A108172 -id:A108172 - cp's OEIS frontend

A108166 Semiprimes p*q where both p and q are primes of the form 6n-1 (A007528).

25, 55, 85, 115, 121, 145, 187, 205, 235, 253, 265, 289, 295, 319, 355, 391, 415, 445, 451, 493, 505, 517, 529, 535, 565, 583, 649, 655, 667, 685, 697, 745, 781, 799, 835, 841, 865, 895, 901, 913, 943, 955, 979, 985, 1003, 1081, 1111, 1135, 1165, 1177, 1189

Offset: 1

Jonathan Vos Post, Jun 13 2005

Every semiprime not divisible by 2 or 3 must be in one of these three disjoint sets:
A108164 - the product of two primes of the form 6n + 1 (A002476),
A108166 - the product of two primes of the form 6n - 1 (A007528),
A108172 - the product of a prime of the form 6n + 1 and a prime of the form 6n - 1.
The product of two primes of the form 6n - 1 is a semiprime of the form 6n + 1.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001358, A007528.

Module[{nn = 150, pf}, pf = Select[6Range[nn] - 1, PrimeQ]; Take[Union[Times@@@Tuples[pf, 2]], nn/2]] (* Harvey P. Dale, Dec 09 2013 *)
Select[6Range[200] + 1, PrimeOmega[#] == 2 && Mod[FactorInteger[#][[1, 1]], 6] == 5 &] (* Alonso del Arte, Aug 24 2017 *)

{a(n)} = {p*q where both p and q are in A007528}.

Edited and extended by Ray Chandler, Oct 15 2005

A108164 Semiprimes p*q where both p and q are primes of the form 6n+1 (A002476).

Original entry on oeis.org

49, 91, 133, 169, 217, 247, 259, 301, 361, 403, 427, 469, 481, 511, 553, 559, 589, 679, 703, 721, 763, 793, 817, 871, 889, 949, 961, 973, 1027, 1057, 1099, 1141, 1147, 1159, 1261, 1267, 1273, 1333, 1339, 1351, 1369, 1387, 1393, 1417, 1477, 1501, 1561, 1591

Offset: 1

Jonathan Vos Post, Jun 13 2005

These are the products of terms from A107890 excluding multiples of 3.
Every semiprime not divisible by 2 or 3 must be in one of these three disjoint sets:
A108164 = the product of two primes of the form 6n+1 (A002476),
A108166 = the product of two primes of the form 6n-1 (A007528),
A108172 = the product of a prime of the form 6n+1 and a prime of the form 6n-1.
The product of two primes of the form 6n+1 is a semiprime of the form 6n+1.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Robert Israel, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.

Cf. A001358, A002476, A107890.

N:= 2000: # To get all terms <= N
P:= select(isprime, [seq(i,i=7..N/7, 6)]):
sort(select(`<=`,[seq(seq(P[i]*P[j],j=1..i),i=1..nops(P))],N)); # Robert Israel, Dec 27 2018

With[{nn=50},Take[Times@@@Tuples[Select[6*Range[nn]+1,PrimeQ],2]// Union,nn]] (* Harvey P. Dale, May 20 2021 *)

{a(n)} = {p*q where both p and q are in A002476}.

Edited and extended by Ray Chandler, Oct 15 2005

A112776 Numbers k such that 6k+5 is semiprime.

Original entry on oeis.org

5, 10, 12, 15, 19, 23, 25, 26, 30, 33, 34, 35, 36, 47, 49, 50, 53, 54, 55, 56, 60, 61, 62, 65, 67, 68, 72, 78, 80, 82, 85, 87, 88, 90, 91, 96, 101, 103, 104, 105, 111, 114, 115, 117, 118, 121, 122, 124, 125, 127, 129, 130, 131, 133, 135, 141, 144, 148, 149, 150

Offset: 1

Jonathan Vos Post and Ray Chandler, Oct 15 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A108172 (resulting semiprimes).

IsSemiprime:=func; [n: n in [2..150] | IsSemiprime(6*n+5)]; // Vincenzo Librandi, Sep 22 2012

Select[Range[3000], Plus@@Last/@FactorInteger[6# + 5] == 2&] (* Vincenzo Librandi, Sep 22 2012 *)

is(n)=bigomega(6*n+5)==2 \\ Charles R Greathouse IV, Feb 08 2017

a(n) = (A108172(n) - 5)/6.

A344872 Semiprimes of the form 3m+2.

Original entry on oeis.org

14, 26, 35, 38, 62, 65, 74, 77, 86, 95, 119, 122, 134, 143, 146, 155, 158, 161, 185, 194, 203, 206, 209, 215, 218, 221, 254, 278, 287, 299, 302, 305, 314, 323, 326, 329, 335, 341, 362, 365, 371, 377, 386, 395, 398, 407, 413, 422, 437, 446, 458, 473, 482, 485, 497

Offset: 1

Peter Munn, May 31 2021

There are no square terms, as squares are congruent to 0 or 1 modulo 3.

Products of a prime of the form 3m+1 and a prime of the form 3m+2 (the former necessarily being of the form 6m+1).

			14 = 2 * 7 has 2 prime factors (counting repetitions) so is a semiprime, and 14 = 3*4 + 2, so has the form 3m+2. So 14 is in the sequence.

Intersection of A001358 and A016789.
Disjoint union of A108172 and A112772.
Complement within A001358 of A001748, A112771 and A112774.
Subsequence of A344703.

Select[Range[2,500,3],PrimeOmega@#==2&] (* Giorgos Kalogeropoulos, Jun 02 2021 *)

isok(m) = bigomega(m) == 2 && m % 3 == 2;

cp's OEIS Frontend

A108166 Semiprimes p*q where both p and q are primes of the form 6n-1 (A007528).

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A108164 Semiprimes p*q where both p and q are primes of the form 6n+1 (A002476).

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A112776 Numbers k such that 6k+5 is semiprime.

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A344872 Semiprimes of the form 3m+2.

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