A194659 - OEIS

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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

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OFFSET

1,13

COMMENTS

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).

LINKS

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

PROG

(PARI)

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(s<n, L[s+1] = k+1)); (L); } \\ From A104272 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

CROSSREFS

Cf. A104272, A194658, A194186, A164288.

Sequence in context: A195926 A195929 A247604 * A194186 A033668 A030239

Adjacent sequences: A194656 A194657 A194658 * A194660 A194661 A194662

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Sep 01 2011

STATUS

approved