A077800 -id:A077800 - cp's OEIS frontend

A342714 Decimal expansion of infinite sum of reciprocals of lesser twin primes, Sum_{n>=1} 1/A001359(n).

1, 0, 5, 9, 0, 6, 4, 2, 6

Offset: 1

Artur Jasinski, Mar 19 2021

Alternative definition: infinite sum of reciprocals of primes whose distance to the next prime is equal to 2.

R. J. Mathar gave an estimate of 1.059064 for this constant in a comment at A209328. Dimitris Valianatos estimated the constant as 1.059064266555685... in a comment at A306539.

			Equals 1.05906426...

Cf. A006512, A065421, A077800, A160910, A209328, A209329, A306539.

Equals 1/3 + 1/5 + 1/11 + 1/17 + 1/29 + 1/41 + 1/59 + ...

Equals (A065421 + A306539)/2.

A088971 Number of twin prime pairs between consecutive prime-indexed primes of order 3. The bounds are included in the calculation.

Original entry on oeis.org

3, 5, 8, 12, 9, 16, 12, 15, 33, 16, 32, 19, 12, 23, 27, 31, 7, 54, 24, 14, 32, 30, 33, 54, 38, 20, 17, 14, 18, 104, 25, 30, 26, 57, 17, 52, 41, 25, 50, 40, 20, 69, 21, 30, 16, 85, 135, 18, 18, 22, 28, 28, 65, 26, 63, 64, 17, 45, 29, 15, 93, 115, 41, 13, 21, 129, 56, 80, 17, 25, 31, 59, 70, 70, 37, 33, 41, 42, 58, 92

Offset: 1

Cino Hilliard, Oct 30 2003

nonn

This sequence contains no 0's between x=1 to 33000. The interval [PIPS3(4133), PIPS3(4134)] contains 1 twin prime pair; the interval [PIPS3(8268), PIPS3(8269)] contains 2 twin prime pairs.

			a(1) = 3, since there are three pairs of twin primes at least PIPS3(1) = 11 and at most PIPS3(2) = 31: (11,13), (17,19), and (29,31).

Cf. A006450, A049090, A077800, A088973.

piptwins3(m,n) = { for(x=m,n, f=1; c=0; p1 = prime(prime(prime(prime(x)))); p2 = prime(prime(prime(prime(x+1)))); forprime(j=p1,p2-2, if(isprime(j+2),f=0; c++) ); print1(c","); ) }

def PIP(n, i): # Returns the n-th prime-indexed prime of order i.
    if i==0:
        return primes_first_n(n)[n-1]
    else:
        return PIP(PIP(n, i-1), 0)
def A088971(n): # Returns a(n)
    return len([i for i in range(PIP(n, 3), PIP(n+1, 3), 2) if (is_prime(i) and is_prime(i+2))])
A088971(1) # Danny Rorabaugh, Mar 30 2015

PIPS3(x) = A049090(x) = the x-th prime-indexed prime of order 3 = prime(prime(prime(prime(x)))) where prime(x) is the x-th prime. a(n) = count of twins in [PIPS3(n), PIPS3(n+1)].

Edited to count twin pairs entirely within [PIPS3(n), PIPS3(n+1)], rather than pairs with the first prime in that interval. - Danny Rorabaugh, Apr 01 2015

A088973 Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation.

Original entry on oeis.org

5, 20, 25, 76, 51, 93, 61, 100, 176, 122, 207, 156, 89, 152, 249, 280, 44, 412, 178, 90, 293, 270, 282, 374, 340, 157, 186, 121, 169, 913, 263, 235, 255, 597, 162, 406, 457, 263, 418, 339, 221, 645, 161, 300, 133, 855, 1235, 236, 162, 240, 256, 243, 786, 261, 514, 590, 156, 481, 374, 211

Offset: 1

Cino Hilliard, Oct 30 2003

nonn

Conjecture: The interval [PIPS4(n), PIPS4(n+1)] always contains at least one twin prime pair. (This implies the Twin Prime Conjecture.)

			a(1) = 5, since there are five pairs of twin primes at least PIPS4(1) = 31 and at most PIPS4(2) = 127: (41,43), (59,61), (71,73), (101,103), and (107,109).

Cf. A006450, A049203, A077800, A088971.

piptwins4(m,n) = { for(x=m,n, f=1; c=0; p1 = prime(prime(prime(prime(prime(x))))); p2 = prime(prime(prime(prime(prime(x+1))))); forprime(j=p1,p2-2, if(isprime(j+2),f=0; c++) ); print1(c","); ) }

def PIP(n,i): # Returns the n-th prime-indexed prime of order i
    if i==0:
        return primes_first_n(n)[n-1]
    else:
        return PIP(PIP(n,i-1),0)
def A088973(n):
    return len([i for i in range(PIP(n,4),PIP(n+1,4),2) if (is_prime(i) and is_prime(i+2))])
A088973(60) # Danny Rorabaugh, Mar 30 2015

PIPS4(x) = A049203(x) = the x-th prime-indexed primes of order 4 = prime(prime(prime(prime(prime(x))))) where prime(x) = A000040(x) is the x-th prime. a(n) = number of twin prime pairs in [PIPS4(n), PIPS(n+1)].

Edited to count twin pairs entirely within [PIPS4(n), PIPS4(n+1)], rather than pairs with the first prime in that interval. - Danny Rorabaugh, Apr 01 2015

A218279 Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (kp(n), k(p(n)+2)), or a(n)=0, if there is no such k.

Original entry on oeis.org

2, 4, 2, 2, 3, 2, 6, 5, 3, 5, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 4, 3, 3, 2, 2, 2, 3, 6, 3, 2, 4, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 5, 2, 2, 2, 3, 2, 3, 3, 6, 3, 4, 9, 5, 2, 5, 4, 2, 3, 2, 3, 3, 2, 4, 3, 2, 2, 5, 3, 4, 4, 4, 4, 3, 2, 6, 2, 7, 4, 2, 6, 4, 2

Offset: 1

Vladimir Shevelev, Oct 25 2012

nonn

Conjecture: a(n)>0 for all n.

			The first pair of twin primes is (3,5). For k=1 and 2, we have the intervals (3,5) and (6,10), such that not the first but the second interval contains exactly one prime(7). Thus a(1)=2. For n=2 and k=1 to 4, we have the intervals (5,7),(10,14),(15,21), and (20,28) and only the last interval contains exactly one prime(23). Thus, a(2)=4.

Zak Seidov, Table of n, a(n) for n = 1..10000
V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A218275, A166251, A217561, A217566, A217577, A001359, A014574, A006512, A077800.

a(6) corrected and terms beyond a(11) contributed by Zak Seidov, Oct 25 2012

A248367 Initial members of prime quadruples (n, n+2, n+36, n+38).

Original entry on oeis.org

5, 71, 101, 191, 311, 821, 1451, 4091, 4481, 4931, 5441, 6791, 12071, 13721, 14591, 17921, 18251, 20441, 20771, 20981, 21521, 21611, 35801, 38711, 41141, 41981, 43541, 46271, 47351, 47741, 48821, 49331, 53231, 64151, 70841

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+36,n+38).
This sequence is a subsequence of A001359 (lesser of twin primes).
Excluding 5, this sequence is a subsequence of A132232 (primes, 11 mod 30).

			For n=71, the numbers 71, 73, 107, 109, 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, A181603 (twin primes, end 1).

a248367[n_] := Select[Prime@Range@n, And[PrimeQ[# + 2], PrimeQ[# + 36], PrimeQ[# + 38]] &]; a248367[8000] (* Michael De Vlieger, Jan 11 2015 *)
Select[Prime[Range[8000]],AllTrue[#+{2,36,38},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 17 2019 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+36) and isprime(n+38): print(n,end=', ')

A259025 Numbers k such that k is the average of four consecutive primes k-11, k-1, k+1 and k+11.

Original entry on oeis.org

420, 1050, 2028, 2730, 3582, 4230, 4242, 4272, 4338, 6090, 6132, 6690, 6792, 8220, 11058, 11160, 11970, 12252, 15288, 19542, 19698, 21588, 21600, 26892, 27540, 28098, 28308, 29400, 30840, 30870, 31080, 32412, 42072, 45318, 47808, 48120

Offset: 1

Karl V. Keller, Jr., Jun 16 2015

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 are congruent to 0 mod 30.
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.

			For n=420: 409, 419, 421, 431 are consecutive primes (n-11=409, n-1=419, n+1=421, n+11=431).
For n=1050: 1039, 1049, 1051, 1061 are consecutive primes (n-11=1039, n-1=1049, n+1=1051, n+11=1061).

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

Cf. A052376, A077800 (twin primes), A014574, A249674 (30n), A256753.

{p, q, r, s} = {2, 3, 5, 7}; lst = {}; While[p < 50000, If[ Differences[{p, q, r, s}] == {10, 2, 10}, AppendTo[lst, q + 1]]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; lst (* Robert G. Wilson v, Jul 15 2015 *)
Mean/@Select[Partition[Prime[Range[5000]],4,1],Differences[#]=={10,2,10}&] (* Harvey P. Dale, Sep 11 2019 *)

is(n)=n%6==0&&isprime(n-11)&&isprime(n-1)&&isprime(n+1)&&isprime(n+11)&&!isprime(n-7)&&!isprime(n-5)&&!isprime(n+5)&&!isprime(n+7) \\ Charles R Greathouse IV, Jul 17 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,50001,2):
  if isprime(i-1) and isprime(i+1):
    if prevprime(i-1) == i-11 and nextprime(i+1) == i+11 :  print (i,end=', ')

a(n) = A052376(n) + 11. - Robert G. Wilson v, Jul 15 2015

A261701 Initial member of four twin prime pairs with gap 210 between them.

Original entry on oeis.org

599, 3917, 5021, 37361, 48779, 81929, 93281, 97157, 263399, 433049, 783149, 821801, 906119, 908669, 1197197, 1308497, 1308707, 1379237, 1464809, 1908449, 2036861, 2341979, 2408561, 2760671, 2804309, 3042491, 3042701, 3042911, 3198197, 4090649, 4543991, 5543927

Offset: 1

K. D. Bajpai, Aug 28 2015

nonn

More precisely, primes p such that p + 2, p + 210, p + 212, p + 420, p + 422, p + 630, p + 632 are all primes.

All the terms in this sequence are congruent to 2 (mod 3).

			599 appears in the sequence because: (a) {599,601}, {809, 811}, {1019, 1021}, {1229, 1231} are four (not consecutive) twin prime pairs; (b) the gap between each twin prime pair (809 - 599) = (1019 -  809) = (1229 - 1019) = 210.

K. D. Bajpai and Dana Jacobsen, Table of n, a(n) for n = 1..10000 [first 865 terms from K. D. Bajpai]
Luis Rodriguez and Carlos Rivera, Gaps between consecutive twin pairs, The Prime Puzzles and Problems Connection.

Cf. A001359 (twin primes), A077800, A113274, A253624.

[p: p in PrimesUpTo (100000) | IsPrime(p+2) and IsPrime(p+210) and IsPrime(p+212) and IsPrime(p+420) and IsPrime(p+422) and IsPrime(p+630) and IsPrime(p+632) ];

select(p -> andmap(isprime, [p, p+2, p+210, p+212, p+420, p+422, p+630, p+632]),[seq(p, p=1..10^5)]);

k = 210; Select[Prime@Range[10^7], PrimeQ[# + 2] && PrimeQ[# + k] && PrimeQ[# + k + 2] && PrimeQ[# + 2 k] && PrimeQ[# + 2 k + 2] && PrimeQ[# + 3 k] &&  PrimeQ[# + 3 k + 2] &]
Select[Prime[Range[400000]],AllTrue[#+{2,210,212,420,422,630,632},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 17 2019 *)

forprime(p= 1, 100000, if(isprime(p+2) && isprime(p+210) && isprime(p+212) && isprime(p+420) && isprime(p+422) && isprime(p+630) && isprime(p+632), print1(p,", ")));

use ntheory ":all"; say join ", ", grep { is_prime($+210) && is_prime($+212) && is_prime($+420) && is_prime($+422) && is_prime($+630) && is_prime($+632) } @{twin_primes(1e8)}; # Dana Jacobsen, Sep 02 2015

use ntheory ":all"; say for sieve_prime_cluster(1, 1e8, 2, 210, 212, 420, 422, 630, 632); # Dana Jacobsen, Oct 03 2015

A275515 Table read by rows: list of prime triples of the form (p, p+2, p+6).

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 41, 43, 47, 101, 103, 107, 107, 109, 113, 191, 193, 197, 227, 229, 233, 311, 313, 317, 347, 349, 353, 461, 463, 467, 641, 643, 647, 821, 823, 827, 857, 859, 863, 881, 883, 887, 1091, 1093, 1097, 1277, 1279, 1283, 1301, 1303, 1307

Offset: 1

Arkadiusz Wesolowski, Jul 31 2016

A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).
Initial members p (other than 5) of prime triples of the form (p, p+2, p+6) are congruent to 11 or 17 (mod 30).
Also called prime triples of the first kind.

			The table starts:
5, 7, 11;
11, 13, 17;
17, 19, 23;
...

C. K. Caldwell, Top Twenty page, Triplet
Eric Weisstein's World of Mathematics, Prime Triplet
Wikipedia, Prime triple
Index entries for primes, gaps between

Cf. A022004, A077800, A136162, A270998, A275516.

&cat[[p, p+2, p+6]: p in PrimesUpTo(1301) | (p le 5 xor p mod 30 in {11, 17}) and IsPrime(p+2) and IsPrime(p+6)];

Prime@ Range[#, # + 2] &@ PrimePi@ Select[Prime@ Range@ 216, Times @@ Boole@ PrimeQ[# + {2, 6}] > 0 &] // Flatten (* Michael De Vlieger, Aug 02 2016 *)

a(3*n-2) = A022004(n).

A283867 Numbers n such that 30n^2 - 1 and 30n^2 + 1 are (twin) primes.

Original entry on oeis.org

1, 3, 10, 14, 18, 38, 62, 73, 116, 118, 143, 183, 221, 232, 242, 330, 333, 413, 430, 455, 470, 496, 507, 533, 538, 556, 606, 622, 645, 675, 687, 701, 720, 777, 792, 819, 846, 879, 881, 895, 913, 1000, 1019, 1030, 1092, 1155, 1214, 1238, 1253, 1261, 1313, 1337, 1350, 1407, 1418, 1429, 1431

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2017

nonn

			3 is in this sequence because 30*3^2 - 1 = 269 and 30*3^2 + 1 = 271 are twin primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A001097, A077800, A158558, A158560.

[n: n in [1..1500] | IsPrime(30*n^2-1) and IsPrime(30*n^2+1)];

Select[Range@ 1431, PrimeQ[30*#^2 + 1] && PrimeQ[30*#^2 - 1] &] (* Indranil Ghosh, Mar 17 2017 *)

is(n)=isprime(30*n^2-1) && isprime(30*n^2+1) \\ Charles R Greathouse IV, Mar 17 2017

from sympy import isprime
[i for i in range(1, 1501) if isprime(30*i**2 - 1) and isprime(30*i**2 + 1)] # Indranil Ghosh, Mar 17 2017

a(n) >> n log^2 n. - Charles R Greathouse IV, Mar 17 2017

A343778 Primes which are two greater than A074040 terms.

Original entry on oeis.org

17, 21800053277, 86984485062381462583582279727, 2948338207972508983453357158259221375675126583677039825367935271466652794027

Offset: 1

Hartmut F. W. Hoft, Apr 29 2021

nonn

a(5) = 3052230...330677 has 17332 digits, the only prime larger than a(4) and among the cumulative products of the first 2000 twin primes pairs plus two.

			a(1) = 17 = A074040(1) + 2 = 3*5 + 2.

Cf. A037074, A077800, A074040, A097490, A097491, A097493.

(* function a074040[ ] is defined in A074040 *)
a343778[n_] := Select[Map[#+2&, a074040[n]], PrimeQ]
a343778[30]

cp's OEIS Frontend

A342714 Decimal expansion of infinite sum of reciprocals of lesser twin primes, Sum_{n>=1} 1/A001359(n).

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A088971 Number of twin prime pairs between consecutive prime-indexed primes of order 3. The bounds are included in the calculation.

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A088973 Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation.

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A218279 Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (k*p(n), k*(p(n)+2)), or a(n)=0, if there is no such k.

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A248367 Initial members of prime quadruples (n, n+2, n+36, n+38).

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A259025 Numbers k such that k is the average of four consecutive primes k-11, k-1, k+1 and k+11.

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A261701 Initial member of four twin prime pairs with gap 210 between them.

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A218279 Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (kp(n), k(p(n)+2)), or a(n)=0, if there is no such k.

A283867 Numbers n such that 30n^2 - 1 and 30n^2 + 1 are (twin) primes.