A209266 a(n) is the number of 3-prime arithmetic progression prime chains surrounding the n-th prime number with 5-smooth intervals

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

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 * A359930 A269400 A130267

Adjacent sequences: A209263 A209264 A209265 * A209267 A209268 A209269

Keyword

nonn,easy

Author

Lei Zhou, Feb 07 2013

Status

approved