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

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

1,1

Comments

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).

Links

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

Example

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

Maple

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

Mathematica

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

Prog

(PARI)
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, ", ")))))}

Crossrefs

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

Sequence in context: A084109 A016105 A187073 * A191683 A032603 A233562

Adjacent sequences: A271098 A271099 A271100 * A271102 A271103 A271104

Keyword

nonn

Author

Antonio Roldán, Mar 30 2016

Status

approved