A023204 -id:A023204 - cp's OEIS frontend

A089530 A023204 indexed by A000040.

1, 3, 4, 6, 7, 8, 10, 14, 15, 16, 19, 21, 24, 25, 30, 31, 33, 34, 37, 39, 40, 44, 45, 46, 48, 49, 50, 57, 59, 61, 63, 68, 70, 71, 75, 76, 78, 80, 85, 90, 91, 93, 96, 97, 99, 101, 102, 103, 109, 111, 112, 113, 117, 118, 120, 131, 132, 137, 139, 140, 144, 147, 149, 154

Offset: 1

Ray Chandler, Nov 07 2003

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

Cf. A023204, A089531, A089532.

[n: n in [1..250] | IsPrime(2*NthPrime(n)+3)]; // Vincenzo Librandi, May 02 2016

Select[Range[160], PrimeQ[2 Prime[#] + 3] &] (* Vincenzo Librandi, May 02 2016 *)

a(n)=k such that A023204(n)=A000040(k).

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

A092109 Primes p such that p+3 is a semiprime.

Original entry on oeis.org

3, 7, 11, 19, 23, 31, 43, 59, 71, 79, 83, 103, 131, 139, 163, 191, 199, 211, 223, 251, 271, 311, 331, 359, 379, 383, 419, 443, 463, 479, 499, 523, 563, 619, 631, 659, 691, 743, 839, 859, 863, 883, 911, 919, 971, 1039, 1091, 1123, 1151, 1171, 1223, 1231, 1259

Offset: 1

Zak Seidov, Feb 21 2004

Primes p such that p-3 is semiprime are in A089531; p and 2p+3 both prime, A023204; p, 2p-3 and 2p+3 prime, A092110.
Primes p such that (p+3)/2 is prime. All these primes are congruent to 3 mod 4. - Artur Jasinski, Oct 11 2008
Subsequence of A131426. - Zak Seidov, Mar 29 2015
Subsequence of A091305. - David Radcliffe, May 22 2022

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

Cf. A023204, A089531, A063908, A092110, A115334, A131426.

IsSemiprime:=func< p | &+[ k[2]: k in Factorization(p)] eq 2 >; [p: p in PrimesUpTo(1300)| IsSemiprime(p+3)]; // Vincenzo Librandi, Feb 21 2014

select(p -> isprime(p) and isprime((p+3)/2), [seq(2*k+1,k=1..1000)]); # Robert Israel, Mar 29 2015

aa = {}; k = 3; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, Prime[n]]], {n, 1, 100}]; aa (* Artur Jasinski, Oct 11 2008 *)
Select[Prime[Range[300]],PrimeOmega[#+3]==2&] (* Harvey P. Dale, Feb 07 2018 *)

is(n)=n%2 && isprime((n+3)/2) && isprime(n) \\ Charles R Greathouse IV, Jul 12 2016

a(n) = 2*A063908(n)-3 = 4*A115334(n)+3. - Artur Jasinski, Oct 11 2008

A089531 Primes p such that (p-3)/2 is also prime.

Original entry on oeis.org

7, 13, 17, 29, 37, 41, 61, 89, 97, 109, 137, 149, 181, 197, 229, 257, 277, 281, 317, 337, 349, 389, 397, 401, 449, 457, 461, 541, 557, 569, 617, 677, 701, 709, 761, 769, 797, 821, 881, 929, 937, 977, 1009, 1021, 1049, 1097, 1117, 1129, 1201, 1217, 1229, 1237

Offset: 1

Ray Chandler, Nov 07 2003

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

Cf. A023204, A089530, A089532.

[p: p in PrimesUpTo(1300) | IsPrime((p-3) div 2)]; // Vincenzo Librandi, Apr 13 2013

Clear[lst,n,f] f[n_]:=PrimeQ[(n-3)/2]; lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009 *)
Select[Prime[Range[300]], PrimeQ[(# - 3) / 2]&] (* Vincenzo Librandi, Apr 13 2013 *)

is(n)=isprime(n) && isprime((n-3)\2) \\ Charles R Greathouse IV, Sep 09 2014

a(n) = 2*A023204(n) + 3.

Comment from Juri-Stepan Gerasimov used as new name (old name moved to formulas). - Charles R Greathouse IV, Sep 09 2014

A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2p and 2q, where q is the next prime after p.

Original entry on oeis.org

59, 71, 101, 107, 149, 263, 311, 347, 461, 499, 521, 569, 673, 757, 821, 823, 857, 881, 883, 907, 967, 977, 1009, 1061, 1091, 1093, 1151, 1213, 1279, 1283, 1297, 1301, 1319, 1433, 1487, 1489, 1493, 1549, 1571, 1597, 1619, 1667, 1697, 1721, 1787, 1871, 1873

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

From Peter Munn, Oct 19 2017: (Start)
This is also a list of the leaf node labels in the tree of primes described in A290183.
For k > 0, the earliest run of k adjacent primes in this sequence starts with the least prime greater than A215238(k+1)/2. Thus we see that A215238(3) = 1637 corresponds to 821 followed by 823 being the first run of 2 adjacent primes in this sequence.
(End)
From Peter Munn, Nov 02 2017: (Start)
If p is in A005384 (a Sophie Germain prime), 2p+1 is therefore a prime, so p cannot be in this sequence. Similarly, any prime p in A023204 has a corresponding prime 2p+3, which (if p>2) likewise implies its absence (and if p=2 it is in A005384).
If p is the lesser of twin primes it is in this sequence if it is neither Sophie Germain nor in A023204.
Conjecture: a(n)/A000040(n) is asymptotic to 3. Reason: I expect the distribution of terms in A102820 to converge to a geometric distribution with mean value 2.
(End)

			59 is a term since 113 is the prime preceding 2*59, 127 is the next prime and 61 is the largest of all prime factors of 114, ..., 122 = 2*61, ..., 126.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

A080191 is the complement of this sequence relative to A000040.
Sequences with related analysis: A005384, A023204, A052248, A102820, A215238, A290183.
Sequences with similar definitions: A195270, A195271, A195325, A195377.

Select[Prime[Range[300]],NextPrime[2#]>2NextPrime[#]&] (* Harvey P. Dale, Jul 07 2011 *)

¯1↓b/⍨(1⌽a)<1πa←2×b←¯2π⍳1E4 ⍝ Michael Turniansky, Dec 29 2020

{forprime(k=2,1873,p=precprime(2*k); q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m

isok(p) = isprime(p) && (primepi(2*p) == primepi(2*nextprime(p+1)));
forprime(p=2, 2000, if (isok(p), print1(p, ", "))) \\ Michel Marcus, Sep 22 2017

first(n) = my(res = vector(n), i = 0); {n==0&&return([]); forprime(p = 2, , if(nextprime(2*p) > 2*nextprime(p + 1), i++; res[i] = p; if(i == n, return(res))))} \\ David A. Corneth, Oct 25 2017

For all k, prime(k) = A000040(k) is a term if and only if A102820(k) = 0. - Peter Munn, Oct 24 2017

A023205 Numbers m such that m and 2*m + 5 are both prime.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 61, 67, 73, 79, 97, 103, 109, 139, 151, 163, 181, 229, 241, 271, 283, 307, 313, 367, 373, 409, 439, 457, 523, 541, 613, 643, 661, 709, 727, 733, 739, 769, 787, 811, 829, 859, 877, 937, 991, 997, 1039, 1063, 1069, 1087, 1117, 1123, 1153, 1171

Offset: 1

David W. Wilson, Jun 14 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of primes of A089038.

Cf. A005384, A023204.

[n: n in [0..100000] | IsPrime(n) and IsPrime(2*n+5)] // Vincenzo Librandi, Aug 04 2010

A023205:=n->`if`(isprime(n) and isprime(2*n+5), n, NULL): seq(A023205(n), n=1..3*10^3); # Wesley Ivan Hurt, Sep 08 2016

Select[Prime@Range@250, PrimeQ[2#+5]&] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2011 *)

a(n) == 1 (mod 6) for n>=2. - John Cerkan, Sep 07 2016

A023206 Numbers m such that m and 2*m + 7 both prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 41, 47, 53, 71, 83, 113, 131, 137, 173, 191, 197, 227, 251, 257, 281, 293, 317, 347, 383, 401, 461, 467, 503, 521, 587, 593, 641, 647, 677, 683, 701, 743, 773, 797, 857, 863, 941, 947, 953, 971, 983, 1031, 1061, 1103, 1151, 1163, 1187, 1193, 1217

Offset: 1

David W. Wilson

Subsequence of A105760. Except for the first two terms, all terms are congruent to 5 mod 6. - John Cerkan, Sep 07 2016

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

Cf. A005384, A023204, A023205.

[n: n in [0..1000] | IsPrime(n) and IsPrime(2*n+7)]; // Vincenzo Librandi, Nov 20 2010

A023206:=n->`if`(isprime(n) and isprime(2*n+7), n, NULL): seq(A023206(n), n=1..3*1000); # Wesley Ivan Hurt, Sep 07 2016

Select[Prime@Range@250,PrimeQ[2#+7]&] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2011 *)

is(n)=isprime(2*n+7) && isprime(n) \\ Charles R Greathouse IV, Sep 07 2016

A023207 Numbers m such that m and 2*m + 9 both prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 31, 37, 47, 59, 61, 71, 79, 101, 107, 109, 127, 131, 137, 149, 151, 179, 211, 227, 229, 239, 241, 257, 269, 277, 281, 311, 317, 337, 359, 367, 389, 401, 409, 439, 449, 479, 487, 491, 521, 541, 547, 557, 571, 577, 607, 641, 647, 659, 709, 719, 739

Offset: 1

David W. Wilson

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

Cf. A005384, A023204, A023205.

[ n: n in PrimesUpTo(1000) | IsPrime(2*n+9) ]; // Vincenzo Librandi, Nov 20 2010

Select[Prime@Range@250,PrimeQ[2#+9]&] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2011 *)

A092110 Primes p such that 2p+3 and 2p-3 are both prime.

Original entry on oeis.org

5, 7, 13, 17, 43, 53, 67, 97, 113, 127, 137, 157, 167, 193, 223, 283, 487, 547, 563, 613, 617, 643, 647, 743, 773, 937, 1033, 1187, 1193, 1277, 1427, 1453, 1483, 1543, 1583, 1627, 1663, 1733, 1847, 2027, 2143, 2297, 2393, 2437, 2467, 2477, 2503, 2617, 2843

Offset: 1

Zak Seidov, Feb 21 2004

Intersection of A023204 and A063908.

All numbers in this sequence end with 3 or 7 (except the first one, which is 5). See A136191 or A136192. - Carlos Alves, Dec 20 2007

			From _K. D. Bajpai_, Sep 08 2020: (Start)
7 is a term because 2*7 + 3 = 17 and 2*7 - 3 = 11 are both prime.
13 is a term because 2*13 + 3 = 29 and 2*13 - 3 = 23 are both prime.
(End)

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A023204, A089531, A063908, A092109.

Cf. A136191, A136192.

[p: p in PrimesUpTo(10000)|IsPrime(2*p-3) and IsPrime(2*p+3)] // Vincenzo Librandi, Nov 16 2010

select(p -> isprime(p) and isprime(2*p+3) and isprime(2*p-3), [seq(2*k+1, k=1..1000)]); # K. D. Bajpai, Sep 08 2020

Select[Prime@Range@1000,PrimeQ[2#-3]&&PrimeQ[2#+3]&] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2011 *)

A258141 Number of ways to write n as p^2 + q with p, q and 2*p + 3 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0

Offset: 1

Zhi-Wei Sun, May 22 2015

nonn

Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5.
We have verified this for n up to 10^8. See also A258139 for a weaker version of this conjecture.
The conjecture is somewhat similar to Goldbach's Conjecture. It implies that there are infinitely many primes p with 2*p + 3 prime.

			a(11) = 1 since 11 = 2^2 + 7 with 2, 7 and 2*2 + 3 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A002375, A023204, A258139, A258140.

Do[r=0;Do[If[PrimeQ[2Prime[k]+3]&&PrimeQ[n-Prime[k]^2],r=r+1],{k,1,PrimePi[Sqrt[n]]}];Print[n," ",r];Continue,{n,1,100}]

cp's OEIS Frontend

A089530 A023204 indexed by A000040.

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

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A092109 Primes p such that p+3 is a semiprime.

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A089531 Primes p such that (p-3)/2 is also prime.

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A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2*p and 2*q, where q is the next prime after p.

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A023205 Numbers m such that m and 2*m + 5 are both prime.

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A023206 Numbers m such that m and 2*m + 7 both prime.

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A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2p and 2q, where q is the next prime after p.