A129521 -id:A129521 - cp's OEIS frontend

A191592 Numbers in A191311 but not in A129521.

4, 11305, 13981, 23001, 30889, 39865, 68101, 88561, 91001, 93961, 107185, 137149, 152551, 157641, 176149, 204001, 228241, 251251, 276013, 401401, 464185, 493697, 493885, 534061, 563473, 574561, 622909, 631351, 683761, 786961, 915981, 950797, 1106785, 1141141

Offset: 1

Jason Holt, Jun 04 2011

nonn

David W. Wilson, Table of n, a(n) for n = 1..697

Cf. A191311, A129521.

More terms from David W. Wilson, Aug 16 2011

A351875 Positive difference of the distinct prime factors of A129521(n).

Original entry on oeis.org

1, 2, 6, 18, 30, 36, 78, 96, 138, 156, 198, 210, 228, 270, 306, 330, 336, 366, 378, 438, 498, 546, 576, 600, 606, 618, 660, 690, 726, 810, 828, 876, 936, 966, 996, 1008, 1068, 1170, 1236, 1278, 1296, 1398, 1428, 1458, 1530, 1608, 1626, 1656, 1758, 1866, 2010, 2028, 2088, 2130

Offset: 1

Wesley Ivan Hurt, Feb 22 2022

nonn

			a(4) = 18; A005383(4) - A005382(4) = 37 - 19 = 18.

Cf. A005382, A005383, A129521.

Select[Prime[Range[350]], PrimeQ[2*# - 1] &] - 1 (* Amiram Eldar, Feb 23 2022 *)

a(n) = A005383(n) - A005382(n).

a(n) = A005382(n) - 1. - Amiram Eldar, Feb 23 2022

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

Original entry on oeis.org

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

A005383 Primes p such that (p+1)/2 is prime.

Original entry on oeis.org

3, 5, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253

Offset: 1

N. J. A. Sloane

Also, n such that sigma(n)/2 is prime. - Joseph L. Pe, Dec 10 2001; confirmed by Vladeta Jovovic, Dec 12 2002
Primes that are followed by twice a prime, i.e., are followed by a semiprime. (For primes followed by two semiprimes, see A036570.) - Zak Seidov, Aug 03 2013, Dec 31 2015
If A005382(n) is in A168421 then a(n) is a twin prime with a Ramanujan prime, A104272(k) = a(n) - 2. - John W. Nicholson, Jan 07 2016
Starting with 13 all terms are congruent to 1 mod 12. - Zak Seidov, Feb 16 2017
Numbers n such that both n and n+12 are terms are 61, 661, 1201, 4261, 5101, 6121, 6361 (all congruent to 1 mod 60). - Zak Seidov, Mar 16 2017
Primes p for which there exists a prime q < p such that 2q == 1 (mod p). Proof: q = (p + 1)/2. - David James Sycamore, Nov 10 2018
Prime numbers n such that phi(sigma(2n)) = phi(2n), excluding n=3 and n=5; as well as phi(sigma(3n)) = phi(3n), excluding n=3 only. - Richard R. Forberg, Dec 22 2020

			Both 3 and (3+1)/2 = 2 are primes, both 5 and (5+1)/2 = 3 are primes. - _Zak Seidov_, Nov 19 2012

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 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Benoit Cloitre, On the fractal behavior of primes, 2011.

Cf. A005382, A057326, A057327, A057328, A057329, A057330, A005603.
A subsequence of A000040 which has A036570 as subsequence.
Cf. A005385, A010051, A065091, A048161, A036570.

a005383 n = a005383_list !! (n-1)
a005383_list = [p | p <- a065091_list, a010051 ((p + 1) `div` 2) == 1]
-- Reinhard Zumkeller, Nov 06 2012

LIMIT = 8000 % Find all members of A005383 less than LIMIT A = primes(LIMIT); n = length(A); %n is number of primes less than LIMIT B = 2*A - 1; C = ones(n, 1)*A; %C is an n X n matrix, with C(i, j) = j-th prime D = B'*ones(1, n); %D is an n X n matrix, with D(i, j) = (i-th prime)*2 - 1 [i, j] = find(C == D); A(j)

[n: n in [1..3300] | IsPrime(n) and IsPrime((n+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012

for n to 300 do
  X := ithprime(n);
Y := ithprime(n+1);
Z := 1/2 mod Y;
  if isprime(Z) then print(Y);
end if:
end do:
# David James Sycamore, Nov 11 2018

Select[Prime[Range[1000]], PrimeQ[(# + 1)/2] &] (* Zak Seidov, Nov 19 2012 *)

A005383_list(n) = select(m->isprime(m\2+1),primes(n)[2..n]) \\ Charles R Greathouse IV, Sep 25 2012

from sympy import isprime
[n for n in range(3, 5000) if isprime(n) and isprime((n + 1)//2)]
# Indranil Ghosh, Mar 17 2017

[n for n in prime_range(3, 1000) if is_prime((n + 1) // 2)]
# F. Chapoton, Dec 17 2019

a(n) = A129521(n)/A005382(n). - Reinhard Zumkeller, Apr 19 2007
A000035(a(n))*A010051(a(n))*A010051((a(n)+1)/2) = 1. - Reinhard Zumkeller, Nov 06 2012
a(n) = 2*A005382(n) - 1. - Zak Seidov, Nov 19 2012
a(n) = A005382(n) + phi(A005382(n)) = A005382(n) + A000010(A005382(n)). - Torlach Rush, Mar 10 2014

More terms from David Wasserman, Jan 18 2002

Name changed by Jianing Song, Nov 27 2021

A020136 Fermat pseudoprimes to base 4.

Original entry on oeis.org

15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071, 2465, 2701, 2821, 3133, 3277, 3367, 3683, 4033, 4369, 4371, 4681, 4795, 4859, 5461, 5551, 6601, 6643, 7957, 8321, 8481, 8695, 8911, 9061, 9131

Offset: 1

David W. Wilson

nonn

If q and 2q-1 are odd primes, then n=q*(2q-1) is in the sequence. So for n>1, A005382(n)*(2*A005382(n)-1) form a subsequence (cf. A129521). - Farideh Firoozbakht, Sep 12 2006
Primes q and 2q-1 are a Cunningham chain of the second kind. - Walter Nissen, Sep 07 2009
Composite numbers n such that 4^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Chris Caldwell, Cunningham chain.
Chris Caldwell et al., Top Twenty Cunningham Chains (2nd kind).
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes

Subsequence of A122781.
Contains A001567 (Fermat pseudoprimes to base 2) as a subsequence.
Cf. A005382, A129521.

Select[Range[9200], ! PrimeQ[ # ] && PowerMod[4, # - 1, # ] == 1 &] (* Farideh Firoozbakht, Sep 12 2006 *)

isok(n) = (Mod(4, n)^(n-1)==1) && !isprime(n) && (n>1); \\ Michel Marcus, Apr 27 2018

A122781 Nonprimes n such that 4^n==4 (mod n).

Original entry on oeis.org

1, 4, 6, 12, 15, 28, 66, 85, 91, 186, 276, 341, 435, 451, 532, 561, 645, 703, 946, 1068, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2044, 2046, 2047, 2071, 2465, 2701, 2821, 2926, 3133, 3277, 3367, 3683, 4033, 4369, 4371, 4681, 4795

Offset: 1

Farideh Firoozbakht, Sep 12 2006

If both numbers q and 2q-1 are prime, then q*(2q-1) is in the sequence. So, A005382(n)*(2*A005382(n)-1) = A129521(n) form a subsequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Contains A020136, A001567, A006935 (except n=2), and A129521 as subsequences.

Cf. A005382.

for n from 1 to 5000 do if 4^n mod n = 4 mod n and not isprime(n) then print(n) fi od; # Gary Detlefs, May 14 2012

Select[Range[4800], ! PrimeQ[ # ] && Mod[4^#, # ] == Mod[4, # ] &]
Join[{1,4},Select[Range[5000],!PrimeQ[#]&&PowerMod[4,#,#]==4&]] (* Harvey P. Dale, Apr 09 2018 *)

A129512 Numbers with at least two pairs of distinct divisors having equal differences.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 138, 140, 144, 150, 153, 156, 160, 162, 165, 168, 174, 176, 180, 182, 186, 189, 190, 192, 195

Offset: 1

Reinhard Zumkeller, Apr 19 2007

nonn

			See example for a(12) = 45 in A129510.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A129510, A066446, A129511 (complement).

Subsequences: A008588, A008597, A129521, A259366.

import Data.List.Ordered (minus)
a129512 n = a129512_list !! (n-1)
a129512_list = minus [1..] a129511_list
-- Reinhard Zumkeller, Aug 10 2015

q[k_] := Count[Tally[Differences /@ Subsets[Divisors[k], {2}] // Flatten][[;; , 2]], ?(# > 1 &)] > 0; Select[Range[200], q] (* _Amiram Eldar, Jan 27 2025 *)

is(n)=my(d=divisors(n)); for(i=1,#d-2, for(j=i+1,#d-1, for(k=1,#d, if(i!=k && setsearch(d, d[j]-d[i]+d[k]), return(1))))); 0 \\ Charles R Greathouse IV, Aug 26 2015

A129510(a(n)) < A066446(a(n)).

A191311 Numbers n such that exactly half of the a such that 0

Original entry on oeis.org

4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, 23001, 30889, 38503, 39865, 49141, 68101, 79003, 88561, 88831, 91001, 93961, 104653, 107185, 137149, 146611, 152551, 157641, 176149, 188191, 204001, 218791, 226801, 228241

Offset: 1

Jason Holt, Jun 04 2011

Values of n for which half the witnesses in the Fermat primality test are false.
When n=pq with p,q=2p-1 prime, a^(n-1) = 1 (mod p) iff a is a quadratic residue mod q. So A129521 is a subsequence. - Gareth McCaughan, Jun 05 2011
From Robert G. Wilson v, Aug 13 2011: (Start)
Number of terms less than 10^n: 2, 4, 5, 7, 22, 60, 129, 303, 690, 1785, …, .
In reference to the numbers in the b-file: (1) number of terms which have k>0 prime factors: 1, 1058, 139, 512, 339, 102, 6; (2) about half of the terms, 1058, are members of A129521, those which have just two prime factors; (3) except for the first term, all terms are squarefree, except for the first two terms, all terms are odd; and (4) most terms, more than 98.5%, are congruent to 1 modulo 6. (End)

David W. Wilson and Robert G. Wilson v, Table of n, a(n) for n = 1..2157
While exploring Carmichael numbers, I noticed a few values on the chart on this page for which exactly half of the relatively prime witnesses to the Fermat primality test were false witnesses.

A063994 gives the number of false witnesses for every n.

A129521 is a subsequence. See also A191592.

fQ[n_] := Block[{pf = First /@ FactorInteger@ n}, 2Times @@ GCD[n - 1, pf - 1] == n*Times @@ (1 - 1/pf)]; Select[ Range@ 250000, fQ] (* Robert G. Wilson v, Aug 08 2011 *)

import math
for x in range(2, 1000):
  false_witnesses = 0
  relatively_prime_values = 0
  for y in range(x):
    if math.gcd(y, x) == 1:
      relatively_prime_values += 1
    if (pow(y, x-1, x) == 1):
      false_witnesses += 1
  if false_witnesses * 2 == relatively_prime_values:
    print(x, "is a Fermat Half-Prime")

Integers, n, such that A063994(n) = 2*A000010(n). - Robert G. Wilson v, Aug 13 2011

Edited by N. J. A. Sloane, Jun 07 2011. I made use of a more explicit definition due to Gareth McCaughan, Jun 05 2011.

A259677 Octagonal numbers (A000567) that are semiprimes (A001358).

Original entry on oeis.org

21, 65, 133, 341, 481, 1541, 4033, 5461, 6533, 8321, 11041, 13333, 14981, 31621, 38081, 48133, 56033, 79381, 83333, 97921, 109061, 111361, 133141, 188501, 197633, 206981, 219781, 229633, 256961, 282133, 293281, 328021, 340033, 360533, 416641, 481601, 556421

Offset: 1

Colin Barker, Jul 03 2015

nonn

			The octagonal number 21 is in the sequence because 21 = 3 * 7.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A129521, A245365, A259676.

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is n*(3*n-2) ]; // Vincenzo Librandi, Jul 04 2015

a={}; Do[If[PrimeOmega[n (3 n - 2)]==2, AppendTo[a, n(3 n - 2)]], {n, 1, 200}]; a (* Vincenzo Librandi, Jul 04 2015 *)
Select[PolygonalNumber[8,Range[500]],PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 15 2019 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
select(n->bigomega(n)==2, vector(2000, n, pg(8, n)))

Equals A000567 intersect A001358.

A226755 Numbers of the form p*q, p and q prime with q=2p-3.

Original entry on oeis.org

9, 35, 77, 209, 299, 527, 989, 1829, 2627, 3239, 3569, 5459, 8777, 9869, 13529, 18527, 20099, 22577, 25199, 31877, 37127, 48827, 55277, 64979, 72389, 73919, 88409, 98789, 107879, 115439, 125249, 137549, 159329, 192509, 200027, 218129, 239777, 277139, 353219

Offset: 1

José María Grau Ribas, Jun 16 2013

nonn

The smaller prime factor of a(n) = p = sopf(a(n))/3 + 1. The larger prime factor of a(n) = q = 2*sopf(a(n))/3 - 1. Furthermore, 2(sopf(a(n))/3 + 1) is representable as the sum of two primes in at least two ways since 2p = p + p = 3 + q. - Wesley Ivan Hurt, Jun 30 2013

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

Cf. A129521, A156592, A226754.

fa = FactorInteger; t[n_]:=Length[fa[n]] == 2 && fa[n][[1,2]]== fa[n][[2, 2]] == 1 && 2 fa[n][[1, 1]]-3 == fa[n][[2, 1]]; Select[1+Range[200000], t]

list(lim)=my(v=List(), q); forprime(p=2, (sqrt(8*lim+9)+3)\4, if(isprime(q=2*p-3), listput(v, p*q))); Vec(v) \\ Charles R Greathouse IV, Nov 19 2013

a(1) added by Charles R Greathouse IV, Nov 19 2013

cp's OEIS Frontend

A191592 Numbers in A191311 but not in A129521.

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A351875 Positive difference of the distinct prime factors of A129521(n).

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

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A005383 Primes p such that (p+1)/2 is prime.

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A020136 Fermat pseudoprimes to base 4.

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A122781 Nonprimes n such that 4^n==4 (mod n).

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A129512 Numbers with at least two pairs of distinct divisors having equal differences.

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