cp's OEIS Frontend

A380953 Numbers m such that the sum of its distinct prime factors and the sum of its nonprime divisors are both squares.

%I A380953 #22 Apr 26 2025 05:31:17
%S A380953 1,323,3887,5183,149903,311790,777923,1327103,6718463,12446783,
%T A380953 14605487,16402499,20373435,28128270,30856494,33144430,37058230,
%U A380953 37380745,68661901,86755609,139557721,159954570,221294682,222538813,229159043,269108440,360590058,412621345
%N A380953 Numbers m such that the sum of its distinct prime factors and the sum of its nonprime divisors are both squares.
%C A380953 Or numbers m such that A008472(m) and (A000203(m) - A008472(m)) are both squares.
%e A380953 s1 is the sum of the prime factors, s2 is the sum of the nonprime divisors.
%e A380953 +----------------------------+-------------------------+-----+-------+
%e A380953 |    m   |  prime factors    |   nonprimedivisors      |  s1 |  s2   |
%e A380953 +----------------------------+-------------------------+-----+-------+
%e A380953 |    323 | {17, 19}          | {1, 323}                | 6^2 |  18^2 |
%e A380953 +----------------------------+-------------------------+-----+-------+
%e A380953 |   3887 | {13, 23}          | {1, 169, 299, 3887}     | 6^2 |  66^2 |
%e A380953 +----------------------------+-------------------------+-----+-------+
%e A380953 |   5183 | {71, 73}          | {1, 5183}               |12^2 |  72^2 |
%e A380953 +----------------------------+-------------------------+-----+-------+
%e A380953 | 149903 | {13, 887}         | {1, 169, 11531, 149903} |30^2 | 402^2 |
%e A380953 +----------------------------+-------------------------+-----+-------+
%p A380953 with(numtheory):nn:=10^8:print(1):
%p A380953 for m from 2 to nn do:
%p A380953  d:=factorset(m):n0:=nops(d):s:=sum('d[i]', 'i'=1..n0):
%p A380953     if issqr(s) and issqr(sigma(m)-s) then print(m):
%p A380953      else
%p A380953     fi:
%p A380953  od:
%o A380953 (PARI) isok(m) = my(f=factor(m), s=sum(k=1, #f~, f[k,1])); issquare(s) && issquare(sigma(f)-s); \\ _Michel Marcus_, Feb 09 2025
%Y A380953 Cf. A000203, A008472, A164722.
%K A380953 nonn
%O A380953 1,2
%A A380953 _Michel Lagneau_, Feb 09 2025
%E A380953 More terms from _Jinyuan Wang_, Feb 11 2025