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

1,1

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

Links

Table of n, a(n) for n=1..41.

Example

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

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

Prog

(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

Crossrefs

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

Sequence in context: A230312 A332646 A306043 * A336230 A130283 A065739

Adjacent sequences: A194266 A194267 A194268 * A194270 A194271 A194272

Keyword

nonn

Author

Michel Lagneau, Aug 27 2011

Extensions

Edited by Max Alekseyev, Nov 22 2024

Status

approved