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A164510 First differences of A071904 (Odd composite numbers).

%I A164510 #29 Aug 29 2025 14:32:37
%S A164510 6,6,4,2,6,2,4,6,4,2,4,2,6,2,4,6,2,4,4,2,4,2,2,4,6,6,4,2,2,2,2,2,4,4,
%T A164510 2,6,2,2,2,6,2,4,2,4,4,2,4,2,6,2,2,2,6,6,2,2,2,2,4,2,2,2,2,4,6,4,2,6,
%U A164510 2,2,2,4,2,4,2,4,2,6,2,4,6,2,2,2,4,2,2,2,2,2,4,6,4,2,2,2,2,2,4,2,4,2,2,2,6
%N A164510 First differences of A071904 (Odd composite numbers).
%C A164510 Are all terms <=6?
%C A164510 This is A067970 without its first term. [_R. J. Mathar_, Aug 17 2009]
%C A164510 Yes, all terms are at most 6. For a value of 8, we have to have p, p+2, p+4 all prime, and this is possible only for p=3. As a result, 1 would have to be an odd composite number, which it is not. Therefore all terms are <=6. [_J. Lowell_, Aug 17 2009]
%H A164510 Michael De Vlieger, <a href="/A164510/b164510.txt">Table of n, a(n) for n = 1..10000</a>
%H A164510 Joel E. Cohen and Dexter Senft, <a href="https://doi.org/10.7546/nntdm.2025.31.3.494-503">Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often</a>, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 494-503. See p. 495.
%t A164510 Differences@ Select[Range[1, 360, 2], CompositeQ] (* _Michael De Vlieger_, Aug 29 2025 *)
%o A164510 (Python)
%o A164510 from sympy import primepi, isprime
%o A164510 def A164510(n):
%o A164510     m, k = n, primepi(n+1) + n + (n+1>>1)
%o A164510     while m != k:
%o A164510         m, k = k, primepi(k) + n + (k>>1)
%o A164510     for d in range(2, 7, 2):
%o A164510         if not isprime(m+d):
%o A164510             return d # _Chai Wah Wu_, Aug 02 2024
%Y A164510 Cf. A071904.
%K A164510 nonn,changed
%O A164510 1,1
%A A164510 _Zak Seidov_, Aug 14 2009