A380052 -id:A380052 - cp's OEIS frontend

A380193 a(n) is the largest number whose sixth power is an n-digit sixth power which has the maximum sum of digits (A373994(n)).

1, 2, 3, 4, 6, 7, 12, 19, 31, 46, 68, 96, 143, 206, 304, 461, 677, 977, 1194, 2136, 2896, 4633, 6373, 9763, 13817, 21542, 30643, 43693, 68123, 99812, 144083, 183967, 311296, 463976, 681017, 994333, 1441977, 2150104, 3022731, 4608562, 6765526, 9258023

Offset: 1

Zhining Yang, Jan 15 2025

			a(11) = 68 because among all 11-digit sixth powers(47^6-68^6), 64^6=68719476736 and 68^6=98867482624 have the maximum sum of digits, 96 = A373994(11) and 68 is the largest number.

Zhining Yang, Table of n, a(n) for n = 1..55 [a(47) corrected by _Kevin Ryde_, Mar 25 2025]

Cf. A373994, A380567.

Other powers: A379298, A380052, A380797, A380566.

C
```
/* See A373994. */
```

Table[t=SortBy[Map[{#,Total@IntegerDigits[#^6]}&,Range[Ceiling[10^((n-1)/6)],Floor[(10^n-1)^(1/6)]]],Last];
Select[t,#[[2]]==t[[-1]][[2]]&][[1,1]],{n,36}]

A380566 a(n) = k is the largest k for which k^5 is n digits long and the sum of digits of k^5 is the maximum for any n digit 5th power (A374025).

Original entry on oeis.org

1, 2, 3, 6, 9, 15, 18, 37, 58, 93, 156, 179, 368, 579, 756, 1379, 2337, 3965, 6006, 9746, 14198, 25046, 38779, 60006, 98746, 151446, 231755, 389658, 585516, 819199, 1584779, 2452779, 3897999, 5400759, 9744998, 15517759, 23936959, 28737498, 62943519, 95635199, 156373159, 225142779, 351816939, 595519999

Offset: 1

Zhining Yang, Jan 26 2025

			a(14) = 579 because among all 14-digit fifth powers(399^5-630^5), 549^5=49872566977749,579^5=65071799758899, both have the maximum sum of digits, 90 = A374025(14) and 579 is the largest.

Zhining Yang, Table of n, a(n) for n = 1..52

Cf. A374025, A379650.

Cf. A380052, A380193.

a[n_]:=Module[{m=Floor[(10^n-1)^(1/5)],max=0},For[k=m,k>=Ceiling[10^((n-1)/5)],k--,t=Total@IntegerDigits[k^5];If[t>max,s=k;max=t]];s];Table[a[n],{n,30}]

A380797 a(n) is the largest number whose fourth power is an n-digit which has the maximum sum of digits (A373914(n)).

Original entry on oeis.org

1, 3, 5, 8, 16, 26, 56, 88, 118, 308, 518, 974, 1768, 2868, 5396, 8979, 17306, 28871, 55368, 97063, 167622, 289146, 562341, 835718, 1727156, 3154276, 5623116, 9397404, 17728256, 27831542, 53129506, 98665756, 166025442, 315265896, 510466356, 904245732, 1188893858, 2298249374, 5315776056

Offset: 1

Zhining Yang, Feb 03 2025

			a(7) = 56 because among all 7-digit fourth powers, 56^4=9834496 is the largest one (another smaller is 47^4=487968) which has the maximum sum of digits, 43 = A373914(7).

Cf. A373914, A380111.

Other powers: A379298, A380052, A380566, A380193.

C
```
/* See A373914. */
```

a[n_]:=Module[{m=Floor[(10^n-1)^(1/4)], max=0},
For[k=m, k>=Ceiling[10^((n-1)/4)], k--, t=Total@IntegerDigits[k^4];
If[t>max, s=k; max=t]]; s];
Table[a[n], {n, 30}]

A379298 Largest number k for which k^2 is n digits long and has the maximum sum of digits possible for such a square (A371728(n)).

Original entry on oeis.org

3, 7, 28, 83, 313, 937, 3114, 9417, 29614, 94863, 298327, 987917, 3162083, 9893887, 29983327, 99483667, 315432874, 994927133, 2999833327, 9486778167, 31464263856, 99497231067, 299998333327, 999949483667, 3160522105583, 9892825177313, 29999983333327

Offset: 1

Zhining Yang, Feb 05 2025

			a(6) = 937 because among all 6-digit squares, 698896 = 836^2, 779689 = 883^2, 877969 = 937^2 have the maximum sum of digits 46 = A371728(6), and 937 is the largest.

Cf. A371728, A348303.

Other powers: A380052, A380797, A380566, A380193.

a[n_] := Module[{s = Floor[Sqrt[(10^n - 1)]], max = 0},
   For[k = s, k >= Ceiling[Sqrt[10^(n - 1)]], k--, t = DigitSum[k^2];
    If[t > max, s = k; max = t]]; s];
Table[a[n], {n, 30}]

Conjecture: a(2*n) = A348303(n).

A379869 a(n) is the least number whose cube is an n-digit cube which has the maximum sum of digits (A373727(n)).

Original entry on oeis.org

2, 4, 9, 19, 31, 92, 157, 423, 927, 1966, 4289, 8782, 12599, 30355, 99829, 215083, 341075, 989353, 2131842, 4081435, 8334082, 20632999, 43967926, 88316866, 190349299, 364929616, 735501679, 1948602829, 3036548692, 9654499999, 17087193298, 31037622999, 99594689449, 181610950229, 426932901019, 956829383603

Offset: 1

Zhining Yang, Jan 11 2025

			For n=7, the maximum sum of digits for a 7-digit cube is A373727(7) = 46 and this is attained by 3 cubes, the smallest of which is 157^3 = 3869893 so that a(7) = 157.

Cf. A373727, A380052.

Other powers: A380111, A379650, A380567.

C
```
/* See A373727. */
```

Table[t =SortBy[Map[{#, Total@IntegerDigits[#^3]} &,
    Range[Ceiling@CubeRoot[10^(n - 1)], CubeRoot[10^n - 1]]], Last];
 Select[t, #[[2]] == t[[-1]][[2]] &][[1, 1]], {n, 18}]

a(26) and a(35) corrected by Kevin Ryde, Apr 03 2025

cp's OEIS Frontend

A380193 a(n) is the largest number whose sixth power is an n-digit sixth power which has the maximum sum of digits (A373994(n)).

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A380566 a(n) = k is the largest k for which k^5 is n digits long and the sum of digits of k^5 is the maximum for any n digit 5th power (A374025).

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A380797 a(n) is the largest number whose fourth power is an n-digit which has the maximum sum of digits (A373914(n)).

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A379298 Largest number k for which k^2 is n digits long and has the maximum sum of digits possible for such a square (A371728(n)).

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A379869 a(n) is the least number whose cube is an n-digit cube which has the maximum sum of digits (A373727(n)).

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