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A064113 Indices of primes prime(i) such that (1/3)*(prime(i)+prime(i+1)+prime(i+2)) is a prime. 13
2, 15, 36, 39, 46, 54, 55, 73, 102, 107, 110, 118, 129, 160, 164, 184, 187, 194, 199, 218, 239, 271, 272, 291, 339, 358, 387, 419, 426, 464, 465, 508, 520, 553, 599, 605, 621, 629, 633, 667, 682, 683, 702, 709, 710, 733, 761, 791, 813, 821, 822, 829, 830 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

n such that d(n) = d(n+1), where d(n)=prime(n+1)-prime(n), or A001223(n).

Of interest because when I generalize it to d(n)=d(n+2), d(n)=d(n+3), etc. I am unable to find any positive number k such that d(n)=d(n+k) has no solution.

From Lei Zhou, Dec 06 2005: (Start)

When (1/3)*(prime(i)+prime(i+1)+prime(i+2)) is prime, then it is equal to prime(i+1).

Alternative definition of this sequence: Indices of primes prime(i) such that (prime(i)+prime(i+2)/2 = prime(i+1).

The mathematica program is based on the alternative definition. (End)

A036263(a(n)) = 0; A122535(n) = A000040(a(n)); A006562(n) = A000040(a(n) + 1); A181424(n) = A000040(a(n) + 2). - Reinhard Zumkeller, Jan 20 2012

A262138(2*a(n)) = 0. - Reinhard Zumkeller, Sep 12 2015

LINKS

Harry J. Smith, Table of n, a(n) for n=1..1000

EXAMPLE

a(2) = 15 because (p(15)+p(16)+p(17)) = 1/3(47 + 53 + 59) = 53 (prime average of three successive primes).

MATHEMATICA

ct = 0; Do[If[(Prime[k] + Prime[k + 2] - 2*Prime[k + 1]) == 0, ct++; n[ct] = k], {k, 1, 2000}]; Table[n[k], {k, 1, ct}] (* Lei Zhou, Dec 06 2005 *)

PROG

(PARI) d(n) = prime(n+1)-prime(n); j=[]; for(n=1, 1500, if(d(n)==d(n+1), j=concat(j, n))); j

(PARI) d(n)= { prime(n + 1) - prime(n) } { n=0; for (m=1, 10^9, if (d(m)==d(m+1), write("b064113.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 07 2009

(Haskell)

import Data.List (elemIndices)

a064113 n = a064113_list !! (n-1)

a064113_list = map (+ 1) $ elemIndices 0 a036263_list

-- Reinhard Zumkeller, Jan 20 2012

CROSSREFS

Cf. A001223, A006562, A024675, A075540, A075541.

Cf. A262138.

Sequence in context: A271909 A075541 A075542 * A007217 A295367 A214541

Adjacent sequences:  A064110 A064111 A064112 * A064114 A064115 A064116

KEYWORD

easy,nonn

AUTHOR

Jason Earls, Sep 08 2001

STATUS

approved

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Last modified September 15 16:44 EDT 2019. Contains 327078 sequences. (Running on oeis4.)