cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A194674 Positions of nonzero terms of A194659(n)-A194186(n+1), n>=1.

Original entry on oeis.org

20, 27, 73, 77, 85, 95, 106, 116, 117, 122, 125, 132, 137, 144, 145, 152, 162, 167, 168, 189, 191, 192, 193, 198, 201, 208, 213, 234, 235, 236, 243, 249, 258, 259, 265, 275, 279, 286, 287, 291, 318, 319, 321, 329, 330, 331, 340
Offset: 1

Views

Author

Vladimir Shevelev, Sep 01 2011

Keywords

Comments

The sequence (together with A194953) 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.

Links

Crossrefs

Cf. A104272, A080359, A193507, A194186, A164368, A194598, A194658, A194659, A194953, A164288, A164294.

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

Views

Author

Vladimir Shevelev, Sep 06 2011

Keywords

Comments

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.

Crossrefs

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

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.

Original entry on oeis.org

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

Views

Author

Vladimir Shevelev, Sep 14 2011

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 1.5*p and 1.5*q, 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

Views

Author

Vladimir Shevelev, Sep 14 2011

Keywords

Comments

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.

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Vladimir Shevelev, Sep 15 2011

Keywords

Comments

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.

Crossrefs

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 7*prime(k)/2 and 7*prime(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

Views

Author

Vladimir Shevelev, Sep 17 2011

Keywords

Crossrefs

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

Programs

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

Extensions

Corrected by R. J. Mathar, Sep 20 2011
Showing 1-6 of 6 results.

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