A045835 -id:A045835 - cp's OEIS frontend

A373976 a(n) = A001222(n) - A001222(A001414(n)), where A001222 is bigomega, the number of prime factors with multiplicity, and A001414 is sopfr, sum of prime factors with multiplicity. a(1) = 0 by convention.

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, -1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 2, 0, 1, 0, 3, 0, 1, -1, 2, 0, 0, -2, 3, 0, 0, 0, 1, 2, 0, 0, 4, 0, 0, -1, 2, 0, 3, -2, 3, 0, 1, 0, 1, 0, 0, 2, 3, -1, -1, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, -1, 0, 0, 4, 1, 1, 0, 2, 0, -1, -3, 3, 0, 3, -1, 0, 0, 0, -2, 5, 0, -1, 2, 2, 0

Offset: 1

Antti Karttunen, Jun 24 2024

sign

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

Cf. A001222, A001414, A342956.

Cf. also A045835.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A373976(n) = if(1==n, 0, bigomega(n)-bigomega(A001414(n)));

a(n) = A001222(n) - A342956(n).

A271101 Squarefree semiprimes (A006881) whose average prime factor is prime.

Original entry on oeis.org

21, 33, 57, 69, 85, 93, 129, 133, 145, 177, 205, 213, 217, 237, 249, 253, 265, 309, 393, 417, 445, 469, 489, 493, 505, 517, 553, 565, 573, 597, 633, 669, 685, 697, 753, 781, 793, 813, 817, 865, 889, 913, 933, 949, 973, 985, 993, 1057, 1077, 1137, 1149, 1177, 1257, 1273, 1285, 1329

Offset: 1

Antonio Roldán, Mar 30 2016

nonn

Sum of factors of a(n) if semiprime (product 2*p, with p prime).
This sequence is subsequence of A006881, A089765, A187073, A108633 and A213015.
This sequence is also subsequence of A045835, because sopfr(omega(a(n))) = omega(sopfr(a(n))): sopfr(omega(a(n)))=sopfr(2)=2, and omega(sopfr(a(n)))=omega(2*p)=2  (p prime,  p>2, average prime factor).

			133 is in the sequence because 133 is a squarefree semiprime: 133=7*19, and (7+19)/2=13, a prime number.

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

Cf. A006881, A046315, A046388, A115585, A187073, A089765, A108633, A213015.

N:= 10000: # for terms <= N
Primes:= select(isprime, [seq(i, i=3..N/3)]):
SP:= [seq(seq([p, q], q = select(`<=`, Primes, min(p-1, N/p))), p=Primes)]:
B:= select(t -> isprime((t[1]+t[2])/2), SP):
sort(map(t -> t[1]*t[2], B)); # Robert Israel, Dec 14 2019

Select[Select[Range@ 1330, SquareFreeQ@ # && PrimeOmega@ # == 2 &], PrimeQ@ Mean[First /@ FactorInteger@ #] &] (* Michael De Vlieger, Mar 30 2016 *)

sopf(n)= { local(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{for (n=6, 2*10^3,  if(bigomega(n)==2&&omega(n)==2, m=sopf(n)/2;if(m==truncate(m),if(isprime(m), print1(n, ", ")))))}

cp's OEIS Frontend

A373976 a(n) = A001222(n) - A001222(A001414(n)), where A001222 is bigomega, the number of prime factors with multiplicity, and A001414 is sopfr, sum of prime factors with multiplicity. a(1) = 0 by convention.

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A271101 Squarefree semiprimes (A006881) whose average prime factor is prime.

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