A036350 -id:A036350 - cp's OEIS frontend

A036349 Numbers whose sum of prime factors (taken with multiplicity) is even.

1, 2, 4, 8, 9, 15, 16, 18, 21, 25, 30, 32, 33, 35, 36, 39, 42, 49, 50, 51, 55, 57, 60, 64, 65, 66, 69, 70, 72, 77, 78, 81, 84, 85, 87, 91, 93, 95, 98, 100, 102, 110, 111, 114, 115, 119, 120, 121, 123, 128, 129, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 154, 155

Offset: 1

Patrick De Geest, Dec 15 1998

nonn

A multiplicative semigroup; if m and n are in the sequence then so is m*n. - David James Sycamore, Jul 17 2018
From Peter Munn, Jul 19 2020: (Start)
Also closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897.
A number is listed if and only if it has an even number of odd prime factors, counting repetitions; equivalently, if and only if it is the product of a term of A046337 and a power of 2 (term of A000079).
(End)

			141 = 3 * 47 is a term since the sum 3 + 47 = 50 is even.

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

Cf. A001414 (sopfr), A059897.
Complement of A335657.
Sequences with similar definitions: A036350, A046363, A289142.
Subsequences: A000079, A028982, A046337, A056913.

filter:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2])::even end proc:
select(filter, [$1..200]); # Robert Israel, Jul 15 2020

Select[Range[160],EvenQ[Total[Times@@@FactorInteger[#]]]&] (* Harvey P. Dale, Sep 21 2011 *)

isok(n) = my(f=factor(n)); (sum(k=1, #f~, f[k,1]*f[k,2]) % 2) == 0; \\ Michel Marcus, Jul 19 2018

Sum_{n>=1} 1/a(n)^s = (zeta(s) + ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s))/2 for Re(s)>1. - Amiram Eldar, Nov 02 2020

First term (2) from Harvey P. Dale, Sep 21 2011

First term (1) from David James Sycamore, Jul 17 2018

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

Original entry on oeis.org

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

David James Sycamore, Jun 26 2017

nonn

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)

			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.

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

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.

select(n -> add(t[1]*t[2],t=ifactors(n)[2]) mod 3 = 0, [$1..1000]); # Robert Israel, Jul 03 2017

Join[{1},Select[Range[250],Mod[Total[Times@@@FactorInteger[#]],3]==0&]] (* Harvey P. Dale, Mar 16 2020 *)

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

isA289142 = A373371; \\ Antti Karttunen, Jun 08 2024

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

Corrected by Robert Israel, Jul 03 2017

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A036349 Numbers whose sum of prime factors (taken with multiplicity) is even.

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

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions