A098420 -id:A098420 - cp's OEIS frontend

A098418 Number of prime triples (p,q,r) with p

0, 0, 1, 2, 3, 3, 3, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

0 <= a(n) <= 3;

a(A098419(n))=0; a(A098420(n))>0; a(A098421(n))=1; a(A098422(n))=2; a(A098423(n))=3.

			A000040(13)=41: A007529(7)=41, A098414(6)=41 and
A098415(k)<>41 for all k, therefore a(13)=2.

Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A007529, A098414, A098415.

A098419 Primes not occurring in prime triples (p,q,r) with p

Original entry on oeis.org

2, 3, 29, 31, 53, 59, 61, 79, 83, 89, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 211, 239, 241, 251, 257, 263, 269, 271, 293, 331, 337, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 479, 487, 491, 499, 503

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

A098418(a(n)) = 0; complement of A098420 in A000040.

Eric Weisstein's World of Mathematics, Prime Triplet

A098421 Primes occurring in exactly one prime triple (p,q,r) with p

Original entry on oeis.org

5, 23, 37, 47, 67, 71, 73, 97, 113, 191, 199, 223, 233, 277, 281, 283, 307, 317, 347, 349, 353, 457, 467, 613, 617, 619, 641, 643, 647, 821, 829, 853, 863, 877, 887, 1087, 1097, 1277, 1279, 1283, 1297, 1307, 1423, 1433, 1447, 1451, 1453, 1481, 1493, 1607, 1609

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

A098418(a(n)) = 1; subsequence of A098420.

Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098419, A098422, A098423.

A098422 Primes occurring in exactly two prime triples (p,q,r) with p

Original entry on oeis.org

7, 19, 41, 43, 101, 109, 193, 197, 227, 229, 311, 313, 461, 463, 823, 827, 857, 859, 881, 883, 1091, 1093, 1301, 1303, 1427, 1429, 1483, 1489, 1871, 1877, 1997, 1999, 2083, 2087, 2687, 2689, 3253, 3257, 3461, 3467, 4517, 4519, 4787, 4789, 5231, 5233, 5651

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

A098418(a(n)) = 2; subsequence of A098420.

Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098419, A098421, A098423.

Select[Tally[Flatten[Select[Partition[Prime[Range[800]],3,1],#[[3]]- #[[1]] == 6&]]],#[[2]]==2&][[All,1]] (* Harvey P. Dale, Jun 01 2019 *)

A098423 Primes occurring in exactly three prime triples (p,q,r) with p

Original entry on oeis.org

11, 13, 17, 103, 107, 1487, 1873, 3463, 5653, 15733, 16063, 16067, 19423, 19427, 21017, 22277, 43783, 43787, 55337, 79693, 88813, 101113, 144167, 165707, 166847, 195737, 201827, 225347, 247607, 257863, 266683, 268817, 276043, 284743

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

A098418(a(n)) = 3; subsequence of A098420.

This sequence consists of all integers of the form (prime(m)*prime(m+4)+36)/prime(m+2), for m>0, where prime(m) = A000040(m). Also note that the integers resulting from that rule equal prime(m+2), therefore a(n) also consists of all integers of the form sqrt[prime(m)*prime(m+4)+36]. - Richard R. Forberg, Jan 11 2016

			A000040(27)=103: A007529(11)=103, A098414(10)=103 and A098415(9)=103, therefore 103 is a term.

Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098419, A098421, A098422.

A309354 Primes of the form p+q+r where p < q < r = p+6 are consecutive primes.

Original entry on oeis.org

23, 31, 41, 59, 131, 211, 311, 941, 1049, 1381, 1931, 2579, 3271, 3911, 4289, 4451, 4999, 6421, 6719, 8059, 8069, 9769, 10391, 10399, 10589, 11551, 12011, 14369, 16249, 20479, 23269, 23629, 26591, 27031, 28309, 31379, 33349, 33521, 35339, 35491, 39019, 41081

Offset: 1

Philip Mizzi, Jul 25 2019

nonn

			P = 5 (prime),
P + 2 = 7 (prime),
P + 6 = 11 (prime),
and 5 + 7 + 11 = 23 is prime and is a term.
P = 7 (prime),
P + 4 = 11 (prime),
P + 6 = 13 (prime)
and 7 + 11 + 13 = 31 is prime and is a term.
However, (p,q,r) = (13,17,19) fails because the sum is not a prime.

Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A098420.

Select[Total /@ Select[Partition[Prime@Range[2000], 3, 1], #[[3]] == 6 + #[[1]] &], PrimeQ] (* Giovanni Resta, Jul 25 2019 *)

More terms from Giovanni Resta, Jul 25 2019

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A098418 Number of prime triples (p,q,r) with p

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A098419 Primes not occurring in prime triples (p,q,r) with p

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A098421 Primes occurring in exactly one prime triple (p,q,r) with p

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A098422 Primes occurring in exactly two prime triples (p,q,r) with p

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A098423 Primes occurring in exactly three prime triples (p,q,r) with p

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A309354 Primes of the form p+q+r where p < q < r = p+6 are consecutive primes.

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