A115334 -id:A115334 - 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

A206037 Values of the difference d for 3 primes in arithmetic progression with the minimal start sequence {3 + j*d}, j = 0 to 2.

Original entry on oeis.org

2, 4, 8, 10, 14, 20, 28, 34, 38, 40, 50, 64, 68, 80, 94, 98, 104, 110, 124, 134, 154, 164, 178, 188, 190, 208, 220, 230, 238, 248, 260, 280, 308, 314, 328, 344, 370, 418, 428, 430, 440, 454, 458, 484, 518, 544, 560, 574, 584, 610, 614, 628, 638, 640, 644, 650

Offset: 1

Sameen Ahmed Khan, Feb 03 2012

The computations were done without any assumptions on the form of d.
Numbers k such that k+3 and 2k+3 are both primes.
Equivalently, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly 3 elements (see example). These 3 elements are not necessarily consecutive primes. In fact, for each term d, there exists only one such AP of primes, and this one starts always with A342309(d) = 3, so this AP is (3, 3+d, 3+2d). - Bernard Schott, Jan 15 2023

			d = 8 then {3, 3 + 1*8, 3 + 2*8} = {3, 11, 19}, which is 3 primes in arithmetic progression.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000.
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Sameen A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A040976, A123556, A206038, A206039, A206040, A206041, A206042, A206043, A206044, A206045, A342309.

Largest AP of prime numbers with k elements: A007921 (k=1), A359408 (k=2), this sequence (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7).

[n: n in [1..700] | IsPrime(3+n) and IsPrime(3+2*n)]; // Vincenzo Librandi, Dec 28 2015

filter := d -> isprime(3+d) and isprime(3+2*d) : select(filter, [$(1 .. 650)]); # Bernard Schott, Jan 16 2023

t={}; Do[If[PrimeQ[{3, 3 + d, 3 + 2*d}] == {True, True, True}, AppendTo[t, d]], {d, 1000}]; t
Select[Range[2,700,2],And@@PrimeQ[{3+#,3+2#}]&] (* Harvey P. Dale, Sep 25 2013 *)

for(n=1, 1e3, if(isprime(n + 3) && isprime(2*n + 3), print1(n, ", "))); \\ Altug Alkan, Dec 27 2015

a(n) = 2 * A115334(n). - Wesley Ivan Hurt, Feb 06 2014

m is a term iff A123556(m) = 3. - Bernard Schott, Jan 15 2023

A088420 Number of primes in arithmetic progression starting with 3 and with d = 2n.

Original entry on oeis.org

3, 3, 1, 3, 3, 1, 3, 2, 1, 3, 1, 1, 2, 3, 1, 1, 3, 1, 3, 3, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 1, 3, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 3, 1, 3, 2, 1, 3, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 3, 3, 1, 2, 2, 1, 1

Offset: 1

Zak Seidov, Sep 29 2003

The arithmetic progression is stopped when the next term is not prime. E.g., for n=5, a=3, the numbers 3, 13, and 23 are prime, while the next term, 33, is not prime.

a(n) <= 3 because 3+3*d is divisible by 3. - Klaus Brockhaus, May 14 2009

Cf. A088421, A088422, A088423, A088424, A088425, A088426, A088427, A088428, A088429.

Cf. A115334. - Klaus Brockhaus, May 14 2009

npap3:=function(d) c:=1; p:=3+d; while IsPrime(p) do c+:=1; p+:=d; end while; return c; end function; [ npap3(2*n): n in [1..105] ]; // Klaus Brockhaus, May 14 2009

A160394 Numbers n = pqr (p, q, r prime) congruent to 0 mod p+q+r.

Original entry on oeis.org

27, 30, 70, 105, 231, 286, 627, 646, 805, 897, 1581, 1798, 2967, 3055, 3526, 4543, 5487, 6461, 6745, 7198, 7881, 9717, 10366, 10707, 14231, 16377, 20806, 21091, 23326, 26331, 29607, 33901, 35905, 37411, 38086, 38843, 40587, 42211, 44998, 55581

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 12 2009

nonn

Also numbers n = p*q*r where r = p*q-(p+q) and p, q, r are prime.
For each twin prime pair (q, q+2) the number n = 2*p*(p+2) is in the sequence, since 2+p+(p+2) divides n.
In some cases the factors of n are in arithmetic progression; occurring common differences are 2, 4, 8, 10, 14, 20, 28, 34, 38, 40, 50, 68, 80, 94, 98, ...
All those arithmetic progressions have first term 3, their common differences are the numbers d such that A088420(d/2) = 3. - Klaus Brockhaus, May 17 2009

			27 = 3*3*3 = (3+3+3)*3, hence 27 is in the sequence; r = 3*3-(3+3).
30 = 2*5*3 = (2+5+3)*3, hence 30 is in the sequence; r = 2*5-(2+5).
165 = 3*5*11 is not a multiple of 3+5+11 = 19, hence 165 is not in the sequence.
627 = 3*11*19 = (3+11+19)*19, hence 627 is in the sequence; r = 3*11-(3+11). The factors 3, 11, 19 are in arithmetic progression (d=8).
40587 = 3*83*163 = (3+83+163)*163, hence 40587 is in the sequence; r = 3*83-(3+83). The factors 3, 83, 163 are in arithmetic progression (d=80).

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

Cf. A014612 (3-almost primes, numbers that are divisible by exactly 3 primes (counted with multiplicity)).

Cf. A001359 (lesser of twin primes), A115334 (numbers n such that 3+2n and 3+4n are prime), A088420 (number of primes in arithmetic progression starting with 3 and with d=2n). [From Klaus Brockhaus, May 17 2009]

[ n: n in [2..56000] | &+[ d[2]: d in f ] eq 3 and n mod &+[ d[1]*d[2]: d in f ] eq 0 where f is Factorization(n) ]; // Klaus Brockhaus, May 17 2009

list(lim)=my(v=List()); forprime(p=2,lim\4, forprime(q=2,lim\(2*p), my(pq=p*q, r=pq-p-q); if(isprime(r), listput(v, pq*r)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Partially edited by N. J. A. Sloane, May 14 2009
Missed entry 27 contributed by Zak Seidov, May 14 2009
Further edited by Klaus Brockhaus, May 17 2009

A210504 Numbers n for which 2n+5, 4n+5, 6n+5, and 8n+5 are primes.

Original entry on oeis.org

0, 3, 6, 21, 24, 48, 63, 126, 213, 237, 297, 318, 402, 609, 657, 714, 783, 864, 948, 1053, 1287, 1347, 1449, 1581, 1683, 2166, 2184, 2358, 2457, 2463, 2481, 2736, 2793, 2898, 2919, 3024, 3723, 3786, 3909, 4017, 4479, 4584, 4602, 4857, 5169, 5634, 5733, 7101

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 25 2013

nonn

All terms are multiple of 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A115334.

Select[Range[0, 10000], PrimeQ[2*# + 5] && PrimeQ[4*# + 5] && PrimeQ[6*# + 5] && PrimeQ[ 8*# + 5] &] (* T. D. Noe, Jan 25 2013 *)
Select[Range[0,8000],AllTrue[#{2,4,6,8}+5,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 14 2016 *)

A210505 Numbers k for which 2k+7, 4k+7, 6k+7, 8k+7, 10k+7 and 12k+7 are primes.

Original entry on oeis.org

0, 75, 1380, 1725, 4575, 7095, 10020, 10620, 31800, 38355, 58710, 61170, 67125, 92235, 92310, 94845, 118530, 137415, 156000, 168765, 189705, 238815, 249450, 257370, 339375, 353925, 507270, 584265, 590040, 617265, 625845, 631740, 761760, 845295, 866910, 943605

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 25 2013

nonn

Conjecture. For every odd prime p there exist infinitely many numbers k for which 2*k+p, 4*k+p, ..., 2*(p-1)*k+p are primes.
For p=3, cf. A115334, for p=5, cf. A210504. This sequence corresponds to p=7.
In general case of prime p, every k == 0 (mod Product{p_2*p_3*...*p_k)), where p_k is the maximal prime < p.

Harvey P. Dale, Table of n, a(n) for n = 0..500

Cf. A115334, A210504.

Select[Range[0, 1000000], PrimeQ[2*# + 7] && PrimeQ[4*# + 7] && PrimeQ[6*# + 7] && PrimeQ[ 8*# + 7] && PrimeQ[ 10*# + 7] && PrimeQ[ 12*# + 7] &] (* T. D. Noe, Jan 25 2013 *)
Select[Range[0,950000],AllTrue[#*Range[2,12,2]+7,PrimeQ]&] (* Harvey P. Dale, Aug 16 2024 *)

a(n) == 0 (mod 15).

A238699 Primes p such that 2p + 3 and 4p + 3 are both prime.

Original entry on oeis.org

2, 5, 7, 17, 19, 47, 67, 89, 157, 227, 229, 307, 349, 439, 467, 487, 509, 599, 647, 797, 929, 1039, 1187, 1217, 1237, 1259, 1307, 1427, 1447, 1567, 1789, 2027, 2309, 2467, 2539, 2707, 2789, 2819, 3167, 3457, 3499, 3659, 3877, 3919, 4057, 4079, 4157, 4289, 4297

Offset: 1

Ilya Lopatin, Mar 03 2014, following a suggestion by Juri-Stepan Gerasimov

nonn

Intersection of A023204 and A023213.

Primes in A115334.

			89 is in the sequence because 2*89 + 3 = 181 and 4*89 + 3 = 359 are both prime.

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

Cf. A023204, A023213; A115334.

[p: p in PrimesUpTo(4500) | IsPrime(2*p+3) and IsPrime(4*p+3)]; // Bruno Berselli, Mar 03 2014

Select[Prime[Range[600]],AllTrue[{2#+3,4#+3},PrimeQ]&] (* Harvey P. Dale, Dec 24 2023 *)

select(p->isprime(2*p+3)&&isprime(4*p+3), primes(1000)) \\ Charles R Greathouse IV, Mar 06 2014

Edited by Bruno Berselli, Mar 03 2014

A240233 a(n) is the smallest prime number such that both a(n) + 6n and a(n) + 12n are prime numbers.

Original entry on oeis.org

5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 5, 7, 11, 5, 11, 5, 7, 23, 13, 11, 5, 5, 41, 5, 7, 37, 29, 11, 5, 13, 7, 5, 13, 23, 13, 7, 5, 5, 23, 11, 11, 5, 5, 13, 7, 5, 29, 23, 13, 7, 5, 19, 41, 13, 17, 11, 7, 5, 19, 7, 7, 7, 5, 5, 7, 5, 7, 11, 29, 13, 5, 17, 5, 19, 7, 7, 5

Offset: 1

Lei Zhou, Apr 02 2014

a(n), a(n) + 6n, and a(n) + 12n form an arithmetic progression with a common difference of 6n.
If the interval is not a multiple of six, such an arithmetic progression of primes cannot exist unless a(n)=3. For example, 3,5,7 has an interval of 2; 3,7,11 has an interval of 4; and 3,11,19 has an interval of 8, as in A115334 and A206037.
Conjecture: a(n) is defined for all n > 0.

			n=1, 6n=6. 5,11,17 are all prime numbers with an interval of 6. So a(1)=5;
...
n=13, 6n=78. 5+78=83, 5+2*78=161=7*23(x); 7+78=85(x); 11+78=89, 11+78*2=167. 11,89,167 are all prime numbers with an interval of 78. So a(13)=11.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A000040, A115334, A206037, A240087.

Table[diff = n*6; k = 1; While[k++; p = Prime[k]; cp1 = p + diff; cp2 = p + 2*diff; ! ((PrimeQ[cp1]) && (PrimeQ[cp2]))]; p, {n, 77}]

cp's OEIS Frontend

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

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A206037 Values of the difference d for 3 primes in arithmetic progression with the minimal start sequence {3 + j*d}, j = 0 to 2.

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A088420 Number of primes in arithmetic progression starting with 3 and with d = 2n.

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A160394 Numbers n = p*q*r (p, q, r prime) congruent to 0 mod p+q+r.

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A210504 Numbers n for which 2*n+5, 4*n+5, 6*n+5, and 8*n+5 are primes.

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A210505 Numbers k for which 2*k+7, 4*k+7, 6*k+7, 8*k+7, 10*k+7 and 12*k+7 are primes.

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A238699 Primes p such that 2p + 3 and 4p + 3 are both prime.

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A160394 Numbers n = pqr (p, q, r prime) congruent to 0 mod p+q+r.

A210504 Numbers n for which 2n+5, 4n+5, 6n+5, and 8n+5 are primes.

A210505 Numbers k for which 2k+7, 4k+7, 6k+7, 8k+7, 10k+7 and 12k+7 are primes.