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

Author

Lei Zhou, Feb 07 2013

Keywords

• nonn
• easy

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

a(46)=2; a(10680)=3; a(32293)=4; a(212493)=5

Examples

			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.
		

Links

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

Crossrefs

Cf. A078611, A051037, A001031, A211376

Programs

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}]

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