cp's OEIS Frontend

A323918 Numbers k with exactly two distinct prime divisors and such that cototient(k) is a square, where: k = p^(2s) * q^(2t+1) with s >= 1, t >= 0, p <> q primes and such that p * (p+q-1) = M^2.

%I A323918 #30 Mar 19 2019 01:15:54
%S A323918 28,68,112,124,272,284,388,448,496,508,657,796,964,1025,1088,1136,
%T A323918 1348,1372,1552,1792,1796,1984,2032,2169,2308,2588,3184,3524,3856,
%U A323918 3868,4352,4544,4604,4996,5392,5488,5913,6025,6057,6208,6268,7168,7184,7936,8128,9232,9244
%N A323918 Numbers k with exactly two distinct prime divisors and such that cototient(k) is a square, where: k = p^(2s) * q^(2t+1) with s >= 1, t >= 0, p <> q primes and such that p * (p+q-1) = M^2.
%C A323918 This is the second subsequence of A323916, the first one is A323917.
%C A323918 Some values of (k,p,q,M): (28,2,7,2), (68,2,17,3), (124,2,31,4), (284,2,71,6), (388,97,7), (657,3,73,5).
%C A323918 The primitive terms of this sequence are the products p^2 * q, with p,q which satisfy p*(p+q-1) = M^2; the first ones are 28, 68, 124, 284, 388, 508, 657, 796. Then, the integers (p^2 * q) * p^2 and (p^2 * q) * q^2 are new terms of the general sequence.
%C A323918 Except 6, all the even perfect numbers of A000396 belong to this sequence.
%C A323918 See the file "Subfamilies of terms" in A063752 for more details, proofs with data, comments, formulas and examples.
%F A323918 cototient(p^2 * q) = p * (p + q - 1) = M^2;
%F A323918 cototient(k) = (p^(s-1) * q^t * M)^2 with k as in the name of this sequence.
%e A323918 272 = 2^4 * 17, M = 2*(2+17-1) = 6^2 and cototient(272) = (2^1 * 17^0 * 6)^2 = 12^2.
%e A323918 1025 = 5^2 * 41 and cototient(1025) = 5 * (5+41-1) = 15^2.
%e A323918 Perfect number: 8128 = 2^6 * 127 and cototient(8128) = 64^2.
%o A323918 (PARI) isok(n) = (omega(n)==2) && issquare(n - eulerphi(n)) && ((factor(n)[1,2] % 2) != (factor(n)[2,2] % 2)); \\ _Michel Marcus_, Feb 10 2019
%Y A323918 Cf. A000396, A051953, A063752, A246551, A323916, A323917, A306670.
%K A323918 nonn
%O A323918 1,1
%A A323918 _Bernard Schott_, Feb 09 2019