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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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Lei Zhou, Table of n, a(n) for n = 1..10000

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

Cf. A078611, A051037, A001031, A211376

Sequence in context: A127953 A260418 A280919 * A269400 A130267 A060610

Adjacent sequences:  A209263 A209264 A209265 * A209267 A209268 A209269

KEYWORD

nonn,easy

AUTHOR

Lei Zhou, Feb 07 2013

STATUS

approved

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Last modified August 10 11:52 EDT 2020. Contains 336379 sequences. (Running on oeis4.)