A194674 -id:A194674 - cp's OEIS frontend

A195270 3-gap primes: Prime p is a term iff there is no prime between 3p and 3q, where q is the next prime after p.

71, 107, 137, 281, 347, 379, 443, 461, 557, 617, 641, 727, 809, 827, 853, 857, 991, 1031, 1049, 1091, 1093, 1289, 1297, 1319, 1433, 1489, 1579, 1607, 1613, 1697, 1747, 1787, 1867, 1871, 1877, 1931, 1987, 1997, 2027, 2237, 2269, 2309, 2377, 2381, 2473, 2591

Offset: 1

Vladimir Shevelev, Sep 14 2011

nonn

For a real r>1, a prime p is called an r-gap prime, if there is no prime between r*p and r*q, where q is the next prime after p. In particular, 2-gap primes are in A080192.

In many cases, q=p+2. E.g., among first 1000 terms there are 509 such cases. - Zak Seidov, Jun 29 2015

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A080192, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

filter:= p -> isprime(p) and nextprime(3*p)>3*nextprime(p):
select(filter, [2,seq(2*i+1,i=1..2000)]); # Robert Israel, Jun 29 2015

pQ[p_, r_] := Block[{q = NextPrime@ p}, Union@ PrimeQ@ Range[r*p, r*q] == {False}]; Select[ Prime@ Range@ 380, pQ[#, 3] &] (* Robert G. Wilson v, Sep 18 2011 *)
k = 3; p = 71; Reap[Do[While[NextPrime[k*p] < k*(q = NextPrime[p]), p = q]; Sow[p]; p = q, {1000}]][[2, 1]] (* for first 1000 terms. - Zak Seidov, Jun 29 2015 *)
Prime/@SequencePosition[PrimePi[3*Prime[Range[400]]],{x_,x_}][[;;,1]] (* Harvey P. Dale, Nov 29 2023 *)

A195271 1.5-gap primes: Prime p is a term iff there is no prime between 1.5p and 1.5q, where q is the next prime after p.

Original entry on oeis.org

2, 5, 17, 29, 41, 79, 101, 137, 149, 163, 191, 197, 227, 269, 281, 313, 349, 353, 461, 463, 521, 541, 569, 593, 599, 613, 617, 641, 757, 769, 809, 821, 827, 857, 881, 887, 941, 1009, 1049, 1061, 1087, 1093, 1097, 1117, 1151, 1223, 1229, 1277, 1279, 1289

Offset: 1

Vladimir Shevelev, Sep 14 2011

nonn

For a real r>1, a prime p is called an r-gap prime, if there is no prime between r*p and r*q, where q is the next prime after p. In particular, 2-gap primes form A080192 and 3-gap primes form A195270.

Cf. A080192, A195270, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

Select[Prime[Range[400]], PrimePi[3*NextPrime[#]/2] == PrimePi[3*#/2] &] (* T. D. Noe, Sep 14 2011 *)

A195329 Records of A195325.

Original entry on oeis.org

2, 59, 71, 149, 191, 641, 809, 3371, 5849, 9239, 20507, 20981, 32117, 48779, 176777, 191249, 204509, 211061, 223679, 245129, 358877, 654161, 2342771, 3053291, 4297961, 4755347, 6750221, 8019509, 9750371, 10196759, 11237981, 23367077, 34910219, 93929219, 186635747

Offset: 1

Vladimir Shevelev, Sep 15 2011

nonn

The sequence is infinite. Conjecture. For n>=2, all terms are in A001359. This conjecture (weaker than the conjecture in comment to A195325) also implies the twin prime conjecture.

Cf. A195325, A080192, A195270, A195271, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7prime(k)/2 and 7prime(k+1)/2.

Original entry on oeis.org

2, 137, 281, 521, 641, 883, 937, 1087, 1151, 1229, 1277, 1301, 1489, 1567, 1607, 1697, 2027, 2081, 2237, 2381, 2543, 2591, 2657, 2687, 2729, 2801, 2851, 2969, 3119, 3257, 3301, 3359, 3463, 3467, 3529, 3673, 3733, 3793, 3821, 3851, 4073, 4217, 4229, 4241, 4259, 4283, 4337, 4421, 4481

Offset: 1

Vladimir Shevelev, Sep 17 2011

nonn

Cf. A080192, A195270, A195271, A195377, A195325, A195329, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

Select[Prime[Range[1000]], PrimePi[7*NextPrime[#]/2] == PrimePi[7*#/2] &] (* T. D. Noe, Sep 20 2011 *)

Corrected by R. J. Mathar, Sep 20 2011

A194953 Nonzero values of |A194659(n)-A194186(n+1)|.

Original entry on oeis.org

2, 6, 2, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 6, 2, 2, 4, 2, 2, 10, 6, 6, 2, 2, 2, 6, 2, 8, 8, 4, 6, 4, 2, 8, 4, 8, 4, 4, 6, 4, 2, 4, 2, 4, 2, 2, 22, 2, 2, 6, 4, 4, 8, 2, 2, 10, 2, 2, 2, 2, 4, 4, 4, 2, 2, 2, 2, 2, 10, 2, 2, 8, 18, 2, 2, 4, 4, 2, 12, 6, 6, 8, 20

Offset: 1

Vladimir Shevelev, Sep 06 2011

nonn

The sequence (together with A194674) characterizes a right-left symmetry in the distribution of primes over intervals (2*p_n, 2*p_(n+1)), n=1,2,..., where p_n is the n-th prime.

Cf. A104272, A080359, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

cp's OEIS Frontend

A195270 3-gap primes: Prime p is a term iff there is no prime between 3p and 3q, where q is the next prime after p.

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A195271 1.5-gap primes: Prime p is a term iff there is no prime between 1.5p and 1.5q, where q is the next prime after p.

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A195329 Records of A195325.

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A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7prime(k)/2 and 7prime(k+1)/2.

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Author

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Mathematica

Extensions

A194953 Nonzero values of |A194659(n)-A194186(n+1)|.

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cp's OEIS Frontend

A195270 3-gap primes: Prime p is a term iff there is no prime between 3*p and 3*q, where q is the next prime after p.

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Author

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Programs

Maple

Mathematica

A195271 1.5-gap primes: Prime p is a term iff there is no prime between 1.5*p and 1.5*q, where q is the next prime after p.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A195329 Records of A195325.

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Author

Keywords

Comments

Crossrefs

A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7*prime(k)/2 and 7*prime(k+1)/2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A194953 Nonzero values of |A194659(n)-A194186(n+1)|.

Views

Author

Keywords

Comments

Crossrefs

A195270 3-gap primes: Prime p is a term iff there is no prime between 3p and 3q, where q is the next prime after p.

A195271 1.5-gap primes: Prime p is a term iff there is no prime between 1.5p and 1.5q, where q is the next prime after p.

A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7prime(k)/2 and 7prime(k+1)/2.