A036347 Numbers k for which the parity of k and the parity of sopfr(k) differ, where sopfr is the sum of prime factors with repetition.

1, 6, 9, 10, 12, 14, 15, 20, 21, 22, 24, 25, 26, 28, 33, 34, 35, 38, 39, 40, 44, 46, 48, 49, 51, 52, 54, 55, 56, 57, 58, 62, 65, 68, 69, 74, 76, 77, 80, 81, 82, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 104, 106, 108, 111, 112, 115, 116, 118, 119, 121, 122, 123, 124, 126, 129, 133, 134, 135, 136

Offset: 1

Author

Patrick De Geest, Dec 15 1998

Keywords

• nonn
• base

Comments

Parity of n and its sum of prime factors differs (counted with multiplicity). - The original name.

Examples

			111 = 3 * 37 -> sum = 40 so 111 is odd while 40 is even.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A030141, A359768 (characteristic function).

Union of A036348 (even terms) and A046337 (odd terms).

Positions of odd terms in A075254 and in A075255.

Cf. also A359771, A359821.

Programs

PARI

isA036347(n) = A359768(n); \\ Antti Karttunen, Jan 15 2023

Python

from itertools import count, islice
from functools import reduce
from operator import ixor
from sympy import factorint
def A036347_gen(startvalue=1): # generator of terms
    return filter(lambda n:(reduce(ixor,(p*e for p, e in factorint(n).items()),0)^n)&1, count(max(startvalue,1)))
A036347_list = list(islice(A036347_gen(),20)) # Chai Wah Wu, Jan 15 2023

Formula

{k | k+A001414(k) == 1 mod 2}. - Antti Karttunen, Jan 16 2023

Extensions

Missing initial term a(1) = 1 prepended, offset corrected, name edited and more terms added by Antti Karttunen, Jan 15 2023