A258722 a(n) is the smallest k (powers of 10 excluded) such that sod(k), sod(k^2),..., sod(k^n) is an arithmetic progression, where sod(m) = A007953(m) is the sum of the digits of m.

4, 4, 16, 16, 16, 7972, 7972, 134242, 59716233, 1844284813, 77298251764

Offset: 3

Author

Giovanni Resta, Jun 08 2015

Keywords

• nonn
• base
• more

Comments

Offset is 3 because an AP of one or two elements makes little sense. Powers of 10 are excluded because they form trivial infinite progressions.

a(14), if it exists, is greater than 4*10^12.

Examples

			a(5) = 16 because sod(16), sod(16^2),..., sod(16^5) are equal to 7, 13, 19, 25, 31, which is an AP with common difference 6 and 16 is the smallest number with this property.
		

Crossrefs

Cf. A007953.

Programs

Mathematica

sod[n_] := Plus @@ IntegerDigits@n; a[n_] := If[n >= 3, Block[{k = 2},
   While[ Mod[k, 10] == 0 || 1 < Length@ Union@ Differences[ sod /@ (k^ Range[n])], k++]; k]]; a /@ Range[3, 10]