A131426 -id:A131426 - cp's OEIS frontend

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

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

A139291 a(n) = 2^(2*prime(n) - 3) - 1.

Original entry on oeis.org

1, 7, 127, 2047, 524287, 8388607, 2147483647, 34359738367, 8796093022207, 36028797018963967, 576460752303423487, 2361183241434822606847, 604462909807314587353087, 9671406556917033397649407, 2475880078570760549798248447, 10141204801825835211973625643007

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Cf. A139286, A139287, A139288, A139289, A139290, A139293, A139294, A139306.

(2^(2#-1))/4-1&/@Prime[Range[20]] (* Harvey P. Dale, May 08 2011 *)

a(n) = A139286(n)/4 - 1.
a(n) = A000225(A131426(n)). - Max Alekseyev, Mar 07 2020
a(n) = A139290(n) - 1. - Omar E. Pol, Mar 10 2020

Edited by Max Alekseyev, May 03 2009, Mar 07 2020

A131424 Triangle read by rows: T(n,k) = prime(n) + prime(k) - 3, 1 <= k <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 7, 9, 11, 10, 11, 13, 15, 19, 12, 13, 15, 17, 21, 23, 16, 17, 19, 21, 25, 27, 31, 18, 19, 21, 23, 27, 29, 33, 35, 22, 23, 25, 27, 31, 33, 37, 39, 43, 28, 29, 31, 33, 37, 39, 43, 45, 49, 55, 30, 31, 33, 35, 39, 41, 45, 47, 51, 57, 59

Offset: 1

Gary W. Adamson, Jul 10 2007

Left border = A006093, (primes - 1): (1, 2, 4, 6, 10, 12, ...). Right border = A131426 (2*primes - 3): (1, 3, 7, 11, 19, 23, 31, ...). Row sums = A131425: (1, 5, 16, 33, 68, 101, 156, ...).

			First few rows of the triangle are:
   1;
   2,  3;
   4,  5,  7;
   6,  7,  9, 11;
  10, 11, 13, 15, 19;
  12, 13, 15, 17, 21, 23;
  16, 17, 19, 21, 25, 27, 31;
  18, 19, 21, 23, 27, 29, 33, 35;
  22, 23, 25, 27, 31, 33, 37, 39, 43;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A131425.

Cf. A127640, A131426, A006093, A000040.

T[n_,k_]:=Prime[n]+Prime[k]-3;Table[T[n,k],{n,11},{k,n}]//Flatten (* James C. McMahon, Feb 20 2025 *)

T(n,k) = if(k<=n, prime(n) + prime(k) - 3, 0) \\ Andrew Howroyd, Sep 01 2018

Equals (A000012 * A127640) + (A127640 * A000012) - 3*A000012 as infinite lower triangular matrices.

Name clarified and terms a(56) and beyond from Andrew Howroyd, Sep 01 2018

A131425 Row sums of triangle A131424.

Original entry on oeis.org

1, 5, 16, 33, 68, 101, 156, 205, 280, 389, 468, 605, 732, 841, 988, 1181, 1392, 1545, 1784, 1999, 2182, 2463, 2714, 3027, 3410, 3709, 3964, 4283, 4554, 4893, 5564, 5947, 6410, 6751, 7386, 7755, 8282, 8827, 9310, 9887, 10482, 10923, 11722, 12191, 12758, 13243

Offset: 1

Gary W. Adamson, Jul 10 2007

nonn

			a(4) = 33 = sum of row 4 terms of triangle A131424: (6 + 7 + 9 + 11).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A131424, A131426, A127640, A000040.

Table[n (Prime[n]-3)+Sum[Prime[k],{k,n}],{n,50}] (* Harvey P. Dale, Nov 28 2024 *)

a(n)={n*(prime(n) - 3) + sum(k=1, n, prime(k))} \\ Andrew Howroyd, Aug 28 2018

a(n) = n*(prime(n) - 3) + Sum_{k=1..n} prime(k). - Andrew Howroyd, Aug 28 2018

Terms a(10) and beyond from Andrew Howroyd, Aug 28 2018

A172256 Primes p such that 2*p+-3 are both nonprimes.

Original entry on oeis.org

59, 61, 79, 103, 109, 131, 149, 151, 163, 179, 239, 257, 271, 281, 293, 313, 359, 367, 389, 401, 419, 449, 479, 491, 499, 541, 569, 571, 593, 601, 619, 673, 677, 683, 691, 709, 719, 733, 761, 769, 821, 823, 829, 839, 857, 877, 883, 911, 919, 947, 953, 971, 983, 1009

Offset: 1

Juri-Stepan Gerasimov, Jan 30 2010

nonn

In the first 10000 primes there are 5698 terms (~57% of the primes). In the 10000 primes from prime(1,000,000,000) to prime(1,000,010,000) there are 8432 primes in this sequence or ~84%. It seems likely the density of these terms within the primes slowly approaches 100%. This indicates the density of "Prime Septets", as defined in A268593 (which rely upon primes in the complement of this sequence), declines steadily at larger n. - Richard R. Forberg, Feb 12 2016

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

Cf. A000040, A131426, A141468.

[p: p in PrimesUpTo(1100)| not IsPrime(2*p+3)and not IsPrime(2*p-3)] // Vincenzo Librandi, Dec 08 2010

npQ[n_]:=Module[{c=2n},!PrimeQ[c+3]&&!PrimeQ[c-3]]; Select[Prime[ Range[ 200]],npQ] (* Harvey P. Dale, Jan 21 2013 *)
Select[Prime[Range[200]],NoneTrue[2#+{3,-3},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 25 2019 *)

isok(p) = isprime(p) && !isprime(2*p+3) && !isprime(2*p-3); \\ Michel Marcus, Feb 12 2016

Corrected and extended by Vincenzo Librandi, Apr 01 2010

A169605 Numbers x of the form x = 2y - 3 = 3z - 2 where y and z are primes.

Original entry on oeis.org

7, 19, 31, 55, 91, 139, 175, 199, 211, 379, 391, 451, 499, 535, 631, 715, 919, 931, 1039, 1135, 1291, 1315, 1399, 1435, 1639, 1711, 1759, 1819, 1855, 1891, 1939, 2179, 2215, 2359, 2431, 2515, 2575, 2719, 2731, 2899, 2971, 3115, 3271, 3691, 3775, 3955, 4195

Offset: 1

Juri-Stepan Gerasimov, Dec 03 2009

nonn

			a(1)=7 because 5*2 - 3 = 3*3 - 2;
a(2)=19 because 11*2 - 3 = 7*3 - 2.

Cf. A000040, A131426.

isA169605 := proc(x) if type(x+3,'even') then if (x+2) mod 3 = 0 then isprime( (x+3)/2) and isprime((x+2)/3) ; else false ; end if else false; end if; end proc: for x from 1 to 10000 do if isA169605(x) then printf("%d,",x) ; end if; end do: # R. J. Mathar, Jan 27 2010

Select[3Prime[Range[250]]-2,PrimeQ[(3+#)/2]&] (* Harvey P. Dale, May 11 2011 *)

is_prime_Q = lambda x: x.is_integral() and Integer(x).is_prime()
A169605 = list(x for x in range(1, 10**4) if is_prime_Q((x+3)/2) and
is_prime_Q((x+2)/3))
A169605[:36]
# D. S. McNeil, Dec 21 2009

Corrected and extended by Jim Nastos and D. S. McNeil, Dec 21 2009

A few more terms from R. J. Mathar, Jan 27 2010

A172258 Primes p such that exactly one of the numbers 2p-3 and 2p+3 is prime.

Original entry on oeis.org

2, 3, 11, 19, 23, 29, 31, 37, 41, 47, 71, 73, 83, 89, 101, 107, 139, 173, 181, 191, 197, 199, 211, 227, 229, 233, 241, 251, 263, 269, 277, 307, 311, 317, 331, 337, 347, 349, 353, 373, 379, 383, 397, 409, 421, 431, 433, 439, 443, 457, 461, 463, 467, 503, 509

Offset: 1

Juri-Stepan Gerasimov, Jan 30 2010

nonn

			a(1)=2 because 2*2-3=1 (nonprime) and 2*2+3=7 (prime);
a(2)=29 because 2*29-3=55 (nonprime) and 2*29+3=61 (prime).

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

Cf. A000040, A092110, A131426.

a := proc (n): if isprime(n) = true and isprime(2*n-3) = true and isprime(2*n+3) = false then n elif isprime(n) = true and isprime(2*n-3) = false and isprime(2*n+3) = true then n else end if end proc: seq(a(n), n = 1 .. 700); # Emeric Deutsch, Feb 15 2010

Select[Prime[Range[100]],Total[Boole[PrimeQ[2#+{3,-3}]]]==1&] (* Harvey P. Dale, Mar 27 2021 *)

Definition edited by Emeric Deutsch, Feb 15 2010

Corrected and extended by Emeric Deutsch, Feb 15 2010

cp's OEIS Frontend

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

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A139291 a(n) = 2^(2*prime(n) - 3) - 1.

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A131424 Triangle read by rows: T(n,k) = prime(n) + prime(k) - 3, 1 <= k <= n.

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A131425 Row sums of triangle A131424.

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A172256 Primes p such that 2*p+-3 are both nonprimes.

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A169605 Numbers x of the form x = 2*y - 3 = 3*z - 2 where y and z are primes.

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A172258 Primes p such that exactly one of the numbers 2p-3 and 2p+3 is prime.

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Maple

A169605 Numbers x of the form x = 2y - 3 = 3z - 2 where y and z are primes.