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A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

%I A325981 #26 Jun 18 2019 18:12:23
%S A325981 45,495,585,765,855,1305,18837,21525,31635,38295,45315,50445,51255,
%T A325981 60435,63495,68085,77265,96615,1403115,2446353,3411975,3999465,
%U A325981 4091745,4233537,4287255,4631319,10813425,10967085,11490345,15578199,16143309,16329645,16633071,17179515,17311203,17355915,21159075,21933975,22579725
%N A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).
%C A325981 Provided that A325977 and A325978 are never zero on same n, these are odd composite numbers n such that A325977(n) is not zero and divides A325978(n).
%C A325981 Based on the first 147 terms it seems that this sequence is a subsequence of A072357, that is each term has exactly one prime factor with exponent 2, with one or more other prime factors that are all unitary (i.e., each term satisfies A001222(x) - A001221(x) = 1). On the other hand, it has been proved that no odd perfect number, if such numbers exist at all, can have such a factorization (see A326137 and a link to P. P. Nielsen's paper there).
%C A325981 Nineteen initial terms factorize as:
%C A325981   45 = 3^2 * 5^1,
%C A325981   495 = 3^2 * 5^1 * 11^1,
%C A325981   585 = 3^2 * 5^1 * 13^1,
%C A325981   765 = 3^2 * 5^1 * 17^1,
%C A325981   855 = 3^2 * 5^1 * 19^1,
%C A325981   1305 = 3^2 * 5^1 * 29^1,
%C A325981   18837 = 3^2 * 7^1 * 13^1 * 23^1,
%C A325981   21525 = 3^1 * 5^2 * 7^1 * 41^1,
%C A325981   31635 = 3^2 * 5^1 * 19^1 * 37^1,
%C A325981   38295 = 3^2 * 5^1 * 23^1 * 37^1,
%C A325981   45315 = 3^2 * 5^1 * 19^1 * 53^1,
%C A325981   50445 = 3^2 * 5^1 * 19^1 * 59^1,
%C A325981   51255 = 3^2 * 5^1 * 17^1 * 67^1,
%C A325981   60435 = 3^2 * 5^1 * 17^1 * 79^1,
%C A325981   63495 = 3^2 * 5^1 * 17^1 * 83^1,
%C A325981   68085 = 3^2 * 5^1 * 17^1 * 89^1,
%C A325981   77265 = 3^2 * 5^1 * 17^1 * 101^1,
%C A325981   96615 = 3^2 * 5^1 * 19^1 * 113^1,
%C A325981   1403115 = 3^1 * 5^1 * 7^2 * 23^1 * 83^1,
%C A325981 and the 62nd term as a(62) = 2919199437 = 3^2 * 7^1 * 11^1 * 43^1 * 163^1 * 601^1.
%C A325981 If we select a subsequence of terms for which the quotient A325978(n)/A325977(n) is positive, then we are left with the following numbers: 495, 585, 31635, 38295, 45315, 51255, 60435, 63495, 1403115, 3999465, etc. which is a subsequence of A326070.
%H A325981 Antti Karttunen (terms 1-62) & Giovanni Resta, <a href="/A325981/b325981.txt">Table of n, a(n) for n = 1..147</a>
%H A325981 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%o A325981 (PARI)
%o A325981 A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
%o A325981 A034460(n) = (A034448(n) - n);
%o A325981 A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
%o A325981 A325313(n) = (A048250(n) - n);
%o A325981 A325977(n) = ((A034460(n)+A325313(n))/2);
%o A325981 A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
%o A325981 A325314(n) = (n - A162296(n));
%o A325981 A048146(n) = (sigma(n)-A034448(n));
%o A325981 A325814(n) = (n-A048146(n));
%o A325981 A325978(n) = ((A325314(n)+A325814(n))/2);
%o A325981 A325975(n) = gcd(A325977(n), A325978(n));
%o A325981 isA325981(n) = ((n%2)&&!isprime(n)&&(A325975(n)==abs(A325977(n))));
%o A325981 \\ Or alternatively as:
%o A325981 isA325981(n) = if(!(n%2)||isprime(n),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(x&&!(y%x)));
%Y A325981 Cf. A072357, A228058, A325973, A325974, A325975, A325977, A325978, A325979, A326064, A326070, A326071, A326137.
%K A325981 nonn
%O A325981 1,1
%A A325981 _Antti Karttunen_, Jun 02 2019