A289142 Numbers whose sum of prime factors (taken with multiplicity) is divisible by 3.

1, 3, 8, 9, 14, 20, 24, 26, 27, 35, 38, 42, 44, 50, 60, 62, 64, 65, 68, 72, 74, 77, 78, 81, 86, 92, 95, 105, 110, 112, 114, 116, 119, 122, 125, 126, 132, 134, 143, 146, 150, 155, 158, 160, 161, 164, 170, 180, 185, 186, 188, 192, 194, 195, 196, 203, 204

Offset: 1

Author

David James Sycamore, Jun 26 2017

Keywords

• nonn

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)

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002476, A003627, A036349, A036350, A046363, A373371 (characteristic function).

Positions of multiples of 3 in A001414 (sopfr) and in A118503.

Subsequences that are formed by intersecting this sequence with other multiplicative semigroups: A102217, A369659, A373373, A373473, A373475, A373478, A373597.

Cf. also A373385, A373602, A374052.

Programs

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 *)

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

Formula

For n >= 2, a(n) = A102217(n-1)/3. - Antti Karttunen, Jun 08 2024

Extensions

Corrected by Robert Israel, Jul 03 2017