A036348 -id:A036348 - cp's OEIS frontend

A359773 Dirichlet inverse of A356163, where A356163 is the characteristic function of the numbers with an even sum of prime factors (counted with multiplicity).

1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Jan 13 2023

sign

a(225) = 2 is the first term with absolute value larger than 1.
As A356163 is not multiplicative, neither is this sequence.
For all numbers n with an odd number of odd prime factors (with mult.), a(n) = 0. Proof: Numbers with an odd number of odd prime factors is sequence A335657 (equal to numbers whose odd part is in A067019). In the convolution formula, when n is any term of A335657, either the divisor (n/d) or d (but not both) is also a term of A335657. As A356163 is zero for all A335657, it is easy to show by induction that also a(n) is zero for all such numbers.
Therefore, nonzero values (including any odd values, see A359775) occur only on a subset of A036349, and A359774(n) <= A356163(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A036347, A036348, A036349, A067019, A335657, A356163, A359774 (parity of terms), A359775 (positions of odd terms), A359776 (of even terms), A359777.

Cf. also A359155, A359763 [= a(A003961(n))], A359780.

A356163(n) = (1-(((n=factor(n))[, 1]~*n[, 2])%2)); \\ After code in A001414.
memoA359773 = Map();
A359773(n) = if(1==n,1,my(v); if(mapisdefined(memoA359773,n,&v), v, v = -sumdiv(n,d,if(dA356163(n/d)*A359773(d),0)); mapput(memoA359773,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA356163(n/d) * a(d).

A100368 Numbers of the form 2^k * p where k > 0 and p is an odd prime.

Original entry on oeis.org

6, 10, 12, 14, 20, 22, 24, 26, 28, 34, 38, 40, 44, 46, 48, 52, 56, 58, 62, 68, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 122, 124, 134, 136, 142, 146, 148, 152, 158, 160, 164, 166, 172, 176, 178, 184, 188, 192, 194, 202, 206, 208, 212, 214, 218, 224

Offset: 1

Labos Elemer, Nov 22 2004

Even numbers with 2 distinct prime factors where the odd factor is prime.
A proper subset of A098202. E.g., 210 is not here, but it is there. Also differs from A100367: 36, 100, 108, 196, etc. are missing here. Different also from A036348 because 90 and 180 are not here.
A128691 is a subsequence; A078834(a(n)) = A006530(a(n)). - Reinhard Zumkeller, Sep 19 2011
Composite numbers k having the property that the number of divisors of 2k equals the number of divisors of k + 2. All primes satisfy this property. - Gary Detlefs, Jan 23 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001221, A038550, A098902, A100367, A036348, A100484.

a:=Filtered([1..224],n->Tau(2*n)=Tau(n)+2 and not IsPrime(n));; Print(a); # Muniru A Asiru, Jan 22 2019

import Data.Set (singleton, deleteFindMin, insert)
a100368 n = a100368_list !! (n-1)
a100368_list = f (singleton 6) (tail a065091_list) where
f s ps'@(p:ps) | mod m 4 > 0 = m : f (insert (2*p) $ insert (2*m) s') ps
| otherwise = m : f (insert (2*m) s') ps'
where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 19 2011

N:= 1000: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..N/2,2)]):
S:= {seq(seq(2^i*p,i=1..ilog2(N/p)),p=P)}:
sort(convert(S,list)); # Robert Israel, Jul 09 2017
with(numtheory): for n from 1 to 224 do if tau(2*n)=tau(n)+2 and not isprime(n) then print(n) fi od # Gary Detlefs, Jan 22 2019

Mathematica
```
<Harvey P. Dale, Sep 03 2016 *)
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is(n)=n%2==0 && isprime(n>>valuation(n,2)) \\ Charles R Greathouse IV, Jul 09 2017

list(lim)=my(v=List()); for(k=1,logint(lim\3,2), forprime(p=3,lim>>k, listput(v,p<Charles R Greathouse IV, Jul 09 2017

Numbers of the form 2^k*p where k > 0, p is an odd prime.

a(n) = 2*A038550(n). - Amiram Eldar, Dec 21 2020

Name edited by Charles R Greathouse IV, Jul 09 2017

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.

Original entry on oeis.org

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

Patrick De Geest, Dec 15 1998

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

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

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

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.

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

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

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

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

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A359773 Dirichlet inverse of A356163, where A356163 is the characteristic function of the numbers with an even sum of prime factors (counted with multiplicity).

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A100368 Numbers of the form 2^k * p where k > 0 and p is an odd prime.

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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.

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