cp's OEIS Frontend

A301374 Values of A023900 which occur only at indices which are powers of a prime.

%I A301374 #66 Oct 22 2021 23:43:37
%S A301374 -1,-2,-4,-6,-10,-16,-18,-22,-28,-30,-42,-46,-52,-58,-66,-70,-78,-82,
%T A301374 -100,-102,-106,-126,-130,-136,-138,-148,-150,-162,-166,-172,-178,
%U A301374 -190,-196,-198,-210,-222,-226,-228,-238,-250,-256,-262,-268,-270,-282,-292,-306
%N A301374 Values of A023900 which occur only at indices which are powers of a prime.
%C A301374 Terms are equal to A023900(p) = A023900(p^2) = A023900(p^3) = ... with p prime, but is never equal to A023900(m*p) with m not a power of p. [Corrected by _M. F. Hasler_, Sep 01 2021]
%C A301374 abs(a(n)) + 1 is prime (A301590).
%C A301374 For n > 1, if and only if n can't be factored into 2*m factors, m > 0, distinct factors f > 1 where f + 1 is prime then -n is a term. - _David A. Corneth_, Mar 25 2018
%C A301374 The values are of the form a(n) = 1 - p with prime p = A301590(n). These are exactly the values A023900(x) = 1 - p occurring only if x = p^j for some j >= 1. (See counterexample for p = 13 in EXAMPLE section.) - _M. F. Hasler_, Sep 01 2021
%e A301374 a(1) = -1 = A023900(2^m), m > 0.
%e A301374 a(2) = -2 = A023900(3^m), m > 0.
%e A301374 a(3) = -4 = A023900(5^m), m > 0.
%e A301374 a(4) = -6 = A023900(7^m), m > 0.
%e A301374 a(5) = -10 = A023900(11^m), m > 0.
%e A301374 a(6) = -16 = A023900(17^m), m > 0.
%e A301374 A023900(13) = -12 is not a term as A023900(42) = -12, and 42 is the product of three prime factors.
%e A301374 From _David A. Corneth_, Mar 25 2018: (Start)
%e A301374 10 can't be factored in an even number of distinct factors f > 1 such that f + 1 is prime, so -10 is in the sequence.
%e A301374 12 can be factored in an even number of distinct factors f > 1; 12 = 2 * 6 and both 2 + 1 and 6 + 1 are prime, hence -12 is not a term. (End)
%t A301374 Keys@ Select[Union /@ PrimeNu@ PositionIndex@ Array[DivisorSum[#, # MoebiusMu[#] &] &, 310], # == {1} &] (* _Michael De Vlieger_, Mar 26 2018 *)
%o A301374 (PARI) f(n) = sumdivmult(n, d, d*moebius(d));
%o A301374 isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0));); return (1);}
%o A301374 lista(nn) = {forprime(p=2, nn, vp = f(p); if (isok(p, vp), print1(vp, ", ")););} \\ _Michel Marcus_, Mar 23 2018
%Y A301374 Cf. A000040, A001055, A023900, A301590, A301591.
%K A301374 sign,easy
%O A301374 1,2
%A A301374 _Torlach Rush_, Mar 19 2018