A063639 -id:A063639 - cp's OEIS frontend

A063640 Primes of form pqr + 1, where p, q and r are primes.

13, 19, 29, 31, 43, 53, 67, 71, 79, 103, 131, 139, 149, 173, 191, 223, 239, 269, 283, 293, 311, 317, 367, 389, 419, 431, 439, 443, 499, 509, 557, 599, 607, 619, 643, 647, 653, 659, 683, 743, 773, 787, 797, 823, 827, 907, 947, 971, 1031, 1039, 1087, 1091

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Odd primes p such that (p-1)/2 is a semiprime. - Robert G. Wilson v, Sep 01 2007

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

A090866 is a subsequence.

Cf. A014612, A063639.

q:= n-> isprime(n) and numtheory[bigomega](n-1)=3:
select(q, [$2..1100])[];  # Alois P. Heinz, Mar 08 2023

Take[ Select[ Union@ Flatten@ Table[ Prime@p*Prime@q*Prime@r + 1, {p, 48}, {q, p}, {r, q}], PrimeQ@ # &], 53] (* Or *)
semiPrimeQ[x_] := Plus @@ Last /@ FactorInteger[x] == 2; Select[Prime@ Range@ 182, semiPrimeQ[(# - 1)/2] &] (* Robert G. Wilson v, Sep 01 2007 *)
2#+1&/@Select[Table[(n-1)/2,{n,Prime[Range[200]]}],PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 11 2018 *)

n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p - 1) == 3, write("b063640.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

A063641 Primes of form pqr - 2, where p, q and r are primes (not necessarily distinct).

Original entry on oeis.org

43, 61, 73, 97, 103, 151, 163, 173, 193, 229, 271, 277, 283, 331, 367, 383, 397, 421, 433, 463, 547, 593, 601, 607, 613, 643, 661, 709, 739, 757, 773, 859, 883, 907, 929, 967, 1013, 1021, 1063, 1093, 1103, 1129, 1171, 1181, 1231, 1237, 1249, 1279, 1307

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

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

Cf. A014612, A063639.

{ n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p + 2) == 3, write("b063641.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

A335652 Numbers k such that Omega(k+1) = Omega(k) + 2, where Omega(k) = A001222(k) is the number of prime factors of k with multiplicity.

Original entry on oeis.org

7, 11, 15, 17, 19, 29, 35, 39, 41, 43, 55, 67, 87, 97, 101, 109, 113, 134, 137, 155, 163, 173, 175, 181, 183, 203, 207, 209, 211, 219, 229, 241, 242, 247, 249, 257, 259, 279, 281, 283, 295, 305, 314, 317, 327, 329, 331, 337, 339, 341, 351, 353, 371, 373, 401, 404, 409, 413, 433, 455

Offset: 1

Zak Seidov, Jun 16 2020

nonn

			7 is prime, Omega(7) = 1, 7 + 1 = 8 = 2*2*2, Omega(8) = 3.

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

Omega(k+1) = Omega(k) + m: A045920 (m = 0), A076156 (m = 1).

Cf. A001222, A335655. Contains A063639.

Omegas:= [seq(numtheory:-bigomega(i),i=1..1001)]:
select(i -> Omegas[i]+2=Omegas[i+1], [$1..1000]); # Robert Israel, Jun 16 2020

m = 2; s = {}; Do[If[PrimeOmega[x + 1] == PrimeOmega[x] + m, AppendTo[s, x]], {x, 500}]; s
Position[Partition[PrimeOmega[Range[500]],2,1],?(#[[1]]==#[[2]]-2&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Jul 04 2021 *)

A386295 Primes p such that p+1 is a triprime and 2*p+1 is prime.

Original entry on oeis.org

11, 29, 41, 113, 173, 281, 641, 653, 761, 1901, 2273, 2693, 2741, 3413, 3593, 5441, 6053, 6113, 6521, 6581, 7121, 7841, 9293, 9473, 10253, 10733, 12101, 12821, 14081, 14621, 15233, 16493, 19301, 19373, 19553, 19913, 20441, 20693, 21341, 21701, 22433, 24473, 27281, 27581, 27893, 28793, 28901

Offset: 1

Zak Seidov and Robert Israel, Jul 17 2025

nonn

Sophie Germain primes of the form p*q*r - 1, where p, q and r are primes.

Except for 11, all terms == 5 (mod 12).

			a(3) = 41 is a term because it is prime, 41 + 1 = 42 = 2 * 3 * 7 is a triprime, and 41 * 2 + 1 = 83 is prime.

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

Cf. A014612.

Intersection of A005384 and A063639.

select(p -> isprime(p) and isprime(2*p+1) and numtheory:-bigomega(p+1) = 3, [seq(i,i=3..30000,2)]);

s= {}; Do[p = Prime[k]; If[3 == PrimeOmega[p + 1] && PrimeQ[2*p +1], AppendTo[s, p]], {k, 2000}];s

cp's OEIS Frontend

A063640 Primes of form p*q*r + 1, where p, q and r are primes.

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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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A063641 Primes of form p*q*r - 2, where p, q and r are primes (not necessarily distinct).

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A335652 Numbers k such that Omega(k+1) = Omega(k) + 2, where Omega(k) = A001222(k) is the number of prime factors of k with multiplicity.

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A386295 Primes p such that p+1 is a triprime and 2*p+1 is prime.

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Programs

Maple

Mathematica

A063640 Primes of form pqr + 1, where p, q and r are primes.

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

A063641 Primes of form pqr - 2, where p, q and r are primes (not necessarily distinct).