A164368 -id:A164368 - cp's OEIS frontend

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

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 *)

A164966 Primes which are obtained at least by two ways using the iterations of the S operator (see A164960) beginning with primes of the union of {2,3} and A164333.

Original entry on oeis.org

127, 149, 211, 223, 257, 307, 431, 449

Offset: 1

Vladimir Shevelev, Sep 02 2009

The sequence is connected with our sieve selecting the primes of the union of {2,3} and A164333 from all primes.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A055496, A164333 A164960, A164962, A164920, A006992, A164917, A164918, A104272, A080359, A164368, A164288.

A166575 Primes p>=5 with the property: if Prime(k)
=Prime(k)+ Prime(k+1).

Original entry on oeis.org

5, 13, 19, 31, 37, 43, 53, 61, 71, 73, 79, 101, 103, 113, 131, 139, 157, 163, 173, 191, 193, 199, 211, 223, 241, 251, 269, 271, 277, 293, 311, 313, 331, 353, 373, 379, 397, 419, 421, 439, 443, 457, 463, 499, 509, 521, 523, 541, 577, 601, 607, 613, 619, 631, 653, 659, 661, 673, 691

Offset: 1

Vladimir Shevelev, Oct 17 2009

nonn

If A(x) is the counting function of a(n) not exceeding x, then, in view of the symmetry, it is natural to conjecture that A(x)~pi(x)/2.

			Let p=13. Then we have 5<13/2<7. Since 13>5+7, then 13 is in the sequence.

Cf. A166574, A182365, A166307, A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554

Reap[Do[p=Prime[n]; k=PrimePi[p/2]; If[p>=Prime[k]+Prime[k+1], Sow[p]], {n,3,PrimePi[1000]}]][[2,1]]

A182451 Numbers of A166252 which are not in A164554.

Original entry on oeis.org

109, 151, 191, 229, 233, 283, 311, 571, 643, 683, 727, 941, 991, 1033, 1051, 1373, 1493, 1667, 1697, 1741, 1747, 1783, 1787, 1801, 1931

Offset: 1

Vladimir Shevelev, Apr 29 2012

nonn

All Ramanujan primes (A104272) are in A164368 and all Labos primes (A080359) are in A194598. Peculiar primes (see comment in A164554)are simultaneously Ramanujan and Labos primes, while central primes (A166252) are in the intersection of A164368 and A194598 for n>=2. Hence, for n>=2, all peculiar primes are central primes, but conversely is not true. The sequence lists non-peculiar central numbers.

Cf. A164554, A166252, A104272, A080359, A164368, A194598.

cp's OEIS Frontend

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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Mathematica

A164966 Primes which are obtained at least by two ways using the iterations of the S operator (see A164960) beginning with primes of the union of {2,3} and A164333.

Views

Author

Keywords

Comments

Links

Crossrefs

A166575 Primes p>=5 with the property: if Prime(k)
=Prime(k)+ Prime(k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A182451 Numbers of A166252 which are not in A164554.

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Author

Keywords

Comments

Crossrefs

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

A164966 Primes which are obtained at least by two ways using the iterations of the S operator (see A164960) beginning with primes of the union of {2,3} and A164333.

Views

Author

Keywords

Comments

Links

Crossrefs

A166575 Primes p>=5 with the property: if Prime(k)=Prime(k)+ Prime(k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A182451 Numbers of A166252 which are not in A164554.

Views

Author

Keywords

Comments

Crossrefs

A166575 Primes p>=5 with the property: if Prime(k)
=Prime(k)+ Prime(k+1).