A053685 -id:A053685 - cp's OEIS frontend

A005382 Primes p such that 2p-1 is also prime.

2, 3, 7, 19, 31, 37, 79, 97, 139, 157, 199, 211, 229, 271, 307, 331, 337, 367, 379, 439, 499, 547, 577, 601, 607, 619, 661, 691, 727, 811, 829, 877, 937, 967, 997, 1009, 1069, 1171, 1237, 1279, 1297, 1399, 1429, 1459, 1531, 1609, 1627, 1657, 1759, 1867, 2011

Offset: 1

N. J. A. Sloane

Sequence gives values of p such Sum_{i=1..p} gcd(p,i) = A018804(p) is prime. - Benoit Cloitre, Jan 25 2002
Let q = 2n-1. For these n (and q), the sum of two cyclotomic polynomials can be written as a product of cyclotomic polynomials and as a cyclotomic polynomial in x^2: Phi(q,x) + Phi(2q,x) = 2 Phi(n,x) Phi(2n,x) = 2 Phi(n,x^2). - T. D. Noe, Nov 04 2003
Primes in A006254. - Zak Seidov, Mar 26 2013
If a(n) is in A168421 then A005383(n) is a twin prime with a Ramanujan prime, A005383(n) - 2. If this sequence has an infinite number of terms in A168421, then the twin prime conjecture can be proved. - John W. Nicholson, Dec 05 2013
Records subsequence of A023509 (n >= 2). - David James Sycamore, May 05 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Ajeet Kumar, Subhamoy Maitra, and Chandra Sekhar Mukherjee, On approximate real mutually unbiased bases in square dimension, Cryptography and Communications (2020) Vol. 13, 321-329.
Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025. See p. 8.
Marius Tărnăuceanu, Arithmetic progressions in finite groups, arXiv:2003.10060 [math.GR], 2020.
Wikipedia, Cunningham chain

Cf. A005383, A005384 (2p+1), A057326, A057327, A057328, A057329, A057330, A005603, A063908 (2p-3), A063909 (2p-5), A023204 (2p+3), A000384, A001358.
Cf. A010051, A000040, A053685 (subsequence), A006254.
Cf. A023509.

a005382 n = a005382_list !! (n-1)
a005382_list = filter
   ((== 1) . a010051 . (subtract 1) . (* 2)) a000040_list
-- Reinhard Zumkeller, Oct 03 2012

[n: n in [0..1000] | IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Nov 18 2010

f := proc(Q) local t1,i,j; t1 := []; for i from 1 to 500 do j := ithprime(i); if isprime(2*j-Q) then t1 := [op(t1),j]; fi; od: t1; end; f(1);
# second Maple program:
q:= p-> andmap(isprime, [p, 2*p-1]):
select(q, [$2..2500])[];  # Alois P. Heinz, Dec 16 2024

Select[Prime[Range[300]], PrimeQ[2#-1]&]

select(p->isprime(2*p-1),primes(500)) \\ Charles R Greathouse IV, Apr 26 2012

forprime(n=2, 10^3, if(ispseudoprime(2*n-1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

a(n) = A129521(n) / A005383(n). - Reinhard Zumkeller, Apr 19 2007

a(n) = (A005383(n) + 1)/2. - Zak Seidov, Nov 04 2010

A047211 Numbers that are congruent to {2, 4} mod 5.

Original entry on oeis.org

2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 27, 29, 32, 34, 37, 39, 42, 44, 47, 49, 52, 54, 57, 59, 62, 64, 67, 69, 72, 74, 77, 79, 82, 84, 87, 89, 92, 94, 97, 99, 102, 104, 107, 109, 112, 114, 117, 119, 122, 124, 127, 129, 132, 134, 137, 139, 142, 144, 147, 149, 152, 154, 157, 159, 162, 164, 167, 169, 172, 174, 177, 179, 182, 184

Offset: 1

N. J. A. Sloane, Dec 11 1999

Conjecture: n such that the characteristic polynomial of M(n) is irreducible over the rationals where M(n) is an n X n matrix with ones on the skew diagonal and below it and the skew line two positions above it and otherwise zeros; see example for one such matrix. Tested up to n=177. - Joerg Arndt, Aug 10 2011

			The 7 X 7 matrix (dots for zeros):
[....1.1]
[...1.11]
[..1.111]
[.1.1111]
[1.11111]
[.111111]
[1111111]
has the characteristic polynomial x^7 - 5*x^6 - 4*x^5 + 15*x^4 + 5*x^3 - 11*x^2 - x + 1 which is irreducible over the field of rational numbers, and 7 is a term of the sequence. - _Joerg Arndt_, Aug 10 2011

David Lovler, Table of n, a(n) for n = 1..1000
Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A047209.

Cf. A053685 (subsequence).

a047211 n = a047211_list !! (n-1)
a047211_list = filter ((`elem` [2,4]) . (`mod` 5)) [1..]
-- Reinhard Zumkeller, Oct 03 2012

[Floor((5*n-1)/2): n in [1..50]]; // Wesley Ivan Hurt, May 25 2014

seq(5*floor((n-1)/2) +3 +(-1)^n, n=1..50); # Gary Detlefs, Mar 02 2010

Select[Range[0, 200], MemberQ[{2, 4}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)
LinearRecurrence[{1,1,-1},{2,4,7},80] (* Harvey P. Dale, Mar 26 2024 *)

a(n)=(5*n-1)\2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = a(n-1) +a(n-2) -a(n-3).
a(n) = (10*n-3-(-1)^n)/4, (n>=1). [Corrected by Bruno Berselli, Sep 20 2010]
a(n) = 5*floor((n-1)/2) +3 +(-1)^n. - Gary Detlefs, Mar 02 2010
G.f.: x*(2+2*x+x^2)/((1+x)*(1-x)^2). - Paul Barry, Sep 11 2008
a(n) = 5*n-a(n-1)-4 (with a(1)=2). - Vincenzo Librandi, Nov 18 2010
a(n) = floor((5*n-1)/2). - Gary Detlefs, May 14 2011
a(n) = 2*n + floor((n-1)/2). - Arkadiusz Wesolowski, Sep 19 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2+2/sqrt(5))*Pi/10 - sqrt(5)*log(phi)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: 1 + ((10*x - 3)*exp(x) - exp(-x))/4. - David Lovler, Aug 23 2022

Conjecture corrected by John M. Campbell, Aug 25 2011

A053635 a(n) = Sum_{d|n} phi(d)*2^(n/d).

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 84, 140, 288, 540, 1080, 2068, 4224, 8216, 16548, 32880, 65856, 131104, 262836, 524324, 1049760, 2097480, 4196412, 8388652, 16782048, 33554600, 67117128, 134218836, 268452240, 536870968, 1073777040, 2147483708, 4295033472, 8589938808

Offset: 0

N. J. A. Sloane, Mar 23 2000

nonn

Dirichlet convolution of phi(n) and 2^n. - Richard L. Ollerton, May 06 2021

Seiichi Manyama, Table of n, a(n) for n = 0..3321
James East and Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
James East and Ron Niles, Integer polygons of given perimeter, Bull. Aust. Math. Soc. 100(1) (2019), 131-147.
T. Pisanski, D. Schattschneider, and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180. See v(n).

Cf. A000031, A053634, A053636. Twice A034738.

Column k=2 of A185651.

[0] cat  [&+[EulerPhi(d)*2^(n div d): d in Divisors(n)]: n in [1..40]]; // Vincenzo Librandi, Jul 20 2019

with(numtheory); A053685:=n->add( phi(n/d)*2^d, d in divisors(n)); # N. J. A. Sloane, Nov 21 2009

a[0] = 0; a[n_] := Sum[EulerPhi[d] 2^(n/d), {d, Divisors[n]}];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Aug 30 2018 *)

a(n) = if (n, sumdiv(n, d, eulerphi(d)*2^(n/d)), 0); \\ Michel Marcus, Sep 20 2017

a(n) = n * A000031(n).
a(n) = Sum_{k=1..n} 2^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021
a(n) = Sum_{k=1..n} 2^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021

cp's OEIS Frontend

A005382 Primes p such that 2p-1 is also prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Formula

A047211 Numbers that are congruent to {2, 4} mod 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A053635 a(n) = Sum_{d|n} phi(d)*2^(n/d).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula