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A209266
a(n) is the number of 3-prime arithmetic progression prime chains surrounding the n-th prime number with 5-smooth intervals
2
0, 0, 1, 1, 2, 2, 2, 1, 3, 3, 1, 3, 3, 4, 3, 5, 4, 2, 5, 4, 4, 4, 4, 3, 3, 6, 6, 4, 4, 3, 4, 5, 6, 3, 6, 5, 4, 5, 5, 6, 4, 3, 4, 5, 5, 2, 5, 4, 6, 4, 6, 6, 3, 7, 5, 7, 6, 4, 7, 6, 5, 5, 7, 5, 4, 5, 8, 6, 7, 6, 8, 6, 7, 9, 4, 6, 5, 5, 8, 3, 6, 6, 5, 4, 6, 5, 7, 7, 8
OFFSET
1,5
COMMENTS
Based on the conjecture in A211376, a(n) > 0.
Last appearance of positive integers in a(n) at n<220000
a(11)=1 (a(n) > 1 for 11<n<220000, the same hereinafter);
a(46)=2; a(10680)=3; a(32293)=4; a(212493)=5
EXAMPLE
n=3: prime(3)=5, 3,5,7 form a 3-prime arithmetic progression prime chain with the interval of 2, a 5-smooth number. And this is the only case. So a(3)=1;
...
n=43: prime(43)=191, the following 3-prime arithmetic progression prime chains exists:
149,191,233 (gap 42=2*3*7, not 5-smooth)
131,191,251 (gap 60=2^2*3*5, 5-smooth)
113,191,269 (gap 78=2*3*13, not 5-smooth)
101,191,281 (gap 90=2*3^2*5, 5-smooth)
89,191,293 (gap 102=2*3*17, not 5-smooth)
71,191,311 (gap 120=2^3*3*5, 5-smooth)
29,191,353 (gap 162=2*3^4, 5-smooth)
23,191,359 (gap 168=2^3*3*7, not 5-smooth)
3,191,379 (gap 188=2^2*47, not 5-smooth)
Among these groups, there are 4 5-smooth gaps. So, a(43)=4.
MATHEMATICA
Table[p = Prime[i]; ct = 0; Do[If[(PrimeQ[p - j]) && (PrimeQ[p + j]),
f = Last[FactorInteger[j]][[1]]; If[f <= 5, ct++]], {j, 2, p,
2}]; ct, {i, 3, 89}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Feb 07 2013
STATUS
approved