A379298 -id:A379298 - 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}]

A380052 a(n) is the largest 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, 46, 92, 208, 453, 942, 1966, 4289, 9949, 12599, 43795, 99829, 215083, 446423, 989353, 2131842, 4081435, 9850783, 20714797, 43967926, 92827483, 190349299, 464110759, 989554129, 2132590453, 4559677342, 9654499999, 21253161559, 31037622999, 99594689449, 181610950229

Offset: 1

Zhining Yang, Jan 11 2025

			For n=7, among cubes which are 7 digits long the maximum sum of digits is A373727(7) = 46 and this is attained by 3 cubes, the largest of which is 208^3 = 8998912 so that a(7) = 208.

Cf. A373727, A379869.

Other powers: A379298, A380797, A380566, A380193.

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

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

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}]

A371728 a(n) is the largest number that is the digit sum of an n-digit square number.

Original entry on oeis.org

9, 13, 19, 31, 40, 46, 54, 63, 70, 81, 88, 97, 106, 112, 121, 130, 136, 148, 154, 162, 171, 180, 187, 193, 205, 211, 220, 229, 235, 244, 253, 262, 271, 277, 286, 297, 301, 310, 319, 331, 334, 343, 355, 360, 367, 378, 388, 396

Offset: 1

Zhining Yang, Apr 04 2024

a(n) appears to be approximately equal to (33*n-11)/4.

			a(6) = 46 because 46 is the largest digital sum encountered among all 6-digit squares (698896, 779689, 877969).

Shouen Wang, Chinese BBS: How many of these A's are there?

Cf. A004159, A049415, A069665, A069666.

Cf. A056991, A348300, A379298.

Array[Max@Map[Total@IntegerDigits[#^2] &, Range[Floor@Sqrt[10^(#)]], Floor@Sqrt[10^(# + 1) - 1]] &, 15]

a(22)-a(48) from Zhao Hui Du, Apr 05 2024
a(49)-a(62) from Zhining Yang, May 08 2024
a(63)-a(64) from Zhining Yang, May 23 2024
Incorrect a(61) and unverified a(49) onward deleted by Zhining Yang, Mar 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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Mathematica

A380052 a(n) is the largest number whose cube is an n-digit cube which has the maximum sum of digits (A373727(n)).

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C

Mathematica

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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C

Mathematica

A371728 a(n) is the largest number that is the digit sum of an n-digit square number.

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