A077800 -id:A077800 - cp's OEIS frontend

A356793 Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

1, 6, 5, 6, 1, 8, 4, 6, 5, 3, 9, 5

Offset: 0

Artur Jasinski, Sep 04 2022

Alternative definition: sum of squares of reciprocals of primes whose distance from the next prime is equal to 2.
Convergence table:
   k       A001359(k)      Sum_{j=1..k} 1/A001359(j)^2
10000000   3285916169 0.165618465394273171950874120818
20000000   7065898967 0.165618465394707600197099741096
30000000  11044807451 0.165618465394836120901019351544
40000000  15151463321 0.165618465394895965582366015390
50000000  19358093939 0.165618465394930089884704869090
60000000  23644223231 0.165618465394951950670948192842
Using the Hardy-Littlewood prediction of the density of twin primes (see A347278), the contribution to the sum after the last entry in the table above can be estimated as 9.056*10^(-14), making the infinite sum ~= 0.16561846539504... . - Hugo Pfoertner, Sep 28 2022

			0.165618465395...

Jeffrey P.S. Lay, Sign changes in Mertens' first and second theorems, arXiv:1505.03589 [math.NT], 2015.
Mark B. Villarino, Mertens' Proof of Mertens' Theorem, arXiv:math/0504289 [math.HO], 2005.
Marek Wolf, Generalized Brun's constants, IFTUWr 910/97 (1998), 1-15.
Marek Wolf, Some heuristics on the gaps between consecutive primes, arXiv:1102.0481 [math.NT]. 2011.

Cf. A006512, A065421, A077800, A078437, A085548, A096247, A160910, A194098, A209328, A209329, A242301, A242302, A242303, A242304, A306539, A342714.

Cf. A347278.

Data extended to ...3, 9, 5 by Hugo Pfoertner, Sep 28 2022

A138329 List of strictly non-palindromic twin primes {p, p+2}.

Original entry on oeis.org

137, 139, 4337, 4339, 8291, 8293, 9419, 9421, 10937, 10939, 13757, 13759, 19427, 19429, 20981, 20983, 36011, 36013, 38327, 38329, 43397, 43399, 59441, 59443, 71327, 71329, 74717, 74719, 76871, 76873, 90437, 90439, 91571, 91573, 117239

Offset: 1

Karl Hovekamp, Mar 14 2008

The strictly non-palindromic twin primes are a part of the normal twin primes. See the list of twin primes A077800 and A016038 for the strictly non-palindromic numbers.

Karl Hovekamp, Palindromzahlen in adischen Zahlensystemen, 2004

K. Hovekamp, Palindromic numbers in different bases.
K. Hovekamp, Strictly non-palindromic prime twins up to 1 billion.
Wikipedia, Strictly non-palindromic numbers.

Cf. A016038, A047811, A016038, A077800, A099609.

Twin primes, where both numbers {p} and {p+2} are strictly non-palindromic.

A140445 List of prime pairs of form p, p + 10.

Original entry on oeis.org

3, 13, 7, 17, 13, 23, 19, 29, 31, 41, 37, 47, 43, 53, 61, 71, 73, 83, 79, 89, 97, 107, 103, 113, 127, 137, 139, 149, 157, 167, 163, 173, 181, 191, 223, 233, 229, 239, 241, 251, 271, 281, 283, 293, 307, 317, 337, 347, 349, 359, 373, 383, 379, 389, 409, 419, 421

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A023203 (1st bisection), A092146 (2nd bisection).

Cf. prime pairs of the form (p, p+k): A077800 (k=2), A094343 (k=4), A156274 (k=6), A156320 (k=8), this sequence (k=10), A156323 (k=12), A140446 (k=14), A272815 (k=16), A156328 (k=18), A272816 (k=20), A140447 (k=22).

i: 1: for k from 1 to 1200 do if isprim (k) and isprim (k+10) then a [ i ] : = k : a [ i + 1]: = k + 10 : i = i + 2 fi od : seq (a [ n ], n=1..i-1);

Flatten[{#,#+10}&/@Select[Prime[Range[100]],PrimeQ[#+10]&]]  (* Harvey P. Dale, Apr 11 2011 *)

Corrected by D. S. McNeil, Dec 10 2009

A176821 List of 4-tuples of twin primes q, p, p+2 and q+2 such that 2*q

Original entry on oeis.org

5, 11, 13, 7, 29, 59, 61, 31, 659, 1319, 1321, 661, 809, 1619, 1621, 811, 2129, 4259, 4261, 2131, 2549, 5099, 5101, 2551, 3329, 6659, 6661, 3331, 3389, 6779, 6781, 3391, 5849, 11699, 11701, 5851, 6269, 12539, 12541, 6271, 10529, 21059, 21061, 10531

Offset: 1

Juri-Stepan Gerasimov, Apr 26 2010, May 01 2010, May 07 2010

The first number q in each quadruplet is in A069142 (equivalent to selecting twin primes q which are also Sophie-Germain primes). [From R. J. Mathar, May 06 2010]

Cf. A077800, A045536.

Corrected (2131 replaced by 3331) by R. J. Mathar, May 06 2010

A182483 a(n) is the least m such that A182482(m) = A001359(n), the n-th twin prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 4, 17, 9, 23, 25, 15, 8, 11, 19, 20, 45, 47, 13, 29, 14, 24, 77, 87, 95, 50, 103, 107, 22, 27, 137, 46, 143, 21, 34, 43, 175, 59, 91, 48, 41, 71, 215, 31, 44, 119, 121, 247, 62, 67, 54, 139, 283, 287, 149, 39, 313, 161, 65, 37, 169, 347, 116

Offset: 2

Vladimir Shevelev, May 01 2012

nonn

a(n) exists for every n>=2.

Ray Chandler, Table of n, a(n) for n = 2..10001

Cf. A182481, A182482, A001359, A006512, A014574, A001097, A077800, A002822.

t = Table[k = 0; While[p = 6*k*n - 1; ! (PrimeQ[p] && PrimeQ[p + 2]), k++]; p, {n, 1000}]; tp = Select[Prime[Range[1000]], PrimeQ[# + 2] &]; t2 = {}; found = True; n = 2; While[found, pos = Position[t, tp[[n]], 1, 1]; If[pos == {}, found = False, AppendTo[t2, pos[[1, 1]]]; n++]]; t2 (* T. D. Noe, May 02 2012 *)

A232878 Twin prime pairs which sum to perfect squares.

Original entry on oeis.org

17, 19, 71, 73, 881, 883, 1151, 1153, 2591, 2593, 3527, 3529, 4049, 4051, 15137, 15139, 20807, 20809, 34847, 34849, 46817, 46819, 69191, 69193, 83231, 83233, 103967, 103969, 112337, 112339, 149057, 149059, 176417, 176419, 179999, 180001, 206081, 206083

Offset: 1

Gary Croft, Dec 01 2013

nonn

All square roots of twin prime sums in this sequence (see A152786) are multiples of 6.
Digital roots of all pairs in this sequence are {8,1}.
Twin primes of the form 18n^2 +- 1. - Charles R Greathouse IV, Aug 26 2014

			17+19 = 36, square root of 36 = 6; 71+73 = 144, square root of 144 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A001097, A054735, A069496, A077800.

t = {}; Do[ps = {2 n^2 - 1, 2 n^2 + 1}; If[PrimeQ[ps[[1]]] && PrimeQ[ps[[2]]], AppendTo[t, ps]], {n, 1000}]; Flatten[t] (* T. D. Noe, Dec 03 2013 *)

for(n=1,1e3, if(isprime(t=18*n^2-1) && isprime(t+2), print1(t", "t+2", "))) \\ Charles R Greathouse IV, Aug 26 2014

a(2*n) = a(2*n-1) + 2, a(2*n+1) = A069496(n).

A247856 Decimal expansion of the value of the continued fraction [0; 3, 5, 5, 7, 11, 13, 17, 19, ...] generated by twin primes.

Original entry on oeis.org

3, 1, 3, 2, 3, 3, 0, 8, 0, 9, 8, 6, 9, 4, 5, 9, 1, 2, 6, 3, 0, 7, 8, 6, 4, 8, 6, 4, 7, 2, 1, 7, 2, 8, 0, 0, 4, 3, 9, 2, 5, 1, 1, 7, 4, 5, 0, 5, 2, 2, 5, 3, 9, 7, 1, 7, 3, 2, 6, 8, 4, 8, 5, 4, 0, 4, 4, 3, 1, 4, 5, 9, 0, 8, 5, 0, 2, 0, 1, 0, 3, 6, 8, 3, 6, 9, 3, 8, 1, 0, 8, 9, 1, 2, 6, 2, 6, 7, 6, 9, 8

Offset: 0

Jean-François Alcover, Sep 25 2014

			0.313233080986945912630786486472172800439251174505225397...

Eric Weisstein's MathWorld, Twin Primes
Marek Wolf, Continued fractions constructed from prime numbers, arxiv.org/abs/1003.4015, p. 6.

Cf. A077800, A108170, A127552.

twinPrimes = Select[{#, #+2}& /@ Prime[Range[100]], PrimeQ[Last[#]]&] // Flatten; u2 = FromContinuedFraction[Join[{0}, twinPrimes]]; RealDigits[u2, 10, 101] // First

A256386 Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 59, 1319, 1619, 4259, 5099, 6659, 6779, 11699, 12539, 21059, 66359, 83219, 88259, 107099, 110879, 114659, 127679, 130199, 140759, 141959, 144539, 148199, 149519, 157559, 161339, 163859, 175079, 186479, 204599, 230939, 249539, 267959, 273899, 312839

Offset: 1

Alex Ratushnyak, Mar 31 2015

nonn

Indices of terms surrounded by pairs of zeros in A255361.
Conjectures:
1. A255361(a(n)) > 0 for n > 4.
2. All terms > 8 are primes.
3. All terms > 8 are terms of these supersequences: A118072, A171667, A176821, A181602, A181669.
From Lamine Ngom, Feb 12 2022: (Start)
For n > 4, a(n) is not a term of A254636. This means that a(n)-2, a(n)-1, a(n)+1 and a(n)+2 are adjacent terms in A254636.
Number of terms < 10^k: 5, 7, 7, 13, 19, 96, 441, 2552, ...
Conjecture 2 would follow if we establish the equivalence "t is in sequence" <=> "t is a term of b(n): lesser of twin primes pair p and q such that (p - 1)/2 and (q + 1)/2 are also a pair of twin primes (A077800)".
It appears that b(n) = a(n) for n > 5. Verified for all terms < 10^9. (End)

			9, 10, 12, 13 cannot be represented as x*y + x + y, where x >= y > 1. Therefore 11 is in the sequence.

Cf. A255361, A254636.
Cf. A118072, A171667, A176821, A181602, A181669.
Cf. A001097, A001359, A006512, A077800, A158870.

a(n) = A158870(n-5) - 2, n > 5 (conjectured). - Lamine Ngom, Feb 12 2022

A272815 Prime pairs of the form (p, p+16).

Original entry on oeis.org

3, 19, 7, 23, 13, 29, 31, 47, 37, 53, 43, 59, 67, 83, 73, 89, 97, 113, 151, 167, 157, 173, 163, 179, 181, 197, 211, 227, 223, 239, 241, 257, 277, 293, 331, 347, 337, 353, 367, 383, 373, 389, 433, 449, 463, 479, 487, 503, 541, 557, 547, 563, 571

Offset: 1

Vincenzo Librandi, May 07 2016

p and p+16 are not necessarily consecutive primes: (1831, 1847) is the first pair of consecutive primes that belongs to the sequence.

			The prime pairs are (3, 19), (7, 23), (13, 29) etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A049488.

Cf. prime pairs of the form (p, p+k): A077800 (k=2), A094343 (k=4), A156274 (k=6), A156320 (k=8), A140445 (k=10), A156323 (k=12), A140446 (k=14), this sequence (k=16), A156328 (k=18), A272816 (k=20), A140447 (k=22).

&cat [[p,p+16]: p in PrimesUpTo(1000) | IsPrime(p+16)];

Flatten[{#, # + 16}&/@Select[Prime[Range[200]], PrimeQ[# + 16] &]]

a(2n+1) = A049488(n+1).

A272816 Prime pairs of the form (p, p+20).

Original entry on oeis.org

3, 23, 11, 31, 17, 37, 23, 43, 41, 61, 47, 67, 53, 73, 59, 79, 83, 103, 89, 109, 107, 127, 131, 151, 137, 157, 173, 193, 179, 199, 191, 211, 251, 271, 257, 277, 263, 283, 293, 313, 311, 331, 317, 337, 347, 367, 353, 373, 359, 379, 389, 409, 401, 421

Offset: 1

Vincenzo Librandi, May 07 2016

p and p+20 are not necessarily consecutive primes: (887, 907) is the first pair of consecutive primes that belongs to the sequence.

			The prime pairs are (3, 23), (11, 31), (17, 37) etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A153419.
Cf. similar sequences listed in A272815.
Prime pairs of the form (p, p+k): A077800 (k=2), A094343 (k=4), A156274 (k=6), A156320 (k=8), A140445 (k=10), A156323 (k=12), A140446 (k=14), A272815 (k=16), A156328 (k=18), this sequence (k=20), A140447 (k=22).

&cat [[p, p+20]: p in PrimesUpTo(1000) | IsPrime(p+20)];

Flatten[{#, # + 20}&/@Select[Prime[Range[200]], PrimeQ[# + 20] &]]

from gmpy2 import is_prime
for n in range(1000):
   if(is_prime(n) and is_prime(n+20)):
      print('{}, {}'.format(n,n+20),end=', ')
# Soumil Mandal, May 14 2016

a(2n+1) = A153419(n+1).

Edited by Bruno Berselli, May 12 2016

cp's OEIS Frontend

A356793 Decimal expansion of sum of squares of reciprocals of lesser twin primes, Sum_{j>=1} 1/(A001359(j))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A138329 List of strictly non-palindromic twin primes {p, p+2}.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A140445 List of prime pairs of form p, p + 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A176821 List of 4-tuples of twin primes q, p, p+2 and q+2 such that 2*q

Views

Author

Keywords

Comments

Crossrefs

Extensions

A182483 a(n) is the least m such that A182482(m) = A001359(n), the n-th twin prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A232878 Twin prime pairs which sum to perfect squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A247856 Decimal expansion of the value of the continued fraction [0; 3, 5, 5, 7, 11, 13, 17, 19, ...] generated by twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A256386 Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A272815 Prime pairs of the form (p, p+16).

Views

Author

Keywords

Comments

Examples