A055566 -id:A055566 - cp's OEIS frontend

A178481 Number of steps of the map x -> A055566(x), starting at n, before reaching the end of the cycle.

0, 0, 5, 3, 4, 4, 2, 3, 2, 2, 1, 3, 2, 4, 2, 1, 2, 1, 2, 2, 5, 3, 3, 1, 1, 3, 1, 1, 0, 1, 3, 1, 2, 1, 1, 0, 0, 1, 3, 1, 3, 2, 2, 2, 1, 1, 0, 3, 2, 3, 4, 2, 4, 2, 1, 2, 3, 1, 5, 4, 2, 4, 1, 2, 2, 3, 1, 4, 4, 1, 4, 1, 2, 2, 3, 2, 3, 4, 2, 4, 2

Offset: 0

Michel Lagneau, May 28 2010

a(n) is the number of times taking the 5th powers of the sums of digits before reaching a sum seen before (reaching the last number of the cycle).
Example:
6 -> 6^5 = 7776 -> (7+7+7+6)^5 = 27^5.
27^5 = 14348907 -> (1+4+3+4+8+9+0+7)^5 = 36^5.
36^5 = 60466176, last number of the cycle because (6+0+4+6+6+1+7+6)^5 = 36^5 = 60466176 belongs to the list.
Generalization for the k-th powers and conjecture: For each k >= 1, iteration of taking the k-th powers of digit sums reaches a cycle.
Example with k = 17; start with 3.
3^17 = 129140163, sum = 27,
27^17 = 2153693963075557766310747, sum = 117,
117^17 = 144264558065210807467328187211661877, sum = 153,
153^17 = 13796036156758195415808856807283698713, sum = 189,
189^17 = 501014933601411817143935347829544613629, sum = 153 is already in the set.
[It remains unclear whether the author wanted to define iterations of (sumofdigits of n)^5, compatible with A177148 and A182128, or sumofdigits(n^5) here. I've taken the latter to be more compliant with the first terms of the original submission. - R. J. Mathar, Jul 08 2012]

			a(0) = 0 and a(1) = 0 because 0 -> 0 and 1 -> 1.
a(15) = 1 because 15^5 = 759375 -> (7+5+9+3+7+5) = 36,
36 ^5 = 60466176 -> (6+0+4+6+6+1+7+6) = 36.

Cf. A177148, A182128.

A178481 := proc(n)
        local traj ,c;
        traj := n ;
        c := [n] ;
        while true do
                traj := A055566(traj) ;
                if member(traj,c) then
                        return nops(c)-1 ;
                end if;
                c := [op(c),traj] ;
        end do:
end proc:
seq(A178481(n),n=0..80) ; # R. J. Mathar, Jul 08 2012

A055565 Sum of digits of n^4.

Original entry on oeis.org

0, 1, 7, 9, 13, 13, 18, 7, 19, 18, 1, 16, 18, 22, 22, 18, 25, 19, 27, 10, 7, 27, 22, 31, 27, 25, 37, 18, 28, 25, 9, 22, 31, 27, 25, 19, 36, 28, 25, 18, 13, 31, 27, 25, 37, 18, 37, 43, 27, 31, 13, 27, 25, 37, 27, 28, 43, 18, 31, 22, 18, 34, 37, 36, 37, 34, 45, 13, 31, 27, 7

Offset: 0

Henry Bottomley, Jun 19 2000

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

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

Cf. A000583, A007953, A055570, A055575 (fixed points), A373914.

Other powers: A004159, A004164, A055566, A055567, A066588.

for i from 0 to 200 do printf(`%d,`,add(j, j=convert(i^4, base, 10))) od;

a[n_Integer]:=Apply[Plus, IntegerDigits[n^4]]; Table[a[n], {n, 0, 100}] (* Vincenzo Librandi, Feb 23 2015 *)

a(n) = sumdigits(n^4); \\ Seiichi Manyama, Nov 16 2021

[sum((n^4).digits()) for n in (0..70)] # Bruno Berselli, Feb 23 2015

a(n) = A007953(A000583(n)). - Michel Marcus, Feb 23 2015

More terms from James Sellers, Jul 04 2000

A055576 Sum of digits of a(n)^5 is equal to a(n).

Original entry on oeis.org

0, 1, 28, 35, 36, 46

Offset: 1

Henry Bottomley, May 26 2000

			a(2) = 28 because 28^5 = 17210368 and 1+7+2+1+0+3+6+8 = 28

Cf. A055566, A055571, A046459, A055575, A055577.

Cf. A152147.

[n: n in [0..50] | &+Intseq(n^5) eq n ]; // Vincenzo Librandi, Feb 23 2015

Select[Range[0, 60], #==Total[IntegerDigits[#^5]] &] (* Vincenzo Librandi, Feb 23 2015 *)

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

Offset changed to 1 by Michel Marcus, Feb 23 2015

A225017 Odd part of digit sum of 5^n divided by maximal possible power of 5.

Original entry on oeis.org

1, 1, 7, 1, 13, 11, 19, 23, 1, 13, 1, 19, 7, 23, 17, 11, 29, 7, 1, 59, 61, 31, 67, 37, 41, 77, 79, 89, 17, 83, 91, 13, 53, 89, 103, 23, 109, 13, 31, 67, 13, 137, 29, 149, 151, 29, 7, 1, 29, 79, 151, 19, 13, 119, 127, 167, 49, 43, 211, 191, 199, 97, 187, 17, 83

Offset: 0

Vladimir Shevelev, Apr 24 2013

Does the sequence contain every prime greater than 5?

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..999 from Peter J. C. Moses)

Cf. A055566, A132740.

a:= proc(n) local m, r; m, r:= 0, 5^n;
      while r>0 do m:= m+irem(r, 10, 'r') od;
      while irem(m, 2, 'r')=0 do m:=r od;
      while irem(m, 5, 'r')=0 do m:=r od; m
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 24 2013

Map[#/(2^IntegerExponent[#,2] 5^IntegerExponent[#,5])&[Total[ IntegerDigits[5^#]]]&,Range[0,99]] (* Peter J. C. Moses, Apr 24 2013 *)

a(n) = my(x = sumdigits(5^n)); x/5^valuation(x, 5) >> valuation(x, 2); \\ Michel Marcus, Dec 10 2018

a(n) = A132740(A055566(n)). - Michel Marcus, Dec 10 2018

A055571 Sum of digits of a(n)^5 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, 19, 22, 23, 24, 25, 26, 27, 28, 31, 33, 35, 36, 37, 38, 46

Offset: 1

Henry Bottomley, May 26 2000

			a(2) = 2 because 2^5 = 32 and 3+2 = 5>= 2

Cf. A055566, A055576, A055568, A055569, A055570, A055572.

Select[Range[0,50],Total[IntegerDigits[#^5]]>=#&] (* Harvey P. Dale, Jul 03 2021 *)

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

A374025 a(n) is the largest digit sum of all n-digit fifth powers.

Original entry on oeis.org

1, 5, 9, 27, 27, 36, 45, 46, 52, 63, 72, 80, 89, 90, 99, 104, 108, 119, 126, 143, 137, 152, 157, 162, 175, 180, 182, 189, 198, 208, 209, 216, 225, 234, 236, 250, 253, 270, 270, 284, 286, 288, 297, 310, 315, 323, 324, 334, 341, 346, 351, 364

Offset: 1

Zhining Yang, Jun 25 2024

			a(5) = 27 because 27 is the largest digital sum encountered among all 5-digit fifth powers (16807, 32768, 59049).

Cf. A055566, A018141, A018143, A348300, A373727, A373914.

Table[Max@Map[Total@IntegerDigits[#^5] &, Range[Ceiling[10^((n - 1)/5)], Floor[(10^n-1)^(1/5)]]], {n, 40}]

from sympy import integer_nthroot
def A374025(n): return max(sum(int(d) for d in str(m**5)) for m in range((lambda x:x[0]+(x[1]^1))(integer_nthroot(10**(n-1),5)),1+integer_nthroot(10**n-1,5)[0])) # Chai Wah Wu, Jun 26 2024

a(41)-a(49) from Chai Wah Wu, Jun 26 2024

a(50)-a(52) from Chai Wah Wu, Jun 27 2024

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.

A328364 a(n) is the smallest number m such that the sum of the digits of m^5 is equal to n^5.

Original entry on oeis.org

0, 1, 47, 13174539

Offset: 0

Seiichi Manyama, Oct 14 2019

			a(2) = 47 as 47^5 = 229345007 is the smallest fifth power whose digit sum = 32 = 2^5.

Cf. A061912, A069648, A067075, A286650, A328374.

Cf. A000584 (n^5), A055566 (sum of digits of n^5).

{a(n) = my(k=0); while(sumdigits(k^5) != n^5, k++); k}

cp's OEIS Frontend

A178481 Number of steps of the map x -> A055566(x), starting at n, before reaching the end of the cycle.

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A055565 Sum of digits of n^4.

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A055576 Sum of digits of a(n)^5 is equal to a(n).

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Magma

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A225017 Odd part of digit sum of 5^n divided by maximal possible power of 5.

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A055571 Sum of digits of a(n)^5 is greater than or equal to a(n).

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Mathematica

A055567 Sum of digits of n^6.

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A374025 a(n) is the largest digit sum of all n-digit fifth powers.

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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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