A194269 - OEIS

%I #36 Nov 26 2024 08:58:19

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

%C The majority of these numbers are squares.

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

%C 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

%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););0;} \\ _Michel Marcus_, Mar 25 2019

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

%K nonn,changed

%O 1,1

%A _Michel Lagneau_, Aug 27 2011

%E Edited by _Max Alekseyev_, Nov 22 2024