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A301374 Values of A023900(k) the only solutions of which have a single distinct prime factor. 3

%I

%S -1,-2,-4,-6,-10,-16,-18,-22,-28,-30,-42,-46,-52,-58,-66,-70,-78,-82,

%T -100,-102,-106,-126,-130,-136,-138,-148,-150,-162,-166,-172,-178,

%U -190,-196,-198,-210,-222,-226,-228,-238,-250,-256,-262,-268,-270,-282,-292,-306

%N Values of A023900(k) the only solutions of which have a single distinct prime factor.

%C 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 <> p.

%C abs(a(n)) + 1 is prime (A301590).

%C 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

%e a(1) = -1 = A023900(2^m), m > 0.

%e a(2) = -2 = A023900(3^m), m > 0.

%e a(3) = -4 = A023900(5^m), m > 0.

%e a(4) = -6 = A023900(7^m), m > 0.

%e a(5) = -10 = A023900(11^m), m > 0.

%e a(6) = -16 = A023900(17^m), m > 0.

%e A023900(13) = -12 is not a term as A023900(42) = -12, and 42 is the product of three prime factors.

%e From _David A. Corneth_, Mar 25 2018: (Start)

%e 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 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 Keys@ Select[Union /@ PrimeNu@ PositionIndex@ Array[DivisorSum[#, # MoebiusMu[#] &] &, 310], # == {1} &] (* _Michael De Vlieger_, Mar 26 2018 *)

%o (PARI) f(n) = sumdivmult(n, d, d*moebius(d));

%o isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0));); return (1);}

%o lista(nn) = {forprime(p=2, nn, vp = f(p); if (isok(p, vp), print1(vp, ", ")););} \\ _Michel Marcus_, Mar 23 2018

%Y Cf. A000040, A001055, A023900, A301590, A301591.

%K sign,easy

%O 1,2

%A _Torlach Rush_, Mar 19 2018

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Last modified July 5 19:02 EDT 2020. Contains 335473 sequences. (Running on oeis4.)