OFFSET
1,2
COMMENTS
U{S(n); 3|n}, where S(n)= {x; sopfr(x)=n}; numbers placed in ascending order.
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Robert Israel, Jul 03 2017
From Antti Karttunen, Jun 11 2024, with minor edits Jun 30 2024: (Start)
Numbers such that the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m-1 (A003627) are equal modulo 3.
For n that is not a multiple of 3, sopfr(n) [= A001414(n)] is a multiple of 3 if and only if the arithmetic derivative of n [= A003415(n)] is a multiple of 3. See A373475 for a proof.
This sequence (as a multiplicative semigroup) is generated by the union of A369659 with {3}.
(End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
For n >= 2, a(n) = A102217(n-1)/3. - Antti Karttunen, Jun 08 2024
EXAMPLE
sopfr(42) = 2 + 3 + 7 = 12 = 4*3, sopfr(95) = 5 + 19 = 24 = 8 * 3, sopfr(180) = 2 + 2 + 3 + 3 + 5 = 15 = 5 * 3.
MAPLE
select(n -> add(t[1]*t[2], t=ifactors(n)[2]) mod 3 = 0, [$1..1000]); # Robert Israel, Jul 03 2017
MATHEMATICA
Join[{1}, Select[Range[250], Mod[Total[Times@@@FactorInteger[#]], 3]==0&]] (* Harvey P. Dale, Mar 16 2020 *)
PROG
(PARI) s(n)=my(f=factor(n), p=f[, 1], e=f[, 2]); sum(k=1, #p, e[k]*p[k]);
for(n=1, 200, if(s(n)%3==0, print1(n, ", "))); \\ Joerg Arndt, Jun 26 2017
(PARI) isA289142 = A373371; \\ Antti Karttunen, Jun 08 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Jun 26 2017
EXTENSIONS
Corrected by Robert Israel, Jul 03 2017
STATUS
approved