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

-1, -2, -4, -6, -10, -16, -18, -22, -28, -30, -42, -46, -52, -58, -66, -70, -78, -82, -100, -102, -106, -126, -130, -136, -138, -148, -150, -162, -166, -172, -178, -190, -196, -198, -210, -222, -226, -228, -238, -250, -256, -262, -268, -270, -282, -292, -306

Offset

1,2

Comments

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]

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

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

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

Links

Table of n, a(n) for n=1..47.

Example

a(1) = -1 = A023900(2^m), m > 0.
a(2) = -2 = A023900(3^m), m > 0.
a(3) = -4 = A023900(5^m), m > 0.
a(4) = -6 = A023900(7^m), m > 0.
a(5) = -10 = A023900(11^m), m > 0.
a(6) = -16 = A023900(17^m), m > 0.
A023900(13) = -12 is not a term as A023900(42) = -12, and 42 is the product of three prime factors.
From David A. Corneth, Mar 25 2018: (Start)
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.
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)

Mathematica

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

Prog

(PARI) f(n) = sumdivmult(n, d, d*moebius(d));
isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0)); ); return (1); }
lista(nn) = {forprime(p=2, nn, vp = f(p); if (isok(p, vp), print1(vp, ", ")); ); } \\ Michel Marcus, Mar 23 2018

Crossrefs

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

Sequence in context: A083814 A073805 A352587 * A130320 A339574 A258599

Adjacent sequences: A301371 A301372 A301373 * A301375 A301376 A301377

Keyword

sign,easy

Author

Torlach Rush, Mar 19 2018

Status

approved