A123921 -id:A123921 - cp's OEIS frontend

A048880 Primes of form pq+2 where p and q are consecutive primes.

17, 37, 79, 223, 439, 4759, 22501, 32401, 53359, 57601, 60493, 72901, 77839, 95479, 99223, 159199, 164011, 176401, 194479, 239119, 324901, 378223, 416023, 497011, 680623, 756853, 804511, 1115113, 1664101, 1742401, 2223079

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net)

All terms > 17 are congruent to 1 mod 6. - Zak Seidov, Dec 03 2010

			487*491+2=239119.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A051507, A123921, A342564.

with (numtheory): for n from 1 to 1000 do if (tau(ithprime(n)*ithprime(n+1)+2)=2) then print(ithprime(n),ithprime(n+1),ithprime(n)*ithprime(n+1)+2); fi; od;

Reap[Do[If[PrimeQ[p=Prime[k]*Prime[k+1]+2],Sow[p]],{k,1,430}]][[2,1]] (* Zak Seidov Dec 03 2010 *)

a(n) = 6*A342564(n-1) + 1 for n >= 2. - Hugo Pfoertner, Jun 24 2021

Corrected and extended by Joe DeMaio (jdemaio(AT)kennesaw.edu).

A048797 Primes p such that pp'-2 is prime, where p' denotes the next prime after p.

Original entry on oeis.org

3, 157, 211, 257, 367, 557, 977, 997, 1381, 1511, 1531, 2467, 2503, 2621, 2777, 2861, 3049, 3307, 3617, 4099, 4357, 4373, 4397, 4463, 4523, 4691, 4831, 4919, 5087, 5209, 5261, 5351, 5407, 5483, 5807, 6173, 6229, 6287, 6619, 6871, 7001, 7187, 7459, 7577

Offset: 1

Joseph L. Pe, Jan 15 2002

nonn

			3*5 - 2 = 13, a prime, so 3 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A123921, A124669.

Prime[ Select[ Range[1000], PrimeQ[ Prime[ # ]Prime[ # + 1] - 2] &]]
Select[Prime[Range[1000]],PrimeQ[# NextPrime[#]-2]&] (* Harvey P. Dale, Aug 05 2024 *)

lista(nn) = {my(p=2); forprime(q=3, nn, if (isprime(p*q-2), print1(p, ", ")); p = q;);} \\ Michel Marcus, Sep 28 2019

A342564 Numbers k such that 6k + 1 is a prime that can be written as pq + 2, with p and q being consecutive primes.

Original entry on oeis.org

6, 13, 37, 73, 793, 3750, 5400, 8893, 9600, 10082, 12150, 12973, 15913, 16537, 26533, 27335, 29400, 32413, 39853, 54150, 63037, 69337, 82835, 113437, 126142, 134085, 185852, 277350, 290400, 370513, 432553, 478837, 531037, 585937, 667333, 768980, 837013, 889350

Offset: 1

Hugo Pfoertner, Jun 20 2021

			a(1) = 6 because 6*6 + 1 = 37 can be written as 5*7 + 2.

Cf. A048880, whose first term 17 = 3*5 + 2 cannot be written as 6*k + 1.

Cf. A123921, A342565.

alist := proc(upto) local L, q, p, n, r; L := []; q := 2;
for n from 1 to upto do
    p := q; q := nextprime(p); r := p * q + 1 ;
    if modp(r, 6) = 0 and isprime(r + 1) then
       L := [op(L), iquo(r, 6)] fi od;
L end: alist(350); # Peter Luschny, Jun 20 2021

(Select[6Range[10^6]+1, PrimeQ[#] && MatchQ[FactorInteger[#-2], {{p_, 1}, {q_, 1}} /; q == NextPrime[p]]&]-1)/6 (* Jean-François Alcover, Jul 07 2021 *)

a342564(plim)={my(p1=5);forprime(p2=7,plim,my(p=p1*p2+2);if(isprime(p),print1((p-1)/6,", "));p1=p2)};
a342564(2400)

from primesieve.numpy import n_primes
from numbthy import isprime
primesarray = numpy.array(n_primes(10000,1))
for i in range (0, 9999):
    totest = int(primesarray[i] * primesarray[i+1] + 2)
    if (isprime(totest)) and  (((totest-1)%6) == 0):
        print((totest-1)//6) # Karl-Heinz Hofmann, Jun 20 2021

a(n) = (A048880(n+1) - 1)/6, excluding A048880(1).

A124669 Product of successive primes minus 2.

Original entry on oeis.org

4, 13, 33, 75, 141, 219, 321, 435, 665, 897, 1145, 1515, 1761, 2019, 2489, 3125, 3597, 4085, 4755, 5181, 5765, 6555, 7385, 8631, 9795, 10401, 11019, 11661, 12315, 14349, 16635, 17945, 19041, 20709, 22497, 23705, 25589, 27219, 28889, 30965, 32397

Offset: 1

Cino Hilliard, Dec 27 2006

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

Cf. A123921.

Times@@@Partition[Prime[Range[50]],2,1]-2 (* Harvey P. Dale, Nov 02 2020 *)

g(n) = forprime(x=1,n,if(isprime(x+2),y=x*(x+2)-2;print1(y",")))

A342565 Numbers k such that 6k - 1 is a prime that can be written as pq - 2, with p and q being consecutive primes.

Original entry on oeis.org

4265, 7842, 11265, 22815, 52265, 160065, 167662, 322003, 383542, 393722, 1016815, 1051677, 1150182, 1290842, 1372803, 1555498, 1826015, 2184065, 2808498, 3168265, 3200307, 3231062, 3333117, 3427680, 3676962, 3913915, 4042598, 4323102, 4537907, 4623542, 4798955

Offset: 1

Hugo Pfoertner, Jun 20 2021

			a(1) = 4265, because the prime 25589 = 6*4265 - 1 can be written as 157*163 - 2, with 157 and 163 being consecutive primes.

Cf. A123921, whose first term 13 = 3*5 - 2 cannot be written as 6*k - 1.

Cf. A048880, A342564.

a342565(plim)={my(p1=5);forprime(p2=7,plim,my(p=p1*p2-2);if(isprime(p),print1((p+1)/6,", "));p1=p2)};
a342565(5400)

from primesieve.numpy import n_primes
from numbthy import isprime
primesarray = numpy.array(n_primes(10000, 1))
for i in range (0, 9999):
    totest = int(primesarray[i] * primesarray[i+1] - 2)
    if (isprime(totest)) and  (((totest+1)%6) == 0):
        print((totest+1)//6) # Karl-Heinz Hofmann, Jun 20 2021

a(n) = (A123921(n+1) + 1)/6, excluding A123921(1).

A124670 (Product of successive primes minus 2) divided by 3 is prime.

Original entry on oeis.org

11, 47, 73, 107, 587, 673, 3467, 3673, 4783, 7499, 10799, 17327, 29983, 33073, 62207, 71147, 72073, 137387, 225227, 243673, 252283, 355007, 442367, 504299, 554699, 567673, 735073, 874799, 924073, 961067, 1062073, 1175627, 1326673, 1486847

Offset: 1

Cino Hilliard, Dec 27 2006

			11*13 = 143, 143-2 = 141, 141/3 = 47 is the second entry.

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

Cf. A123921.

Select[(Times@@@Partition[Prime[Range[500]],2,1]-2)/3,PrimeQ] (* Harvey P. Dale, Oct 02 2012 *)

g(n) = { for(x=1,n, y=prime(x)*prime(x+1)-2; if(y%3==0,if(isprime(y/3), print1(y/3",")))) }

A124684 Primes of the form (p*q - 2)/5 where p and q are successive primes.

Original entry on oeis.org

229, 1153, 14149, 15013, 189733, 214657, 253573, 350593, 514561, 522289, 725041, 853669, 1304581, 1453681, 2027569, 2183281, 2212453, 2469637, 3238513, 4166017, 4331941, 4467013, 5234689, 5510371, 5992933, 6102913, 8100097, 8130673

Offset: 1

Cino Hilliard, Dec 27 2006

These numbers times 5 are semiprimes.

			31*37 = 1147, 1147-2 = 1145, 1145/5 = 229 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A123921.

Select[(Times @@@ Partition[Prime[Range[1000]], 2, 1] - 2) / 5, PrimeQ] (* Amiram Eldar, May 26 2024 *)

g(n,p=5) = { for(x=1,n, y=prime(x)*prime(x+1)-2; if(y%p==0,if(isprime(y/p), print1(y/p, ", ")))) }

A125502 (Product of successive primes minus 2) divided by 7 is prime.

Original entry on oeis.org

4127, 49727, 212627, 565727, 697727, 1152227, 3102227, 3486227, 5742227, 7488227, 8078627, 8848127, 10837727, 14200127, 23041427, 41870627, 50437727, 59044127, 68766227, 70088927, 91008227, 115141727, 118573727, 122641427

Offset: 1

Cino Hilliard, Dec 28 2006

			167*173 = 28891, (28891-2)/7 = 4127 the first entry.

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

Cf. A123921.

Select[(Times@@@Partition[Prime[Range[5000]],2,1]-2)/7,PrimeQ] (* Harvey P. Dale, Sep 03 2015 *)

g(n,p) = { for(x=1,n, y=prime(x)*prime(x+1)-2; if(y%p==0,if(isprime(y/p), print1(y/p",")))) }

Typo in example corrected by Harvey P. Dale, Sep 03 2015

A229613 Primes of the form p*q - 30, where p and q are consecutive primes.

Original entry on oeis.org

5, 47, 113, 191, 293, 1117, 1487, 1733, 4057, 5153, 5737, 9767, 11633, 14321, 16607, 19013, 20681, 22469, 23677, 25561, 27191, 30937, 32369, 36833, 37991, 41959, 50591, 53327, 70717, 72869, 75037, 79493, 82889, 99191, 136861, 148957, 159167, 163979, 171341, 176369

Offset: 1

K. D. Bajpai, Sep 26 2013

nonn

For primes p <= prime(5000) = 48611, the expression p*q - c with p and q consecutive primes yields more primes at c = 30 than at any other positive c <= 100.

For the above range of primes p, c=30 yields 999 primes, but there are values of c > 100 that yield larger counts; e.g., c = 210, 420, 2310, and 9240 yield 1129, 1194, 1295, and 1316, respectively. - Jon E. Schoenfield, Jun 25 2022

			prime(4)*prime(5) - 30 = 7*11 - 30 = 47, which is prime, so 47 is a term.
prime(11)*prime(12) - 30 = 31*37 - 30 = 1117, which is prime, so 1117 is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from K. D. Bajpai)

Cf. A123921.

KD:= proc() local a; a:= ithprime(n)*ithprime(n+1)-30; if isprime((a)) then RETURN((a)):fi; end: seq(KD(),n=1..500);

Select[Table[Prime[n]*Prime[n + 1] - 30, {n, 100}], PrimeQ]

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q = 2, 3
    while True:
        t = p*q-30
        if isprime(t):
            yield t
        p, q = q, nextprime(q)
print(list(islice(agen(), 40))) # Michael S. Branicky, Jun 25 2022

cp's OEIS Frontend

A048880 Primes of form pq+2 where p and q are consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A048797 Primes p such that pp'-2 is prime, where p' denotes the next prime after p.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A342564 Numbers k such that 6*k + 1 is a prime that can be written as p*q + 2, with p and q being consecutive primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A124669 Product of successive primes minus 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A342565 Numbers k such that 6*k - 1 is a prime that can be written as p*q - 2, with p and q being consecutive primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Python

Formula

A124670 (Product of successive primes minus 2) divided by 3 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A124684 Primes of the form (p*q - 2)/5 where p and q are successive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A125502 (Product of successive primes minus 2) divided by 7 is prime.

Views

Author

Keywords

A342564 Numbers k such that 6k + 1 is a prime that can be written as pq + 2, with p and q being consecutive primes.

A342565 Numbers k such that 6k - 1 is a prime that can be written as pq - 2, with p and q being consecutive primes.