A023509 -id:A023509 - 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

A103667 Primes p such that the largest prime divisor of p-1 is greater than the largest prime divisor of p+1.

Original entry on oeis.org

7, 11, 23, 29, 31, 47, 53, 59, 71, 79, 83, 89, 103, 107, 127, 131, 139, 149, 167, 173, 179, 191, 199, 223, 227, 233, 239, 263, 269, 293, 307, 311, 317, 347, 349, 359, 367, 373, 383, 389, 419, 431, 439, 449, 461, 467, 479, 499, 503, 509, 557, 563, 569, 571, 587

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

Primes of the form 2*A070087(n)+1 for some n. - Charles R Greathouse IV, Dec 22 2022

Conjecture: this sequence is of positive relative density in the primes, perhaps even 1/2. - Charles R Greathouse IV, Dec 22 2022

			a(1)=7 because the largest prime divisor of 6 is greater than the largest prime divisor of 8.

Karl Hovekamp, Table of n, a(n) for n = 1..39328

Cf. A023503 (greatest prime divisor of n-th prime - 1), A023509 (greatest prime divisor of n-th prime + 1), A103666, A070087.

filter:= p -> isprime(p) and max(numtheory:-factorset(p-1)) > max(numtheory:-factorset(p+1)):
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jan 15 2024

Select[Prime@Range[2, 107], If[FactorInteger[#-1][[-1, 1]]>FactorInteger[#+1][[-1, 1]], True]&] (* James C. McMahon, Jan 15 2024 *)

A077313 Primes of the form 2^r*5^s - 1.

Original entry on oeis.org

3, 7, 19, 31, 79, 127, 199, 499, 1249, 1279, 1999, 4999, 5119, 8191, 12799, 20479, 31249, 49999, 51199, 79999, 81919, 131071, 199999, 524287, 799999, 1249999, 1310719, 3124999, 3276799, 4999999, 7812499, 12499999, 19999999, 20479999

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

Primes p such that 10^p is divisible by p+1. Primes p whose fractions p/(p+1) are terminating decimals, i.e., primes p such that A158911(p)=0. Primes p such that the prime divisors of p+1 are also prime divisors of the numbers m obtained by the concatenation of p and p+1. For example, for p=19, m = 1920, the prime divisors of 20 are {2, 5} and the prime divisors of 1920 are {2, 3, 5}. - Jaroslav Krizek, Feb 25 2013

For n > 1, all terms are congruent to 1 (mod 6). - Muniru A Asiru, Sep 29 2017

			1250000 = 2*2*2*2*5*5*5*5*5*5*5 and 1250000 - 1 = A000040(96469), therefore 1249999 is a term.
List of (r, s): (2, 0), (3, 0), (2, 1), (5, 0), (4, 1), (7, 0), (3, 2), (2, 3), (1, 4), (8, 1), (4, 3), (3, 4), (10, 1), ...  - _Muniru A Asiru_, Sep 29 2017

Ray Chandler, Table of n, a(n) for n = 1..4222

Cf. A003592, A023509, A077497.

A:=Filtered([1..10^7],IsPrime);;    I:=[5];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A077313:=List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]); # Muniru A Asiru, Sep 29 2017

With[{n = 10^8}, Union@ Select[Flatten@ Table[2^p*5^q - 1, {p, 0, Log[2, n/(1)]}, {q, 0, Log[5, n/(2^p)]}], PrimeQ]] (* Michael De Vlieger, Sep 30 2017 *)

More terms from Reinhard Zumkeller, Nov 15 2002

More terms from Vladeta Jovovic, May 08 2003

A103666 Primes p such that the largest prime divisor of p-1 is less than the largest prime divisor of p+1.

Original entry on oeis.org

5, 13, 17, 19, 37, 41, 43, 61, 67, 73, 97, 101, 109, 113, 137, 151, 157, 163, 181, 193, 197, 211, 229, 241, 251, 257, 271, 277, 281, 283, 313, 331, 337, 353, 379, 397, 401, 409, 421, 433, 443, 457, 463, 487, 491, 521, 523, 541, 547, 577, 601, 613, 617, 631

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

			a(1)=5 because the largest prime divisor of 4 is less than the largest prime divisor of 6.

Karl Hovekamp, Table of n, a(n) for n=1..39168

Cf. A023503 greatest prime divisor of n-th prime - 1, A023509 greatest prime divisor of n-th prime + 1, A103667.

Select[Prime[Range[2,200]],Max[Transpose[FactorInteger[#-1]][[1]]]< Max[Transpose[FactorInteger[#+1]][[1]]]&]  (* Harvey P. Dale, Apr 26 2011 *)

A077314 Primes of the form 2^r*7^s - 1.

Original entry on oeis.org

3, 7, 13, 31, 97, 127, 223, 1567, 3583, 4801, 6271, 8191, 19207, 25087, 33613, 76831, 131071, 401407, 524287, 917503, 1229311, 1605631, 3764767, 6588343, 14680063, 184473631, 737894527, 2147483647, 2259801991, 2877292543, 3758096383

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

In all terms except 3, r is odd. - Robert Israel, Jun 08 2018

Robert Israel, Table of n, a(n) for n = 1..1736

Cf. A023509, A077313.

N:= 10^30: # for all terms <= N
sort(select(isprime, [3, seq(seq(2^r*7^s-1, r=1..ilog2((N+1)/7^s),2),
s=0..floor(log[7]((N+1)/2)))])); # Robert Israel, Jun 08 2018

Corrected and extended by Ray Chandler, Aug 02 2003

A077315 Primes of the form 2^r * 11^s - 1.

Original entry on oeis.org

3, 7, 31, 43, 127, 241, 967, 5323, 8191, 117127, 131071, 524287, 7496191, 10307263, 77948683, 253755391, 428717761, 738197503, 1714871047, 2147483647, 16240345087, 27437936767, 42218553343, 1965081755647, 2414538435583, 7024111812607, 7860327022591, 16630113370111

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..910

Cf. A023509, A077313, A077314, A173062.

lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/11; for(s=0, logint(nn\q, 11), p=11*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ Jinyuan Wang, Feb 23 2020

More terms from Ray Chandler, Aug 02 2003

a(26)-a(28) from Jinyuan Wang, Feb 23 2020

A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 7, 2, 1, 4, 1, 1, 1, 0, 1, 14, 1, 4, 3, 2, 1, 8, 2, 1, 5, 2, 1, 2, 1, 0, 2, 1, 1, 28, 1, 1, 1, 8, 1, 3, 1, 1, 3, 2, 1, 16, 1, 4, 1, 2, 1, 10, 1, 4, 3, 2, 1, 4, 1, 1, 1, 0, 1, 4, 1, 2, 1, 2, 1, 56, 1, 1, 6, 1, 3, 2, 1, 1, 47, 2, 1, 6, 3, 1, 1, 2, 1, 6, 5, 3, 2, 1, 1, 32, 1, 2, 1, 8, 1, 2, 1

Offset: 1

Vladimir Shevelev, Mar 31 2010

nonn

a(n)=1 if the index n is an odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A006530, A103667, A023503, A023509, A174857, A020639, A002618, A036689.

A006530 := proc(n) option remember; if n = 1 then 1; else max(op(numtheory[factorset](n)) ) ; end if; end proc:
A174869 := proc(n) if n <= 2 then 0; else gpf := A006530(n) ; if gpf = 2 then 0; else for k from 1 do if A006530(n+k) < gpf then return k; end if; end do: end if; end if; end proc:
seq(A174869(n),n=1..120) ; # R. J. Mathar, Aug 10 2010

Block[{s = Array[FactorInteger[#][[-1, 1]] &, 120]}, Array[If[IntegerQ@ Log2[#], 0, Block[{k = 1, n = s[[#]]}, While[n <= s[[# + k]], k++; If[# + k > Length[s], AppendTo[s, FactorInteger[# + k][[-1, 1]] ]] ]; k]] &, 102, 2]] (* Michael De Vlieger, Apr 06 2021 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A174869(n) = if(!bitand(n,n-1), 0, my(gpf=A006530(n)); for(k=1,oo,if(A006530(n+k)Antti Karttunen, Apr 06 2021

More terms from R. J. Mathar, Aug 10 2010

A023510 Greatest exponent in prime-power factorization of prime(n) + 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 2, 2, 3, 1, 5, 1, 1, 2, 4, 3, 2, 1, 2, 3, 1, 4, 2, 2, 2, 1, 3, 3, 1, 1, 7, 2, 1, 2, 2, 3, 1, 2, 3, 1, 2, 1, 6, 1, 2, 3, 2, 5, 2, 1, 2, 4, 2, 2, 1, 3, 3, 4, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 3, 4, 1, 2, 7, 1, 1, 1, 1, 2, 1, 4, 1, 3, 2, 2, 1, 1, 4, 2, 5, 3, 2, 3, 3, 1, 2, 2

Offset: 1

Clark Kimberling

nonn

If a(n)=1 then prime(a(n)) is a term in A049097. - Zak Seidov, Jul 20 2016

			For n=5, the fifth prime is 11, and the prime factorization of 11 + 1 = 12 is 2^2*3. This has exponents 2 and 1, so a(5) is the largest of these exponents, 2. - _Michael B. Porter_, Jul 20 2016

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A008864, A023509, A049097, A051903.

Table[Max[#[[2]] & /@ FactorInteger[Prime[k] + 1]], {k, 10000}] (* Zak Seidov, Jul 19 2016 *)

a(n) = vecmax(factor(prime(n)+1)[,2]) \\ Michel Marcus, Jul 20 2016

a(n) = A051903(A008864(n)). - Michel Marcus, Jul 20 2016

A076555 Greatest prime divisor of n-th prime + 2.

Original entry on oeis.org

2, 5, 7, 3, 13, 5, 19, 7, 5, 31, 11, 13, 43, 5, 7, 11, 61, 7, 23, 73, 5, 3, 17, 13, 11, 103, 7, 109, 37, 23, 43, 19, 139, 47, 151, 17, 53, 11, 13, 7, 181, 61, 193, 13, 199, 67, 71, 5, 229, 11, 47, 241, 3, 23, 37, 53, 271, 13, 31, 283, 19, 59, 103, 313, 7, 29, 37, 113, 349

Offset: 1

Zak Seidov, Oct 19 2002

Greatest prime divisor of n-th prime + 1 in A023509. Greatest prime divisor of n-th prime + 3 in A023576.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A023509, A023576.

FactorInteger[#][[-1,1]]&/@(Prime[Range[70]]+2) (* Harvey P. Dale, Sep 05 2014 *)

A379445 a(n) = gpf(prime(n)-1)*gpf(prime(n)+1), where gpf is A006530.

Original entry on oeis.org

4, 6, 6, 15, 21, 6, 15, 33, 35, 10, 57, 35, 77, 69, 39, 145, 155, 187, 21, 111, 65, 287, 55, 21, 85, 221, 159, 33, 133, 14, 143, 391, 161, 185, 95, 1027, 123, 581, 1247, 445, 65, 57, 291, 77, 55, 371, 259, 2147, 437, 377, 85, 55, 35, 86, 1441, 335, 85, 3197, 329, 3337

Offset: 2

Hugo Pfoertner, Dec 28 2024

nonn

Observation: Even terms of A006881 not occurring in this sequence are, e.g., 22, 34, 38, 46, ..., due to the sparseness of Mersenne primes (A000668) and Fermat primes (A000215). Also missing are many multiples of 3, e.g., 3*{31, 67, 79, 83, 101, 103, 113, ...}, as a consequence of the gaps of A058383 and A268640 and the size distribution of prime factors, i.e., the rareness of smooth numbers.

			a(43390) = 146 because 2^19-1 = A000668(5) is the 43390th prime and the greatest prime factor of 2^19-2 is 73.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000
Hugo Pfoertner, 1 million terms of A379445, 7z compressed file (5 MB) (2025).

Cf. A006530, A023503, A023509, A087713.
Each term > 4 is element of A006881.
Cf. A000215, A000668, A058383, A268640.

Table[Times @@ Map[FactorInteger[#][[-1, 1]] &, Prime[n] + {-1, 1}], {n, 2, 61}] (* Michael De Vlieger, Jan 20 2025 *)

a379445(n) = my (p=prime(n), fm=factor(p-1), fp=factor(p+1)); fm[#fm~,1]*fp[#fp~,1]

a(n) = A023503(n)*A023509(n). - Michel Marcus, Jan 21 2025

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

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A103667 Primes p such that the largest prime divisor of p-1 is greater than the largest prime divisor of p+1.

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A077313 Primes of the form 2^r*5^s - 1.

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A103666 Primes p such that the largest prime divisor of p-1 is less than the largest prime divisor of p+1.

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A077314 Primes of the form 2^r*7^s - 1.

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A077315 Primes of the form 2^r * 11^s - 1.

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A174869 a(n) is 0 if n is a power of 2, otherwise the smallest k > 0 such that A006530(n+k) < A006530(n).

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