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A064113
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Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.
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35
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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
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OFFSET
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1,1
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COMMENTS
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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.
When (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is prime, then it is equal to prime(k+1).
Also, indices k such that (prime(k)+prime(k+2)/2 = prime(k+1).
The mathematica program is based on the alternative definition. (End)
Inflection and undulation points of the primes, i.e., positions of zeros in A036263, the second differences of the primes. - Gus Wiseman, Mar 24 2020
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LINKS
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EXAMPLE
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a(2) = 15 because (p(15)+p(16)+p(17)) = 1/3(47 + 53 + 59) = 53 (prime average of three successive primes).
Splitting the prime gaps into anti-runs gives: (1,2), (2,4,2,4,2,4,6,2,6,4,2,4,6), (6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6), (6,4,6), ... Then a(n) is the n-th partial sum of the lengths of these anti-runs. - Gus Wiseman, Mar 24 2020
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MATHEMATICA
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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 *)
Join@@Position[Differences[Array[Prime, 100], 2], 0] (* Gus Wiseman, Mar 24 2020 *)
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PROG
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(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
(Python)
from itertools import count, islice
from sympy import prime, nextprime
def A064113_gen(startvalue=1): # generator of terms >= startvalue
c = max(startvalue, 1)
p = prime(c)
q = nextprime(p)
r = nextprime(q)
for k in count(c):
if p+r==(q<<1):
yield k
p, q, r = q, r, nextprime(r)
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CROSSREFS
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The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for weak ascents is A333230.
The version for weak descents is A333231.
The second differences of the primes are A036263.
A triangle for anti-runs of compositions is A106356.
Lengths of maximal runs of prime gaps are A333254.
Anti-runs of compositions in standard order are A333381.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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