A082300 - OEIS

A082300

Numbers relatively prime to the sum of their prime factors (with repetition).

1, 6, 10, 12, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133

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OFFSET

1,2

COMMENTS

In other words, numbers n such that n and sopfr(n) are relatively prime, where sopfr(n) (A001414) is the sum of the primes (with repetition) dividing n.

Conjecture: a(n) ~ (Pi^2/6)n. - Charles R Greathouse IV, Aug 04 2016

No term is prime since for prime p, p and 2p are not coprime; similarly no term is a prime power. A050703 is a subsequence because then n+sopfr(n) is prime, and so coprime to n. - David James Sycamore, Mar 04 2018

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

EXAMPLE

gcd(2*2*5,2+2+5) = gcd(2*2*5,3*3)=1, therefore 20 is a term;

gcd(3*11,3+11) = gcd(3*11,2*7)=1, therefore 33 is a term.

MATHEMATICA

Select[Range@ 106, CoprimeQ[#, Total@ Flatten@ Map[Table[#1, {#2}] & @@ # &, FactorInteger[#]]] &] (* Michael De Vlieger, Aug 06 2016 *)

PROG

(PARI) sopfr(n)=my(f=factor(n)); sum(i=1, #f~, f[i, 1]*f[i, 2])

is(n)=gcd(sopfr(n), n)==1 \\ Charles R Greathouse IV, Aug 04 2016

CROSSREFS

Cf. A001414, A275665, A050703.