

A301374


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


3



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
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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 <> p.
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


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^21, 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: A327408 A083814 A073805 * A130320 A258599 A101176
Adjacent sequences: A301371 A301372 A301373 * A301375 A301376 A301377


KEYWORD

sign,easy


AUTHOR

Torlach Rush, Mar 19 2018


STATUS

approved



