A194658 -id:A194658 - cp's OEIS frontend

A194659 a(n) = A104272(n) - A194658(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 12, 0, 0, 0, 0, 36, 32, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 18, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 44, 40

Offset: 1

Vladimir Shevelev, Sep 01 2011

nonn

Conjecture 1. The sequence is unbounded.
Records are 0, 18, 36, 48, 64, 84, 114, 138, 184, 202, 214, 268, 282, 366, 374, 378, 412, 444, 528, ... with indices 1, 13, 19, 43, 144, 145, 167, 560, 635, 981, 982, 2605, 3967, 4582, 7422, 7423, 7424, 7425, 10320, ... .
The places of nonzero terms correspond to places of those terms of A194658 which are in A164288. Moreover, for n>=1, places of nonzero terms of A194659 and A194186(n+1) coincide. This means that these sequences have the same lengths of the series of zeros.
Conjecture 2. The asymptotic density of nonzero terms is 2/(e^2+1).

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A104272, A194658, A194186, A164288.

up_to = 65537;
A104272list(n) = { my(L=vector(n), s=0, k=1); for(k=1, prime(3*n)-1, if(isprime(k), s++); if(k%2==0 && isprime(k/2), s--); if(sA104272 by Satish Bysany, Mar 02 2017
v104272 = A104272list(65537);
A104272(n) = v104272[n];
A080359(n) = {my(x = 1); while((primepi(x) - primepi(x\2)) != n, x++; ); x; }; \\ From A080359
A194658(n) = precprime(A080359(1+n)-1);
A194659(n) = (A104272(n) - A194658(n)); \\ Antti Karttunen, Sep 21 2018

A195325 Least n-gap prime: a(n) = least prime p for which there is no prime between np and nq, where q is the next prime after p.

Original entry on oeis.org

2, 59, 71, 29, 59, 149, 191, 641, 149, 347, 809, 461, 3371, 1487, 857, 1301, 1877, 5849, 4721, 9239, 4271, 1619, 1481, 20507, 20981, 32117, 13337, 19379, 24977, 48779, 20441, 25301, 5651, 37991, 17747, 43577, 176777, 145757, 191249, 84809, 150209, 11717

Offset: 1

Vladimir Shevelev, Sep 15 2011

nonn

Such a prime always exists.
The sequence is unbounded.
Conjecture. For n >= 2, a(n) is a lesser of twin primes (A001359). This implies the twin prime conjecture. - Vladimir Shevelev, Sep 15 2011
If a member of this sequence is not the lesser of a twin prime pair, it is greater than 10^10. - Charles R Greathouse IV, Sep 15 2011
A dual sequence: b(n)= least prime p for which there is no prime between n*q and n*p, where q is the previous prime before p. Evidently, b(n) is the next prime after a(n): 3,61,73,31,..., and for n>=2, by the same conjecture, b(n) is a greater of twin primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..169, (first 100 terms from Alois P. Heinz)
Index entries for primes, gaps between

Cf. A080192, A195270, A195271, A164368, A194658, A164294, A110835, A195465.

a:= proc(n) local p, q;
      p:= 2; q:= nextprime(p);
      while nextprime(n*p) < (n*q) do
        p, q:= q, nextprime(q)
      od; p
    end:
seq (a(n), n=1..25); # Alois P. Heinz, Sep 15 2011

pQ[p_, r_] := Block[{q = NextPrime[p]},NextPrime[r*p]> r*q]; f[n_] := Block[{p = 2}, While[ !pQ[p, n], p = NextPrime[p]]; p]; f[1] = 2; Array[f, 42] (* Robert G. Wilson v, Sep 18 2011 *) (* Revised by Zak Seidov, Sep 19 2011 *)

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.

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

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 *)

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

Vladimir Shevelev, Sep 01 2011

nonn

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.

V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes

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

A195377 2.5-gap primes: Prime p is a term if there is no prime between 2.5p and 2.5q, where q is the next prime after p.

Original entry on oeis.org

127, 197, 281, 311, 347, 431, 613, 659, 673, 739, 877, 991, 1049, 1229, 1277, 1289, 1367, 1481, 1579, 1613, 1667, 1721, 1787, 1877, 1907, 2027, 2081, 2087, 2141, 2203, 2213, 2237, 2239, 2269, 2287, 2309, 2377, 2383, 2473, 2657, 2689, 2707, 2749, 2767, 2801

Offset: 1

Vladimir Shevelev, Sep 17 2011

nonn

Cf. A195270, A195271, A195325, A194658, A164294.

Select[Prime[Range[500]], PrimePi[5*NextPrime[#]/2] == PrimePi[5*#/2] &] (* T. D. Noe, Sep 20 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.

A195465 The first a(n) n-gap primes are lessers of twin primes, a(n) maximal.

Original entry on oeis.org

0, 5, 5, 17, 5, 6, 14, 6, 24, 75, 2, 4, 27, 11, 48, 50, 46, 9, 21, 7, 16, 137, 4, 55, 85, 14, 111, 24, 102, 291, 67, 89, 155, 180, 137, 330, 127, 413, 250, 241, 332, 619, 139, 234, 453, 929, 94, 160, 169, 22, 131, 434

Offset: 1

Vladimir Shevelev, Sep 19 2011

nonn

For definition of n-gap primes, see comment to A195270.

Conjecture: a(n)>0 for n>1. This conjecture is equivalent to the conjecture that all terms of A195325 are lessers of twin primes.

Cf. A195325, A080192, A195270, A195271, A164368, A194658, A164294.

a:= proc(n) local i, p, q;
      p, q:= 2, 3;
      for i from 0 do
        while nextprime(n*p) < (n*q) do
          p, q:= q, nextprime(q)
        od;
        if not isprime(p+2) then return i fi;
        p, q:= q, nextprime(q)
      od
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 20 2011

a[n_] := a[n] = Module[{i, p = 2, q = 3}, For[i = 0, True, i++, While[NextPrime[n p] < n q, p = q; q = NextPrime[q]]; If[!PrimeQ[p+2], Return[i]]; p = q; q = NextPrime[q]]];
Array[a, 20] (* Jean-François Alcover, Nov 21 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A194659 a(n) = A104272(n) - A194658(n).

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A195325 Least n-gap prime: a(n) = least prime p for which there is no prime between n*p and n*q, where q is the next prime after p.

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

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A195377 2.5-gap primes: Prime p is a term if there is no prime between 2.5*p and 2.5*q, where q is the next prime after p.

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Mathematica

A195329 Records of A195325.

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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.

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Extensions

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

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A195465 The first a(n) n-gap primes are lessers of twin primes, a(n) maximal.

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Maple

A195325 Least n-gap prime: a(n) = least prime p for which there is no prime between np and nq, where q is the next prime after p.

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.

A195377 2.5-gap primes: Prime p is a term if there is no prime between 2.5p and 2.5q, 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.