A114135 Primitive numbers n such that the sums of the digits of n, n^2 and n^3 coincide (cf. A111434).

1, 468, 585, 5851, 5868, 28845, 58968, 21688965, 29588877, 37848897, 49879981, 58577797, 79898994, 79958368, 79979698, 89757468, 109699677, 159699969, 468957888, 479597652, 479896587, 480749985, 494899398, 497349981, 498678256

Offset: 1

Author

Giovanni Resta and Robert G. Wilson v, Nov 21 2005

Keywords

• base
• nonn

Comments

Members of A111434 not congruent to 0 (mod 10). If k is a member of A111434 then so is 10^e*k.

The authors have calculated all members below 10^11.

The number of members less than 10^n {n=0..11}: 0,1,1,3,5,7,7,7,16,34,57,125.

Number of members congruent to k (mod 10): 0,7,1,0,2,23,8,20,49,15. But more interesting, number of members are congruent to k (mod 9): 66,59,0,0,0,0,0,0,0.

A007953(n) == n mod 9. Since 0 and 1 are the only k in [0,1,...8] with k == k^2 mod 9, all terms are congruent to 0 or 1 mod 9. - Robert Israel, Jan 26 2015

Links

• Toshitaka Suzuki and Nikhil Mahajan, Table of n, a(n) for n = 1..600 (first 325 terms from Toshitaka Suzuki)

Crossrefs

Cf. A007953, A111434, A005188.

Programs

Mathematica

sod[n_] := Plus @@ IntegerDigits@n; lst = {}; Do[ If[(Mod[n, 9] == 0 || Mod[n, 9] == 1) && Mod[n, 10] != 0 && sod@n == sod[n2] == sod[n3], AppendTo[lst, n]], {n, 108/2}]; lst
Select[Range[5*10^8],Length[Union[Total/@IntegerDigits/@{#,#^2,#^3}]]==1 && Mod[#,10]!=0&] (* Harvey P. Dale, Jul 07 2020 *)

PARI

isok(n) = (n % 10) && ((sd=sumdigits(n)) == sumdigits(n^2)) && (sd == sumdigits(n^3)); \\ Michel Marcus, Jan 20 2015