A224610 -id:A224610 - cp's OEIS frontend

A224611 Smallest j such that j2p(n)^3-1=q is prime, j2p(n)q^2-1=r, j2p(n)r^2-1=s, where r and s are also prime.

902, 145, 771, 1060, 3569, 520, 938, 294, 2457, 3911, 1650, 483, 8604, 3450, 2345, 548, 25004, 1635, 5767, 14519, 2518, 6394, 198, 7961, 4272, 8370, 4146, 654, 4489, 6987, 222, 5426, 5250, 17670, 7691, 360, 3994, 20821, 9008, 6525, 9204, 1464, 6111, 6625, 11229, 3315, 62340, 735, 6962, 5236

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

Pierre CAMI, Table of n, a(n) for n = 1..3500

Cf. A224491, A224609, A224610, A224612.

a[n_] := For[j = 1, j < 10^7, j++, p = Prime[n]; If[PrimeQ[q = j*2*p^3 - 1] && PrimeQ[r = j*2*p*q^2 - 1] && PrimeQ[j*2*p*r^2 - 1], Return[j]]]; Table[ Print[an = a[n]]; an, {n, 1, 50}] (* Jean-François Alcover, Apr 12 2013 *)

A224614 Primes p such that q = 2p^3-1 and 2p*q^2-1 are both prime.

Original entry on oeis.org

181, 199, 4363, 4549, 14563, 15073, 15739, 27361, 27901, 33469, 34231, 37123, 46279, 48271, 48673, 54193, 56101, 64591, 64609, 65539, 65731, 70183, 70891, 75703, 75979, 77659, 77863, 80953, 94309, 112573, 114889, 115153, 117361, 118189, 135799, 144751

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

When A224610(i) = 1 then prime(i) is in this sequence.

Subsequence of A177104. - R. J. Mathar, Apr 19 2013

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A224610, A224613.

[p: p in PrimesUpTo(180000) | IsPrime(q) and IsPrime(2*p*q^2-1) where q is 2*p^3-1 ]; // Bruno Berselli, Apr 19 2013

Reap[For[p = 2, p < 200000, p = NextPrime[p], If[PrimeQ[q = 2*p^3 - 1] && PrimeQ[r = 2*p*q^2 - 1], Sow[p]]]][[2, 1]] (* Jean-François Alcover, Apr 19 2013 *)
bpQ[n_]:=Module[{c=2n^3-1},AllTrue[{c,2n*c^2-1},PrimeQ]]; Select[ Prime[ Range[ 15000]],bpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 05 2015 *)

A224612 Let p = prime(n). Smallest j such that q = j2p^3-1, r = jp2q^2-1, s = jp2r^2-1, and jp2*s^2-1 are prime numbers.

Original entry on oeis.org

29952, 12063, 1463, 6102, 11661, 49552, 639179, 2099290, 291248, 393186, 545251, 321303, 436641, 278295, 746832, 237852, 56490, 165901, 152847, 619755, 777177, 3410085, 117513, 2015421, 497170, 14750, 161190, 347039, 251835, 57536, 222, 2083286, 384944, 1228474, 3909960, 344164, 332224, 207751, 14060

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

Conjecture: a(n) exists for all n.

Cf. A224492, A224609, A224610, A224611.

f:= proc(n) local j,p,q,r,s;
    p:= ithprime(n);
    for j from 1 do
      q:= j*2*p^3-1; if not isprime(q) then next fi;
      r:= j*p*2*q^2-1; if not isprime(r) then next fi;
      s:= j*p*2*r^2-1; if not isprime(s) then next fi;
      if isprime(j*p*2*s^2-1) then return j fi;
    od
end proc;
map(f, [$1..25]); # Robert Israel, May 15 2025

a[n_] := For[j = 1, j < 10^7, j++, p = Prime[n]; If[PrimeQ[q = j*2*p^3 - 1] && PrimeQ[r = j*2*p*q^2 - 1] && PrimeQ[s = j*2*p*r^2 - 1] && PrimeQ[j*2*p*s^2 - 1], Return[j]]]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Apr 12 2013 *)

More terms from Jean-François Alcover, Apr 12 2013

Name clarified and more terms from Robert Israel, May 15 2025

cp's OEIS Frontend

A224611 Smallest j such that j2p(n)^3-1=q is prime, j2p(n)q^2-1=r, j2p(n)r^2-1=s, where r and s are also prime.

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A224614 Primes p such that q = 2p^3-1 and 2p*q^2-1 are both prime.

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A224612 Let p = prime(n). Smallest j such that q = j2p^3-1, r = jp2q^2-1, s = jp2r^2-1, and jp2*s^2-1 are prime numbers.

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cp's OEIS Frontend

A224611 Smallest j such that j*2*p(n)^3-1=q is prime, j*2*p(n)*q^2-1=r, j*2*p(n)*r^2-1=s, where r and s are also prime.

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Mathematica

A224614 Primes p such that q = 2*p^3-1 and 2*p*q^2-1 are both prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A224612 Let p = prime(n). Smallest j such that q = j*2*p^3-1, r = j*p*2*q^2-1, s = j*p*2*r^2-1, and j*p*2*s^2-1 are prime numbers.

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Maple

Mathematica

Extensions

A224611 Smallest j such that j2p(n)^3-1=q is prime, j2p(n)q^2-1=r, j2p(n)r^2-1=s, where r and s are also prime.

A224614 Primes p such that q = 2p^3-1 and 2p*q^2-1 are both prime.

A224612 Let p = prime(n). Smallest j such that q = j2p^3-1, r = jp2q^2-1, s = jp2r^2-1, and jp2*s^2-1 are prime numbers.