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%I #29 Mar 25 2019 06:48:13
%S 4,9,25,49,68,121,169,289,361,529,841,961,1369,1681,1849,2209,2809,
%T 3481,3721,4489,5041,5329,6241,6889,7921,9409,10201,10609,11449,11881,
%U 12769,16129,17161,17500,18769,19321,22201,22801,24649,26569,27889
%N 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.
%C The majority of these numbers are squares.
%C The sequence of numbers j such that Sum_{i=1..k} d(i)^i = j generates the numbers 1, 130, 135, 288, 5083, 8064, 10130, ... (no more terms through 10^8).
%C From _Michel Marcus_, Mar 25 2019: (Start)
%C All prime squares p^2 (A001248) are terms because the partial sum 1^1 + p^2 satisfy the condition.
%C Up to 10^8, the terms that are not squares are: 68, 17500, 5053176.
%C (End)
%e The divisors of 68 are 1, 2, 4, 17, 34, 68; 1^1 + 2^2 + 4^3 = 69, so 68 is a term.
%p 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:
%p for n from 1 to 30000 do if isA194269(n) then print(n); end if; end do: # _R. J. Mathar_, Aug 27 2011
%o (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););} \\ _Michel Marcus_, Mar 25 2019
%Y Cf. A001248, A064510, A180851.
%K nonn
%O 1,1
%A _Michel Lagneau_, Aug 27 2011
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