A213015 -id:A213015 - cp's OEIS frontend

A213016 Numbers n such that the sum of prime factors of n (counted with multiplicity) is 3 times a prime.

8, 9, 14, 20, 24, 26, 27, 38, 44, 62, 68, 74, 105, 112, 116, 125, 126, 134, 150, 160, 180, 188, 192, 195, 208, 212, 216, 218, 231, 234, 243, 254, 275, 278, 314, 330, 332, 343, 352, 356, 362, 396, 398, 422, 428, 465, 483, 490, 496, 548, 558, 575, 588, 609, 614

Offset: 1

Michel Lagneau, Jun 02 2012

nonn

The numbers A100118(n)^3 are in the sequence.

			44 is in the sequence because 44 = 2^2 * 11 => sum of prime factors = 2*2+11 = 15 = 3*5 where 5 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001414, A100118, A213015.

with(numtheory):A:= proc(n) local e, j; e := ifactors(n)[2]: add (e[j][1]*e[j][2], j=1..nops(e)) end: for m from 1 to 3000 do: if type(A(m)/3,prime)= true then printf(`%d, `,m):else fi:od:

L = {}; Do[ww = Transpose[FactorInteger[k]]; w = ww[[1]].ww[[2]]; If[PrimeQ[w/3], AppendTo[L, k]], {k, 2, 1000}]; L
Select[Range[700],PrimeQ[Total[Times@@@FactorInteger[#]]/3]&] (* Harvey P. Dale, Nov 23 2022 *)

sopfr(n) = 3*p, p prime.

A213020 Smallest number k such that the sum of prime factors of k (counted with multiplicity) is n times a prime.

Original entry on oeis.org

2, 4, 8, 15, 21, 35, 33, 39, 65, 51, 57, 95, 69, 115, 86, 87, 93, 155, 212, 111, 122, 123, 129, 215, 141, 235, 158, 159, 265, 371, 177, 183, 194, 427, 201, 335, 213, 219, 365, 511, 237, 395, 249, 415, 446, 267, 278, 623, 964, 291, 302, 303, 309, 515, 321, 327

Offset: 1

Michel Lagneau, Jun 02 2012

nonn

Smallest k such that sopfr(k) = n*p, p prime.

			a(19) = 212 because 212 = 2^2 * 53 => sum of prime factors = 2*2+53 = 57 = 19*3 where 3 is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001414, A100118, A213015, A213016.

sopfr:= proc(n) option remember;
          add(i[1]*i[2], i=ifactors(n)[2])
        end:
a:= proc(n) local k, p;
      for k from 2 while irem (sopfr(k), n, 'p')>0 or
        not isprime(p) do od; k
    end:
seq (a(n), n=1..100); # Alois P. Heinz, Jun 03 2012

sopfr[n_] := Sum[Times @@ f, {f, FactorInteger[n]}];
a[n_] := For[k = 2, True, k++, If[PrimeQ[sopfr[k]/n], Return[k]]];
Array[a, 100] (* Jean-François Alcover, Nov 13 2020 *)

sopfr(n) = my(f=factor(n)); sum(k=1,#f~,f[k,1]*f[k,2]); \\ A001414
isok(k, n) = my(dr = divrem(sopfr(k), n)); (dr[2]==0) && isprime(dr[1]);
a(n) = {my(k=2); while (!isok(k, n), k++); k;} \\ Michel Marcus, Nov 13 2020

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

A213016 Numbers n such that the sum of prime factors of n (counted with multiplicity) is 3 times a prime.

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A213020 Smallest number k such that the sum of prime factors of k (counted with multiplicity) is n times a prime.

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

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Maple

Mathematica

PARI