiling[10^((n - 1)/4)], Floor[(10^n-1)^(1/4)]]], {n, 32}]

a(n) = my(m=ceil(10^((n-1)/4)), M=sqrtint(sqrtint(10^n))); vecmax(apply(sumdigits, vector(M-m+1, i, (i+m-1)^4))); \\ Michel Marcus, Jun 23 2024

from sympy import integer_nthroot
def A373914(n): return max(sum(int(d) for d in str(m**4)) for m in range((lambda x:x[0]+(x[1]^1))(integer_nthroot(10**(n-1),4)),1+integer_nthroot(10**n-1,4)[0])) # Chai Wah Wu, Jun 26 2024

A055570 Sum of digits of (a(n)^4) is greater than or equal to a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 28, 36

Offset: 0

Henry Bottomley, May 26 2000

			a(2) = 2 because 2^4 = 16 and 1+6 = 7>= 2

Cf. A055565, A055575, A055568, A055569, A055571, A055572.

Select[Range[0,50],Total[IntegerDigits[#^4]]>=#&] (* Harvey P. Dale, Jan 20 2020 *)

Definition clarified by Harvey P. Dale, Jan 20 2020

A055567 Sum of digits of n^6.

Original entry on oeis.org

0, 1, 10, 18, 19, 19, 27, 28, 19, 18, 1, 28, 45, 37, 37, 27, 37, 37, 18, 37, 10, 36, 37, 46, 36, 28, 46, 45, 37, 37, 18, 46, 37, 54, 37, 46, 45, 46, 37, 45, 19, 28, 45, 37, 46, 45, 64, 46, 36, 37, 19, 54, 55, 37, 54, 46, 55, 54, 55, 37, 27, 37, 46, 36, 64, 55, 45, 55, 64, 45

Offset: 0

Henry Bottomley, May 26 2000

			a(2) = 10 because 2^6 = 64 and 6+4 = 10.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000012, A004159, A004164, A055565, A055566, A055572, A055577, A066588.

DigitSum[Range[0, 100]^6] (* Paolo Xausa, Jul 03 2024 *)

a(n) = sumdigits(n^6); \\ Seiichi Manyama, Nov 16 2021

A286650 a(n) is the smallest number m such that the sum of the digits of m^4 is equal to n^4.

Original entry on oeis.org

0, 1, 11, 1434, 1269681358

Offset: 0

Seiichi Manyama, Aug 15 2017

			a(2) = 11 as 11^4 = 14641 is the smallest fourth power whose digit sum = 16 = 2^4.

Cf. A061912, A067075, A069648, A291145.

Cf. A000583 (n^4), A055565 (sum of digits of n^4).

{a(n) = my(k=0); while(sumdigits(k^4) != n^4, k++); k}

A336225 Table read by antidiagonals: T(n, k) = digitsum(n*k) with n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 1, 3, 3, 1, 6, 0, 0, 7, 3, 6, 7, 6, 3, 7, 0, 0, 8, 5, 9, 2, 2, 9, 5, 8, 0, 0, 9, 7, 3, 6, 7, 6, 3, 7, 9, 0, 0, 1, 9, 6, 10, 3, 3, 10, 6, 9, 1, 0, 0, 2, 2, 9, 5, 8, 9, 8, 5, 9, 2, 2, 0

Offset: 0

Stefano Spezia, Jul 12 2020

			The table T(n, k) begins
0   0   0   0   0   0   0   0 ...
0   1   2   3   4   5   6   7 ...
0   2   4   6   8   1   3   5 ...
0   3   6   9   3   6   9   3 ...
0   4   8   3   7   2   6  10 ...
0   5   1   6   2   7   3   8 ...
0   6   3   9   6   3   9   6 ...
0   7   5   3  10   8   6  13 ...
...

Index entries for sequences related to sum of digits.

Cf. A003991, A004092, A004159 (diagonal), A004164 (digitsum of n^3), A004247, A007953, A055565 (digitsum of n^4), A055566 (digitsum of n^5), A055567 (digitsum of n^6).

T[n_,k_]:=Total[IntegerDigits[n*k]]; Table[T[n-k,k],{n,0,12},{k,0,n}]//Flatten

PARI
```
T(n, k) = sumdigits(n*k);
```

T(n, k) = A007953(A004247(n, k)).
T(n, 1) = T(1, n) = A007953(n).
T(n, 2) = T(2, n) = A004092(n).
T(n, k) = A007953(A003991(n, k)) for n*k > 0. - Michel Marcus, Jul 13 2020.

A118470 Numbers k for which digitsum(k) + digitsum(k^2) + digitsum(k^3) = digitsum(k^4).

Original entry on oeis.org

0, 162, 171, 351, 468, 558, 1620, 1710, 2106, 3321, 3510, 4023, 4680, 5121, 5247, 5544, 5580, 5868, 8001, 10008, 10071, 10224, 10305, 10503, 10818, 11025, 11241, 11511, 12321, 12654, 12888, 13239, 14004, 14301, 15471, 15876, 16011, 16200, 16218, 17100

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 04 2006

If x is a term, then so is 10*x. - Michael S. Branicky, Dec 25 2021

			162 is a term because s(162) = 9, s(162^2) = 18, s(162^3) = 27, s(162^4) = 54 and 9 + 18 + 27 = 54.

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..3721 (all terms < 10^7)

Cf. A007953, A004159, A004164, A055565.

Select[Range[0, 20000], Sum[i*(DigitCount[ # ][[i]] + DigitCount[ #^2][[i]] + DigitCount[ #^3][[i]]), {i, 1, 9}] == Sum[i*DigitCount[ #^4][[i]], {i, 1, 9}] &] (* Stefan Steinerberger, May 04 2006 *)
s[n_] := Plus @@ IntegerDigits@n; Select[ Range[0, 16217], s@# + s[ #^2] + s[ #^3] == s[ #^4] &] (* Robert G. Wilson v, May 04 2006 *)
Parallelize[While[True,If[Total[IntegerDigits[n]]+Total[IntegerDigits[n^2]]+Total[IntegerDigits[n^3]]==Total[IntegerDigits[n^4]],Print[n]];n++];n] (* J.W.L. (Jan) Eerland, Dec 25 2021 *)

is(n)=my(s=sumdigits); s(n)+s(n^2)+s(n^3) == s(n^4) \\ Anders Hellström, Sep 16 2015

select(isA118470(n)={sumdigits(n)+sumdigits(n^2)+sumdigits(n^3) == sumdigits(n^4)}, [0..1000]) \\ J.W.L. (Jan) Eerland, Dec 25 2021

def sd(n): return sum(map(int, str(n)))
def ok(n): return sd(n) + sd(n**2) + sd(n**3) == sd(n**4)
print([k for k in range(20000) if ok(k)]) # Michael S. Branicky, Dec 25 2021

More terms from Joshua Zucker, May 11 2006

A099358 a(n) = sum of digits of k^4 as k runs from 1 to n.

Original entry on oeis.org

1, 8, 17, 30, 43, 61, 68, 87, 105, 106, 122, 140, 162, 184, 202, 227, 246, 273, 283, 290, 317, 339, 370, 397, 422, 459, 477, 505, 530, 539, 561, 592, 619, 644, 663, 699, 727, 752, 770, 783, 814, 841, 866, 903, 921, 958, 1001, 1028, 1059, 1072, 1099, 1124, 1161

Offset: 1

Yalcin Aktar, Nov 16 2004

Partial sums of A055565.

			a(3) = sum_digits(1^4) + sum_digits(2^4) + sum_digits(3^4) = 1 + 7 + 9 = 17.

Cf. k^1 in A037123, k^2 in A071317 & k^3 in A071121.

f[n_] := Block[{s = 0, k = 1}, While[k <= n, s = s + Plus @@ IntegerDigits[k^4]; k++ ]; s]; Table[ f[n], {n, 50}] (* Robert G. Wilson v, Nov 18 2004 *)
Accumulate[Table[Total[IntegerDigits[n^4]],{n,60}]] (* Harvey P. Dale, Jun 08 2021 *)

a(n) = a(n-1) + sum of decimal digits of n^4.
a(n) = sum(k=1, n, sum(m=0, floor(log(k^4)), floor(10((k^4)/(10^(((floor(log(k^4))+1))-m)) - floor((k^4)/(10^(((floor(log(k^4))+1))-m))))))).
General formula: a(n)_p = sum(k=1, n, sum(m=0, floor(log(k^p)), floor(10((k^p)/(10^(((floor(log(k^p))+1))-m)) - floor ((k^p)/(10^(((floor(log(k^p))+1))-m))))))). Here a(n)_p is a sum of digits of k^p from k=1 to n.

Edited and extended by Robert G. Wilson v, Nov 18 2004

Existing example replaced with a simpler one by Jon E. Schoenfield, Oct 20 2013

cp's OEIS Frontend

A055566 Sum of digits of n^5.

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A055575 Sum of digits of n^4 is equal to n.

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A373914 a(n) is the largest digit sum of all n-digit fourth powers.

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A055570 Sum of digits of (a(n)^4) is greater than or equal to a(n).

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A055567 Sum of digits of n^6.

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A286650 a(n) is the smallest number m such that the sum of the digits of m^4 is equal to n^4.

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A336225 Table read by antidiagonals: T(n, k) = digitsum(n*k) with n >= 0 and k >= 0.

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A118470 Numbers k for which digitsum(k) + digitsum(k^2) + digitsum(k^3) = digitsum(k^4).

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