A077800 -id:A077800 - cp's OEIS frontend

A175765 Pairs (t,t+8) of 3-almost primes t separated by 8.

12, 20, 20, 28, 42, 50, 44, 52, 68, 76, 70, 78, 102, 110, 116, 124, 117, 125, 130, 138, 164, 172, 174, 182, 182, 190, 222, 230, 230, 238, 236, 244, 238, 246, 258, 266, 282, 290, 284, 292, 310, 318, 325, 333, 366, 374, 402, 410, 404, 412, 410, 418, 418, 426, 426, 434

Offset: 1

Juri-Stepan Gerasimov, Aug 30 2010

Pairs (12,20), (20,28), (42,50) etc, sorted by the smaller member, such that both numbers are members of A014612 and their difference is 8.

Cf. A014612, A077800, A175612.

Corrected (236 replaced by 238, 258, 266 inserted etc.) by R. J. Mathar, Sep 01 2010

A176811 Number of primes between 2(lesser of n-th twin prime pair) and 2(greater of n-th twin prime pair).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1

Offset: 1

Juri-Stepan Gerasimov, Apr 26 2010

nonn

Number of primes between 2*A001359(n) and 2*A006512(n).
Number of primes between A108605(n) and A176810(n).
Number of primes between 2*A077800(2n-1) and 2*A077800(2n).

			a(1)=1 because 2*3 < 7 (prime) < 2*5;
a(2)=2 because 2*5 < 11 (prime) < 13(prime) < 2*7;
a(3)=1 because 2*11 < 23 (prime) < 2*13.

Cf. A001359, A006512, A077800, A108605, A176810.

A001359 := proc(n) option remember; if n = 1 then 3; else for a from procname(n-1)+2 by 2 do if isprime(a) and isprime(a+2) then return a; end if; end do: end if; end proc:
A006512 := proc(n) A001359(n)+2 ; end proc:
A176811 := proc(n) numtheory[pi](2*A006512(n)) - numtheory[pi](2*A001359(n)) ; end proc:
seq(A176811(n),n=1..120) ; # R. J. Mathar, Apr 27 2010

PrimePi[2*#[[2]]]-PrimePi[2*#[[1]]]&/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]- #[[1]] == 2&] (* Harvey P. Dale, Jul 21 2023 *)

Terms corrected starting at a(34) by R. J. Mathar, Apr 27 2010

A176831 List of all primes p such that 2*A099609(2n-1)A099609(2n).

Original entry on oeis.org

5, 7, 11, 13, 23, 37, 59, 61, 83, 277, 359, 383, 397, 457, 479, 541, 563, 839, 863, 1201, 1237, 1283, 1319, 1321, 1619, 1621, 1657, 2039, 2063, 2099, 2459, 2557, 2579, 2857, 2903, 2963, 3217, 3863, 4057, 4177, 4259, 4261, 4283, 4621, 4679, 5099, 5101, 5581

Offset: 1

Juri-Stepan Gerasimov, Apr 27 2010

nonn

Where A099609 is a naive list of twin primes (A077800 prefixed by 2,3).

			a(1)=5 because 2*A099609(2*1-1)=4<5(prime)<2*A099609(2*1)=6;
a(2)=7 because 2*A099609(2*2-1)=6<7(prime)<2*A099609(2*2)=10;
a(3)=11 and a(4)=13 because 2*A099609(2*3-1)<11(prime)<13(prime)<2*A099609(2*3).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A077800, A099609, A171821.

Flatten@ Map[Select[Range @@ #, PrimeQ] &, 2 Select[Partition[#, 2, 1] &@ Prime@ Range@ 410, First@ Differences@ # <= 2 &]] (* Michael De Vlieger, Mar 18 2017 *)

Entries checked by R. J. Mathar, May 10 2010

A241560 Decimal expansion of the sum of the reciprocals of the averages of the twin prime pairs.

Original entry on oeis.org

9, 2, 8, 8, 3, 5, 8, 2, 7, 1, 3

Offset: 0

Omar E. Pol, May 07 2014

This constant is due to JJGJJG, see link section.

Denominators are in A014574.

			0.92883582713... = Sum_{k>=1} 1/A014574(k) = 1/4 + 1/6 + 1/12 + 1/18 + 1/30 + 1/42 + ...

JJGJJG, La constante entre primos gemelos, gaussianos, Apr 06 2014.

Cf. A001097, A014574, A065421, A077800.

A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

Original entry on oeis.org

5, 17, 617, 857, 1277, 1427, 1697, 2087, 2687, 3557, 4217, 5417, 5477, 7307, 8837, 9437, 10067, 13877, 17657, 18287, 20747, 21587, 23537, 25577, 27917, 28547, 30467, 32117, 32297, 35507, 37337, 37547, 40127, 41177, 41387, 41957

Offset: 1

Karl V. Keller, Jr., Jan 09 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n, n+2, n+24, n+26).
Excluding 5, this is a subsequence of each of the following: A128468 (a(n) = 30*n + 17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).
A253624 is a subsequence of this sequence.

			For n = 17, the numbers 17, 19, 41, 43 are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A253624.

a245568[n_] := Select[Prime@ Range@ n, And[PrimeQ[# + 2], PrimeQ[# + 24], PrimeQ[# + 26]] &]; a245568[5000] (* Michael De Vlieger, Jan 11 2015 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+24) and isprime(n+26): print(n,end=', ')

A247089 Initial members of prime quadruples (p, p+2, p+30, p+32).

Original entry on oeis.org

11, 29, 41, 71, 107, 149, 197, 239, 281, 431, 569, 827, 1019, 1031, 1061, 1289, 1451, 1667, 1997, 2081, 2111, 2237, 2309, 2657, 2969, 3299, 3329, 3359, 3527, 3821, 4019, 4127, 4229, 4241, 4517, 5849, 6269, 6659, 6761, 7457, 7559, 8597

Offset: 1

Karl V. Keller, Jr., Jan 10 2015

nonn

Primes p such that (p, p+2) and (p+30, p+32) are twin prime pairs.
This sequence is a subsequence of A001359 (lesser of twin primes).
The subset of terms ending in 1 in this sequence is a subsequence of A132232 (primes, 11 mod 30).
The subset of terms ending in 7 in this sequence is a subsequence of A141860 (primes, 2 mod 15).
The subset of terms ending in 9 in this sequence is a subsequence of A132236 (primes, 29 mod 30).

			For n=11, the numbers 11, 13, 41, 43, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A001359, A132232, A132236, A141860, A181603 (twins, end 1), A181605 (twins, end 7), A181606 (twins, end 9).

a247089[n_] := Select[Prime@ Range@ n, And[PrimeQ[# + 2], PrimeQ[# + 30], PrimeQ[# + 32]] &]; a247089[1100] (* Michael De Vlieger, Jan 11 2015 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+30) and isprime(n+32): print(n,end=', ')

A248523 Initial members of prime quadruples (n, n+2, n+144, n+146).

Original entry on oeis.org

5, 137, 1787, 1997, 2237, 2657, 3527, 4127, 4337, 4787, 8087, 12107, 13757, 14447, 17987, 19697, 21377, 23057, 23687, 31247, 32297, 34157, 34367, 35447, 37547, 38567, 39227, 43397, 48677, 51197, 51827, 53087, 58907, 65027, 65837

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+144,n+146).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).

			For n=137, the numbers 137, 139, 281, 283, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605.

from sympy import isprime
for n in range(1,10000001,2):
    if isprime(n) and isprime(n+2) and isprime(n+144) and isprime(n+146): print(n,end=', ')

A248661 Initial members of prime quadruples (n, n+2, n+54, n+56).

Original entry on oeis.org

5, 17, 137, 227, 827, 1427, 1667, 1877, 2027, 2087, 2657, 3527, 3767, 4217, 4967, 10037, 11117, 11777, 12107, 13877, 17987, 19697, 20717, 21557, 22037, 23687, 24977, 27527, 27737, 34157, 37307, 41177, 42017, 42407, 47657, 48677

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+54,n+56).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (primes, 30n-13), A181605 (twin primes, end 7), and A092340 (prime n, where n^2+2*n divides (fibonacci(n^2)+fibonacci(2*n))).

			For n=17, the numbers 17, 19, 71, 73, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A092340.

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+54) and isprime(n+56): print(n,end=', ')

A252862 Initial members of prime sextuples (n, n+2, n+6, n+8, n+18, n+20).

Original entry on oeis.org

11, 18041, 97841, 165701, 392261, 663581, 1002341, 1068701, 1155611, 1329701, 1592861, 1678751, 1718861, 1748471, 2159231, 2168651, 2177501, 2458661, 2596661, 3215741, 3295541, 3416051, 3919241, 4353311, 5168921, 5201291, 5205461, 6404771

Offset: 1

Karl V. Keller, Jr., Dec 23 2014

nonn

This sequence is prime n, where there exist three twin prime pairs of (n,n+2), (n+6,n+8) and (n+18,n+20).
This is a subsequence of A132232 (Primes congruent to 11 mod 30 ).
Also, this is a subsequence of A128467 (30k+11).

			For n = 18041, the numbers, 18041, 18043, 18047, 18049, 18059, 18061, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A030430 (primes,10*n+1), A132232, A128467, A172456.

Select[Prime[Range[2500]], Union[PrimeQ[{#, # + 2, # + 6, # + 8, # + 18, # + 20}]] = {True} &] (* Alonso del Arte, Dec 23 2014 *)
Select[Prime[Range[450000]],AllTrue[#+{2,6,8,18,20},PrimeQ]&] (* Harvey P. Dale, Jun 11 2023 *)

forprime(p=1,10^7,if(isprime(p+2) && isprime(p+6) && isprime(p+8) && isprime(p+18) && isprime(p+20), print1(p,", "))) \\ Derek Orr, Dec 31 2014

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+8) and isprime(n+18) and isprime(n+20): print(n,end=', ')

A253627 Initial members of prime sextuples (n, n+2, n+12, n+14, n+18, n+20).

Original entry on oeis.org

179, 809, 5639, 9419, 62969, 88799, 109829, 284729, 452519, 626609, 663569, 855719, 983429, 1003349, 1146779, 1322159, 2116559, 2144489, 2668229, 3153569, 3437699, 4575269, 4606559, 4977419, 5248079, 5436269, 5450099, 5651729

Offset: 1

Karl V. Keller, Jr., Jan 06 2015

nonn

This sequence is prime n, where there exist three twin prime pairs of (n,n+2), (n+12,n+14) and (n+18,n+20).
This is a subsequence of each of the following: A128469(30n+29), A060229(smaller of twin primes of 30n+29).
The prime sextuple does not have to comprise only consecutive primes. - Harvey P. Dale, Aug 15 2016

			For n= 809, the numbers, 809, 811, 821, 823, 827, 829, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A001359, A128469, A060229.

a253627[n_] := Select[Range@n, And[PrimeQ[#], PrimeQ[# + 2], PrimeQ[# + 12], PrimeQ[# + 14], PrimeQ[# + 18], PrimeQ[# + 20]] &]; a253627[10^7] (* Michael De Vlieger, Jan 06 2015 *)
Select[Prime[Range[400000]],AllTrue[#+{2,12,14,18,20},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2016 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+12) and isprime(n+14) and isprime(n+18) and isprime(n+20): print(n,end=', ')

cp's OEIS Frontend

A175765 Pairs (t,t+8) of 3-almost primes t separated by 8.

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A176811 Number of primes between 2*(lesser of n-th twin prime pair) and 2*(greater of n-th twin prime pair).

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A176831 List of all primes p such that 2*A099609(2n-1)A099609(2n).

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A241560 Decimal expansion of the sum of the reciprocals of the averages of the twin prime pairs.

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A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

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A247089 Initial members of prime quadruples (p, p+2, p+30, p+32).

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A248523 Initial members of prime quadruples (n, n+2, n+144, n+146).

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A248661 Initial members of prime quadruples (n, n+2, n+54, n+56).

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A176811 Number of primes between 2(lesser of n-th twin prime pair) and 2(greater of n-th twin prime pair).