A194269 Numbers j such that Sum_{i=1..k} d(i)^i = j+1 for some k where d(i) is the sorted list of divisors of j.

4, 9, 25, 49, 68, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 17500, 18769, 19321, 22201, 22801, 24649, 26569, 27889

Offset: 1

Author

Michel Lagneau, Aug 27 2011

Keywords

• nonn

Comments

Equivalently, numbers j such that Sum_{i=2..k} A027750(j,i)^i = j for some k.

The majority of these numbers are squares.

The sequence of numbers j such that Sum_{i=1..k} A027750(j,i)^i = j for some k is given by A378313.

All prime squares p^2 (A001248) are terms because the partial sum 1^1 + p^2 satisfy the condition. The terms that are not squares are given by A307137. - Michel Marcus, Mar 25 2019

Examples

			The divisors of 68 are 1, 2, 4, 17, 34, 68; 1^1 + 2^2 + 4^3 = 69, so 68 is a term.
		

Crossrefs

Cf. A001248, A027750, A064510, A180851, A190940, A307137, A378313, A378314.

Programs

Maple

isA194269 := proc(n) local dgs ,i,k; dgs := sort(convert(numtheory[divisors](n),list)) ; for k from 1 to nops(dgs) do if add(op(i,dgs)^i,i=1..k) = n+1 then return true; end if; end do; false ; end proc:
for n from 1 to 30000 do if isA194269(n) then print(n); end if; end do: # R. J. Mathar, Aug 27 2011

PARI

isok(n) = {my(d=divisors(n), s=0); for(k=1, #d, s += d[k]^k; if(s == n+1, return(1)); if(s > n+1, break););0;} \\ Michel Marcus, Mar 25 2019

Extensions

Edited by Max Alekseyev, Nov 22 2024