A063640 -id:A063640 - cp's OEIS frontend

A063639 Primes of the form pqr - 1, where p, q and r are primes (not necessarily distinct).

7, 11, 17, 19, 29, 41, 43, 67, 97, 101, 109, 113, 137, 163, 173, 181, 211, 229, 241, 257, 281, 283, 317, 331, 337, 353, 373, 401, 409, 433, 523, 547, 577, 601, 617, 641, 653, 677, 691, 709, 761, 787, 821, 829, 853, 907, 937, 941, 977, 1009, 1021, 1033, 1051

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A014612, A063640.

Cf. A000040, A001222.

a063639 n = a063639_list !! (n-1)
a063639_list = [p | p <- a000040_list, a001222 (p+1) == 3]
-- Reinhard Zumkeller, Feb 04 2012

Take[Select[Union[Times@@#-1&/@Tuples[Prime[Range[60]],3]],PrimeQ],60] (* Harvey P. Dale, Jan 23 2012 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p + 1) == 3, write("b063639.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

A063644 Primes with 2 representations: pqr - 1 = uvw + 1 where p, q, r, u, v and w are primes.

Original entry on oeis.org

19, 29, 43, 67, 173, 283, 317, 653, 787, 907, 1867, 2083, 2693, 2803, 3413, 3643, 3677, 4253, 4363, 4723, 5443, 5717, 6197, 6547, 6653, 8563, 8573, 9067, 9187, 9403, 9643, 10733, 11443, 11587, 12163, 12917, 13997, 14107, 14683, 15187, 17827

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Also, primes sandwiched by 3-almost primes. Primes p such that p-+1 are 3-almost primes (A014612). - Zak Seidov, Jul 06 2015

			4723 is a term because 4723 = A063639(168)= 4724 - 1 = 2*2*1181 - 1, and because 4723 = A063640(158)= 4722 + 1 = 2*3*787 + 1.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A063639, A063640, A014612.

Select[Prime[Range[3000]], 3 == PrimeOmega[# - 1] == PrimeOmega[# + 1] &] (* Vincenzo Librandi, Jul 07 2015 *)

n=0; default(primelimit, 2000000); for (m=2, 10^9, p=prime(m); if (bigomega(p + 1) == 3 && bigomega(p - 1) == 3, write("b063644.txt", n++, " ", p); if (n==1000, break)) ) \\ Harry J. Smith, Aug 27 2009

list(lim)=my(v=List(),u=v,L=(lim+1)\2,t); forprime(p=2,L\2, forprime(q=2,min(p,L\p), listput(u,p*q))); u=Set(u); for(i=2,#u, if(u[i]-u[i-1]==1 && isprime(t=2*u[i]-1), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jan 31 2017

A063642 Primes of form pqr + 2, where p, q and r are primes (not necessarily distinct).

Original entry on oeis.org

29, 47, 101, 107, 127, 149, 167, 173, 197, 233, 257, 263, 277, 281, 347, 359, 389, 401, 431, 457, 467, 479, 509, 541, 557, 563, 577, 607, 617, 641, 647, 653, 659, 727, 743, 761, 797, 863, 887, 911, 929, 937, 971, 983, 1019, 1087, 1097, 1129, 1181, 1187

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A014612, A063640.

With[{nn=50},Take[Select[Times@@@Tuples[Prime[Range[nn]],3]+2,PrimeQ]// Union,nn]] (* Harvey P. Dale, Jan 21 2021 *)

{ n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p - 2) == 3, write("b063642.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

A367947 Semiprimes s such that 2*s+1 is prime.

Original entry on oeis.org

6, 9, 14, 15, 21, 26, 33, 35, 39, 51, 65, 69, 74, 86, 95, 111, 119, 134, 141, 146, 155, 158, 183, 194, 209, 215, 219, 221, 249, 254, 278, 299, 303, 309, 321, 323, 326, 329, 341, 371, 386, 393, 398, 411, 413, 453, 473, 485, 515, 519, 543, 545, 551, 554, 581, 611, 614

Offset: 1

Alexandre Herrera, Dec 05 2023

nonn

Intersection of A001358 and A005097.

Cf. A063640.

isok(s) = (bigomega(s)==2) && isprime(2*s+1); \\ Michel Marcus, Dec 06 2023

import sympy as sp
l = []
for i in range(620):
    if (sum(sp.factorint(i).values()) == 2) and sp.isprime(2*i+1):
        l.append(i)
print(l)

a(n) = (A063640(n) - 1)/2. - Hugo Pfoertner, Dec 05 2023

cp's OEIS Frontend

A063639 Primes of the form p*q*r - 1, where p, q and r are primes (not necessarily distinct).

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A063644 Primes with 2 representations: p*q*r - 1 = u*v*w + 1 where p, q, r, u, v and w are primes.

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A063642 Primes of form p*q*r + 2, where p, q and r are primes (not necessarily distinct).

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A367947 Semiprimes s such that 2*s+1 is prime.

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A063639 Primes of the form pqr - 1, where p, q and r are primes (not necessarily distinct).

A063644 Primes with 2 representations: pqr - 1 = uvw + 1 where p, q, r, u, v and w are primes.

A063642 Primes of form pqr + 2, where p, q and r are primes (not necessarily distinct).