cp's OEIS Frontend

A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.

%I A355571 #17 Jul 15 2022 20:40:18
%S A355571 4,9,12,16,18,20,24,25,28,30,32,36,40,42,44,45,48,49,50,52,54,56,60,
%T A355571 63,66,68,70,72,75,76,78,80,81,84,88,90,92,96,98,99,100,102,104,105,
%U A355571 108,110,112,114,116,117,120,121,124,126,128,130,132,135,136,138,140,147,148,150,152
%N A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.
%C A355571 There are no primes in the sequence, since A007956(p^2) = p for all primes p.
%C A355571 There are infinitely many terms, in fact p^2 is a term for all primes p.
%C A355571 If 8k+1 is not a perfect square, then p^k is a term for all primes p.
%C A355571 Depends only on the prime signature: n is in this sequence if and only if A046523(n) is in this sequence. - _Charles R Greathouse IV_, Jul 08 2022
%C A355571 Contains all the weak numbers (A052485) aside from the primes (A000040) and squarefree semiprimes (A006881). - _Charles R Greathouse IV_, Jul 08 2022
%D A355571 Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.
%F A355571 a(n) = n + O(n log log n/log n). - _Charles R Greathouse IV_, Jul 08 2022
%e A355571 4 is a term of this sequence because there are no numbers k such that A007956(k) = 4.
%e A355571 2^10 is not a term of this sequence because A007956(32) = 1024 (Note that 8*10+1=81=9^2 is a perfect square).
%e A355571 p^4 belongs to this sequence for all primes p, in fact 8*4+1=33 is not a perfect square, so there are no numbers h such that A007956(h) = p^4.
%t A355571 Complement[Complement[Table[n, {n, 2, 1000}], Select[NumericalSort[Table[Times @@ Most[Divisors[n]], {n, 1000000}]], # != 1 && # < 1000 &]], Select[Table[Prime[n], {n, 1, 1000}], # < 1000 &]]
%Y A355571 Subsequences by prime signature: A001248 (p^2), A054753 (p^2*q), A030514 (p^4), A065036 (p^3*q), A007304 (p*q*r), A050997 (p^5), A085986 (p^2*q^2).
%Y A355571 Cf. A007956, A052485, A106543.
%K A355571 nonn
%O A355571 1,1
%A A355571 _Luca Onnis_, Jul 07 2022