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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.

%I A209266 #34 Jul 22 2025 21:41:34
%S A209266 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,
%T A209266 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,
%U A209266 7,6,8,6,7,9,4,6,5,5,8,3,6,6,5,4,6,5,7,7,8
%N A209266 a(n) is the number of 3-prime arithmetic progression prime chains surrounding the n-th prime number with 5-smooth intervals.
%C A209266 Based on the conjecture in A211376, a(n) > 0.
%C A209266 Last appearance of positive integers in a(n) at n<220000
%C A209266 a(11)=1 (a(n) > 1 for 11<n<220000, the same hereinafter);
%C A209266 a(46)=2; a(10680)=3; a(32293)=4; a(212493)=5
%H A209266 Lei Zhou, <a href="/A209266/b209266.txt">Table of n, a(n) for n = 1..10000</a>
%e A209266 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;
%e A209266 ...
%e A209266 n=43: prime(43)=191, the following 3-prime arithmetic progression prime chains exists:
%e A209266   149,191,233 (gap 42=2*3*7, not 5-smooth)
%e A209266   131,191,251 (gap 60=2^2*3*5, 5-smooth)
%e A209266   113,191,269 (gap 78=2*3*13, not 5-smooth)
%e A209266   101,191,281 (gap 90=2*3^2*5, 5-smooth)
%e A209266   89,191,293  (gap 102=2*3*17, not 5-smooth)
%e A209266   71,191,311  (gap 120=2^3*3*5, 5-smooth)
%e A209266   29,191,353  (gap 162=2*3^4, 5-smooth)
%e A209266   23,191,359  (gap 168=2^3*3*7, not 5-smooth)
%e A209266   3,191,379   (gap 188=2^2*47, not 5-smooth)
%e A209266 Among these groups, there are 4 5-smooth gaps.  So, a(43)=4.
%t A209266 Table[p = Prime[i]; ct = 0; Do[If[(PrimeQ[p - j]) && (PrimeQ[p + j]),
%t A209266    f = Last[FactorInteger[j]][[1]]; If[f <= 5, ct++]], {j, 2, p,
%t A209266    2}]; ct, {i, 3, 89}]
%Y A209266 Cf. A078611, A051037, A001031, A211376
%K A209266 nonn,easy
%O A209266 1,5
%A A209266 _Lei Zhou_, Feb 07 2013