A005382 -id:A005382 - cp's OEIS frontend

A038869 Primes p such that both p-2 and 2p-1 are prime.

7, 19, 31, 139, 199, 229, 271, 601, 619, 661, 811, 829, 1279, 1429, 1609, 2029, 2089, 2131, 2311, 2551, 2791, 3169, 3331, 3391, 3529, 3769, 4051, 4159, 4231, 4261, 4339, 4639, 4801, 5419, 5479, 5659, 5851, 6271, 6301, 6361, 6691, 6961, 7561, 7951, 8539

Offset: 1

Thomas Kellar

nonn

Primes p such that A(2*p) - 3*A(p) = 3 (7, 31, 661, 811, 2551, ...) and primes p such that 7*A(p) - A(2*p) = 21 (19, 139, 619, 1429, ...), where A=A288814, are both subsequences of A038869. - David James Sycamore, Aug 07 2017

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A005382, A118071.

[n: n in [0..10000]|IsPrime(n) and IsPrime(n-2) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 18 2010

Transpose[Select[Partition[Prime[Range[1200]],2,1],#[[2]]-#[[1]]==2 && PrimeQ[2#[[2]]-1]&]][[2]] (* Harvey P. Dale, Jun 19 2014 *)

is(n)=n%6==1 && isprime(n-2) && isprime(n) && isprime(2*n-1) \\ Charles R Greathouse IV, Aug 09 2017

A046969 Denominators of coefficients in Stirling's expansion for log(Gamma(z)).

Original entry on oeis.org

12, 360, 1260, 1680, 1188, 360360, 156, 122400, 244188, 125400, 5796, 1506960, 300, 93960, 2492028, 505920, 396, 2418179400, 444, 21106800, 3109932, 118680, 25380, 104700960, 6468, 324360, 2283876, 382800, 40356, 201025024200, 732

Offset: 1

Douglas Stoll, dougstoll(AT)email.msn.com

From Lorenzo Sauras Altuzarra, Oct 13 2020: (Start)
Conjecture I: if n > 2, then a(A005382(n))/12 is prime.
Conjecture II: if a(n)/12 is prime, then a(n-1)/12 - (n-1), a(n)/12 - n and a(n+2)/12 - (n+2) are multiples of 6. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
Thomas Bayes, A letter to John Canton, Phil. Trans. Royal Society London, 53 (1763), 269-271.
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1.
C. Impens, Stirling's series made easy, Am. Math. Monthly, 110 (No. 8, 2003), pp. 730-735.
Gergő Nemes, Generalization of Binet's Gamma function formulas, Integral Transforms and Special Functions, 24:8, pp. 597-606, 2013.
Eric Weisstein's World of Mathematics, Stirling's Series

Numerators are given in A046968. Cf. A005382.

a := n -> denom(bernoulli(2*n)/(2*n*(2*n-1))): # Lorenzo Sauras Altuzarra, Oct 13 2020

Table[ Denominator[ BernoulliB[2n]/(2n(2n - 1))], {n, 31}] (* Robert G. Wilson v, Sep 21 2006 *)
s = LogGamma[z] + z - (z - 1/2) Log[z] - Log[2 Pi]/2 + O[z, Infinity]^62;
DeleteCases[CoefficientList[s, 1/z], 0] // Denominator (* Jean-François Alcover, Jun 13 2017 *)

a(n)=if(n<1,0,denominator(bernfrac(2*n)/(2*n)/(2*n-1)))

From denominator of Jk(z) = (-1)^(k-1)*Bk/(((2k)*(2k-1))*z^(2k-1)), so Gamma(z) = sqrt(2pi)*z^(z-0.5)*exp(-z)*exp(J(z)).

More terms from Frank Ellermann, Jun 13 2001

Bayes reference from Henry Bottomley, Jun 03 2003

A053177 Odd composite k such that (k-1)/2 is prime.

Original entry on oeis.org

15, 27, 35, 39, 63, 75, 87, 95, 119, 123, 135, 143, 147, 159, 195, 203, 207, 215, 219, 255, 275, 279, 299, 303, 315, 327, 335, 363, 387, 395, 399, 423, 447, 455, 459, 483, 515, 527, 539, 543, 555, 567, 615, 623, 627, 635, 663, 675, 695, 699, 707, 735, 747

Offset: 1

Enoch Haga, Feb 29 2000

Composite numbers produced in A053176.

			a(3)=35 and 35-1=34, 34/2=17, prime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
R. P. Boas & N. J. A. Sloane, Correspondence, 1974

Cf. A053176, A005385, A005382, A259978.

Select[2 Prime@ Range@ 74 + 1, CompositeQ] (* Michael De Vlieger, Jul 13 2015 *)
Select[Range[1,801,2],CompositeQ[#]&&PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Apr 01 2019 *)

main(size)={my(v=vector(size),i,t=1);for(i=1, size, while(isprime(2*prime(t)+1), t++); v[i]=2*prime(t)+1;t++;);return(v)} /* Anders Hellström, Jul 13 2015 */

From the composite, subtract 1, divide by 2 and result is a prime.

Definition clarified by Peter Munn, Oct 26 2017

A053685 Primes p > 7 which are congruent to 2 or 4 (mod 5) for which 2p-1 is also prime.

Original entry on oeis.org

19, 37, 79, 97, 139, 157, 199, 229, 307, 337, 367, 379, 439, 499, 547, 577, 607, 619, 727, 829, 877, 937, 967, 997, 1009, 1069, 1237, 1279, 1297, 1399, 1429, 1459, 1609, 1627, 1657, 1759, 1867, 2029, 2089, 2137, 2179, 2467, 2539, 2557, 2617, 2707, 2719

Offset: 1

James Sellers, Feb 15 2000

For such primes p, 2p-1 divides Fibonacci(p). Actually it is also true that (2m-1) divides Fibonacci(m) for *all* m > 7, m = 2 or 4 (mod 5) for which 2m-1 is prime.

Intersection of A047211 and A005382 without terms <= 7. - Reinhard Zumkeller, Oct 03 2012

			Note that 19 is prime and so is 2*19-1 or 37.

T. D. Noe, Table of n, a(n) for n = 1..1000
Vladimir Drobot, On primes in the Fibonacci sequence, Fib. Quart. 38 (1) (2000) 71.

Cf. A000045.

a053685 n = a053685_list !! (n-1)
a053685_list = dropWhile (<= 7) $ i a047211_list a005382_list where
   i xs'@(x:xs) ys'@(y:ys) | x < y     = i xs ys'
                           | x > y     = i xs' ys
                           | otherwise = x : i xs ys
-- Reinhard Zumkeller, Oct 03 2012

okQ[n_]:=Module[{x=Mod[n,5]},PrimeQ[2n-1]&&MemberQ[{2,4},x]]; Select[Prime[Range[5,500]],okQ]  (* Harvey P. Dale, Jan 14 2011 *)

A064812 Smallest prime p such that the infinite sequence {p, p'=2p-1, p''=2p'-1, ...} begins with a string of exactly n primes.

Original entry on oeis.org

5, 3, 2, 2131, 1531, 33301, 16651, 15514861, 857095381, 205528443121, 1389122693971, 216857744866621, 758083947856951, 107588900851484911, 69257563144280941

Offset: 1

David Terr, Oct 21 2002

nonn

Chains of length n of nearly doubled primes.

Smallest prime beginning a complete Cunningham chain of length n of the second kind. (For the first kind see A005602.) - Jonathan Sondow, Oct 30 2015

			a(3) = 2 because 2 is the smallest prime such that the sequence {2, 3, 5, 9, ...} begins with exactly 3 primes, where each term in the sequence is twice the preceding term minus 1.

G. Löh, Long chains of nearly doubled primes, Math. Comp., 53 (1989), 751-759.
Wikipedia, Cunningham chain

A better version of A005603.
Cf. (A005382 and A005383), A057326, A057327, A057328, A057329, A057330, A005602.
Cf. A002808, A006450, A049076-A049081, A050436, A076237-A076239, A050436-A050439, A065860, A181697, A263879.

A110059 Smallest prime ending a complete Cunningham chain of the second kind (2x-1) of length n.

Original entry on oeis.org

11, 13, 5, 17041, 24481, 12338881, 1065601, 1985902081, 219416417281, 105230562877441, 1422461638625281, 444124661486837761, 3105111850422067201

Offset: 1

Alexandre Wajnberg, Sep 04 2005

nonn

"Complete" means that the chain is not part of a longer chain.

A005603 has the first prime of each chain.

			a(4)=17041 because 2131,4261,8521,17041 are prime, but the preceding and following numbers (1066,34081) are not.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A005382, A005384, A057326, A059761, A005603.

Some terms computed by Gilles Sadowski.

Edited by Don Reble, May 16 2006

A110581 Primes p such that 2p-1 is prime, but 4p-3 is not prime.

Original entry on oeis.org

3, 7, 31, 37, 97, 139, 157, 199, 211, 229, 271, 307, 337, 367, 379, 547, 577, 601, 607, 661, 691, 727, 811, 877, 937, 967, 997, 1009, 1171, 1237, 1297, 1399, 1429, 1459, 1609, 1627, 1657, 1759, 1867, 2011, 2029, 2137, 2221, 2281, 2467, 2539, 2551, 2557

Offset: 1

T. D. Noe, Jul 28 2005

A subsequence of A005382, primes p such that 2p-1 is also prime. Note that for all n, 2a(n)-1 is not in this sequence.

T. D. Noe, Table of n, a(n) for n=1..1000
Jeffery J. Holt, The minimal number of solutions to phi(n)=phi(n+k), Math. Comp., 72 (2003), 2059-2061.

[p: p in PrimesUpTo(2600) | IsPrime(2*p-1) and not IsPrime(4*p-3)]; // Vincenzo Librandi, Apr 14 2013

Select[Prime[Range[2, 1000]], PrimeQ[2#-1] && !PrimeQ[4#-3]&]

A117961 Hexagonal numbers with prime indices.

Original entry on oeis.org

6, 15, 45, 91, 231, 325, 561, 703, 1035, 1653, 1891, 2701, 3321, 3655, 4371, 5565, 6903, 7381, 8911, 10011, 10585, 12403, 13695, 15753, 18721, 20301, 21115, 22791, 23653, 25425, 32131, 34191, 37401, 38503, 44253, 45451, 49141, 52975, 55611

Offset: 1

Jonathan Vos Post, Apr 05 2006

See also: A034953 Triangular numbers (A000217) with prime indices. A001248 Squares of primes. A116995 Pentagonal numbers with prime indices. A000384 Hexagonal numbers: n(2n-1). There are no prime hexagonal numbers. The n-th Hexagonal number A000384(n) = n*(2*n-1) is semiprime iff both n and 2*n-1 are prime iff A000384(n) is an element of A001358 iff n is an element of A005382.

Cf. A000005, A000040, A000384, A001021, A034953, A001248, A116995.

With[{hex=Table[n(2n-1),{n,250}]},Flatten[Table[Take[hex,{Prime[n]}],{n, 40}]]] (* Harvey P. Dale, Dec 04 2011 *)

a(n) = A000040(n)*(2*A000040(n)-1). a(n) = A000384(prime(n)). a(n) = number of divisors of 12^(prime(n)-1) = A000005(A001021(A000040(n)-1)).

A125294 Numerator of (Sum_{k=1..n} k^2) / (Product_{k=1..n} k^2).

Original entry on oeis.org

1, 5, 7, 5, 11, 91, 1, 17, 19, 11, 23, 13, 1, 29, 31, 17, 1, 703, 1, 41, 43, 23, 47, 1, 1, 53, 1, 29, 59, 1891, 1, 1, 67, 1, 71, 2701, 1, 1, 79, 41, 83, 43, 1, 89, 1, 47, 1, 97, 1, 101, 103, 53, 107, 109, 1, 113, 1, 59, 1, 61, 1, 1, 127, 1, 131, 67, 1, 137, 139, 71, 1, 73, 1, 149

Offset: 1

Alexander Adamchuk, Jan 17 2007

All a(n) are either 1, semiprime or prime.
a(n) = 1 for n = 1 and n = {7, 13, 17, 19, 24, 25, 27, 31, 32, 34, 37, 38, 43, 45, 47, 49, ...} = A067656 = numbers n such that n!*B(2*n) is an integer, where the B(2*n)'s are the Bernoulli numbers.
p divides a(p-1) for prime p > 3. p divides a((p-1)/2) for prime p > 3.
a(p-1) = p*(2p-1) is a semiprime hexagonal number for prime p = {7, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, ...} = A005382(n) for n > 2, where A005382(n) are the numbers n such that n and 2*n-1 are primes.
a(p-1) = p for prime p = {5, 11, 13, 17, 23, 29, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, ...} = primes that do not belong to A005382(n).
a((p-1)/2) = p for prime p = {5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 259, 271, 281, 283, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 401, ...}, which is apparently the union of {5} and A034849(n).

			The first few fractions are 1, 5/4, 7/18, 5/96, 11/2880, 91/518400, 1/181440, 17/135475200, 19/8778792960, ... = A125294/A334735. - _Petros Hadjicostas_, May 09 2020

Cf. A005382, A034849, A067656, A166602, A334735 (denominators).

Table[Numerator[n(n+1)(2n+1)/6/(n!)^2],{n,1,500}]

a(n) = numerator(sum(k=1, n, k^2)/prod(k=1, n, k^2)); \\ Michel Marcus, May 09 2020

a(n) = numerator((Sum_{k=1..n} k^2) / (Product_{k=1..n} k^2)).

a(n) = numerator(n*(n+1)*(2*n+1)/6/(n!)^2).

A158016 Primes p such that 8*p-1 is also prime.

Original entry on oeis.org

3, 13, 19, 61, 79, 103, 163, 181, 193, 223, 229, 313, 331, 349, 409, 433, 439, 541, 571, 613, 619, 691, 751, 769, 853, 859, 919, 991, 1021, 1033, 1039, 1321, 1381, 1423, 1483, 1543, 1549, 1621, 1699, 1759, 1801, 1861, 1873, 1879, 1951, 1999, 2011, 2029, 2113

Offset: 1

Roger L. Bagula, Mar 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005382 for the type 2p-1, A062737 for 4p-1, A158015 for 6p-1, A158017 for 10p-1.

[p: p in PrimesUpTo(2200) | IsPrime(8*p - 1)]; // Vincenzo Librandi, Apr 14 2013

Select[Prime[Range[600]], PrimeQ[(8 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

Edited by the Associate Editors of the OEIS, Apr 22 2009

cp's OEIS Frontend

A038869 Primes p such that both p-2 and 2p-1 are prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A046969 Denominators of coefficients in Stirling's expansion for log(Gamma(z)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A053177 Odd composite k such that (k-1)/2 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A053685 Primes p > 7 which are congruent to 2 or 4 (mod 5) for which 2p-1 is also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A064812 Smallest prime p such that the infinite sequence {p, p'=2p-1, p''=2p'-1, ...} begins with a string of exactly n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A110059 Smallest prime ending a complete Cunningham chain of the second kind (2x-1) of length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A110581 Primes p such that 2p-1 is prime, but 4p-3 is not prime.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

A117961 Hexagonal numbers with prime indices.

Views

Author

Keywords