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A101328 Recurring numbers in the count of consecutive composite numbers between balanced primes and their lower or upper prime neighbors.

%I A101328 #44 Aug 04 2025 05:39:06
%S A101328 1,5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,
%T A101328 131,137,143,149,155,161,167,173,179,185,191,197,203,209,215,221,227,
%U A101328 233,239,245,251,257,263,269,275,281,287,293,299,305,311,317,323,329
%N A101328 Recurring numbers in the count of consecutive composite numbers between balanced primes and their lower or upper prime neighbors.
%C A101328 Except for the initial term, these numbers appear to differ by 6. Proof?
%C A101328 Numbers that occur in A101597. - _David Wasserman_, Mar 26 2008
%C A101328 There is no proof (yet). Heuristic evidence (Hardy-Littlewood 1923) and extensive computations indicates that the balanced-prime structure is not accidental. A theorem of van der Carput (1939) already guarantees infinitely many 3-term arithmetic progressions of primes exist, although not all of those progressions are consecutive primes. A full proof that every such 6k gap occurs infinitely often (and thus infinitely many balanced primes) remains elusive. - _Hilko Koning_, Apr 15 2025
%F A101328 If the numbers continue to differ by 6, then this is the sum of paired terms of 3n+1: (1, 4, 7, 10, 13, ...); and binomial transform of [1, 4, 2, -2, 2, -2, 2, ...]. - _Gary W. Adamson_, Sep 13 2007
%F A101328 a(n) = nextprime(A054342(n)+1)-A054342(n)-1. - _David Wasserman_, Mar 26 2008
%t A101328 balancedPrimes = {};compositeGaps = {}; Do[pPrev = Prime[i];p = Prime[i + 1]; pNext = Prime[i + 2]; If[p == (pPrev + pNext)/2, AppendTo[balancedPrimes, p];
%t A101328 gap1 = p - pPrev - 1; gap2 = pNext - p - 1; AppendTo[compositeGaps, gap1]; AppendTo[compositeGaps, gap2];], {i, 1, 50000}];recurringCounts = Select[Tally[compositeGaps], #[[2]] > 1 &][[All, 1]]; Sort[recurringCounts](* _Hilko Koning_, Apr 15 2025 *)
%t A101328 (* or with balanced primes *)
%t A101328 targetGaps = {1, 5, 11, 17, 23, 29, 35, 41, 47}; gapToBalancedPrimes = Association @@ (Rule[#, {}] & /@ targetGaps); Do[pPrev = Prime[i]; p = Prime[i + 1]; pNext = Prime[i + 2]; If[p == (pPrev + pNext)/2, gap1 = p - pPrev - 1; gap2 = pNext - p -1;uniqueGaps = DeleteDuplicates[{gap1, gap2}]; Do[If[KeyExistsQ[gapToBalancedPrimes, gap], gapToBalancedPrimes[gap] = Append[gapToBalancedPrimes[gap], p]], {gap,uniqueGaps}];], {i, 1, 50000}]; gapToBalancedPrimes (* _Hilko Koning_, Apr 15 2025 *)
%Y A101328 Cf. A016969, A054342, A101597.
%Y A101328 Conjectured partial sums of A329502.
%K A101328 nonn
%O A101328 2,2
%A A101328 _Cino Hilliard_, Jan 26 2005
%E A101328 More terms from _David Wasserman_, Mar 26 2008