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

1, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329

Offset: 2

Author

Cino Hilliard, Jan 26 2005

Keywords

• nonn

Comments

Except for the initial term, these numbers appear to differ by 6. Proof?

Numbers that occur in A101597. - David Wasserman, Mar 26 2008

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

Crossrefs

Cf. A016969, A054342, A101597.

Conjectured partial sums of A329502.

Programs

Mathematica

balancedPrimes = {};compositeGaps = {}; Do[pPrev = Prime[i];p = Prime[i + 1]; pNext = Prime[i + 2]; If[p == (pPrev + pNext)/2, AppendTo[balancedPrimes, p];
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 *)
(* or with balanced primes *)
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 *)

Formula

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

a(n) = nextprime(A054342(n)+1)-A054342(n)-1. - David Wasserman, Mar 26 2008

Extensions

More terms from David Wasserman, Mar 26 2008