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A339385
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a(n) = (smallest prime >= A002182(n)) - (largest prime <= A002182(n)).
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1
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0, 2, 2, 2, 6, 6, 6, 2, 14, 2, 2, 8, 8, 14, 18, 24, 18, 12, 2, 12, 14, 12, 30, 32, 18, 24, 2, 40, 2, 30, 26, 30, 18, 14, 34, 14, 40, 18, 20, 40, 34, 36, 18, 20, 42, 120, 90, 24, 34, 52, 44, 72, 20, 20, 38, 44, 42, 54, 24, 60, 72, 20, 72, 30, 20, 20, 24, 70
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OFFSET
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2,2
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COMMENTS
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The prime gap size at the n-th highly composite number A002182(n), for n > 2.
The obtained arithmetic mean of the normalized gap size, i.e., a(n)/log(A002182(n)), for the terms 3..10000 is 3.030.
From Gauss's prime counting function approximation, the expected gap size should be approximately log(A002182), however, the observed values seem to be closer to log(A002182(n)^3).
The maximum merit (= a(n)/log(prevprime(A002182))) in the range 3..10000 is 12.96 and is obtained for n = 6911.
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LINKS
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FORMULA
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MATHEMATICA
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s = {}; dm = 1; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[s, NextPrime[n - 1] - NextPrime[n + 1, -1]]], {n, 2, 10^6}]; s (* Amiram Eldar, Dec 02 2020 *)
{0}~Join~Map[Subtract @@ NextPrime[#, {1, -1}] &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[3 ;; 10^3, -1]] ] (* Michael De Vlieger, Dec 10 2020 *)
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PROG
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(PARI) lista(nn) = my(r=1); forstep(n=2, nn, 2, if(numdiv(n)>r, r=numdiv(n); print1(nextprime(n) - precprime(n), ", "))); \\ Michel Marcus, Dec 03 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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