A354604 -id:A354604 - cp's OEIS frontend

A380139 Prime gaps between 10^m and 10^(m+1), m>=0, sorted first by falling number of occurrences and then by rising gap size, written as an irregular triangle.

2, 1, 4, 4, 6, 2, 8, 6, 4, 2, 10, 8, 12, 14, 18, 20, 6, 2, 4, 10, 12, 8, 14, 18, 16, 22, 24, 20, 30, 28, 26, 34, 32, 36, 6, 2, 4, 12, 10, 8, 18, 14, 16, 20, 22, 24, 30, 28, 26, 36, 32, 34, 40, 38, 42, 52, 44, 50, 46, 54, 58, 48, 56, 60, 62, 64, 72

Offset: 1

Hugo Pfoertner based on an idea by Richard Stephen Donovan, Jan 23 2025

A gap between two primes p1 and p2 is assumed to belong to the range [10^m .. 10^(m+1)[ if 10^m <= (p1+p2)/2 < 10^(m+1). Thus the gap between 7 and 11 is counted in the interval 1 .. 10. Gaps symmetric to 10^k occur for k = 17, 45, ... .

			The triangle begins, with corresponding counts in [...]:
  [2, 1, 1]
   2, 1, 4,
  [7, 7, 6, 1]
   4, 6, 2, 8,
  [37, 32, 27, 16, 14,  8,  7,  1,  1]
    6,  4,  2, 10,  8, 12, 14, 18, 20
  [255, 170, 162, 103, 98, 86, 47, 39, 33, 16, 15, 14, 11,  5,  3,  3,  1,  1]
    6,   2,   4,   10, 12,  8, 14, 18, 16, 22, 24, 20, 30, 28, 26, 34, 32, 36,
  [1641, 1018, 1013, 860, 797, 672, 474, 430, 306, 223, 207, 191, 135, 93, 85, ...]
     6,    2,    4,   12,  10,  8,   18,  14,  16,  20,  22,  24,  30, 28, 26, ...
  [11609, 7040, 6945, 6928, 6163, 4796, 4395, 3749, 2542, 2476, 2164, 1949, ...]
     6,    12,    2,    4,   10,    8,   18,   14,   16,   24,   20,   22,  ...
  6, 12, 2, 4, 10, 18, 8, 14, 24, 16, 30, 20, 22, 28, 26, 36, 42, 34, ...
  6, 12, 4, 2, 10, 18, 8, 14, 24, 30, 16, 20, 22, 28, 26, 36, 42, 34, ...
  6, 12, 10, 4, 2, 18, 8, 14, 24, 30, 16, 20, 22, 28, 36, 26, 42, 34, ...
  6, 12, 18, 10, 2, 4, 8, 24, 30, 14, 20, 16, 22, 36, 28, 26, 42, 34, ...

Hugo Pfoertner, Table of n, a(n) for n = 1..583

Cf. A001223, A028334, A038460, A354604.

Cf. A005597, A173557, A305444 for the asymptotic behavior of gap sizes.

A382529 The composite numbers ordered by decreasing reciprocal-distance from the primes (reciprocal distance defined in comments).

Original entry on oeis.org

4, 6, 9, 8, 10, 15, 12, 14, 21, 26, 16, 25, 18, 20, 27, 34, 22, 33, 24, 35, 39, 28, 50, 30, 32, 45, 56, 49, 36, 51, 38, 64, 55, 40, 57, 42, 44, 63, 76, 46, 120, 65, 119, 93, 48, 69, 86, 121, 118, 92, 75, 52, 54, 94, 77, 122, 117, 81, 58, 85, 60, 62, 87, 123, 91, 144

Offset: 1

James Propp, Mar 30 2025

nonn

The reciprocal distance between m and p is defined as d(m,p) = abs(1/m - 1/p).
The distance between a composite number m and the set of primes is d(m) = Min_{p prime} d(m,p), which means considering p which is the next prime below m, and q the next prime above m.
Bertrand's postulate p > m/2 means d(m) < 1/m so that all m with d(m) > epsilon are m < 1/epsilon.
The plot (e.g., ListPlot in Mathematica) shows interesting large-scale structure.

			a(3) and a(4) are 9 and 8 respectively. 9 precedes 8 in the list of composites because min(1/7-1/9,1/9-1/11) is greater than min(1/7-1/8,1/8-1/11).

James Propp, Table of n, a(n) for n = 1..1440
Hugo Pfoertner, Plot of a(n) and 4*n for n<=250000.
James Propp, Plot of a(n) for n = 1..1440

Cf. A000040, A002386, A002808, A005250, A354604.

epsilon = .0005; (* terms < 1/epsilon *);
table = {}; For[m = 2, m <= 1/epsilon, m++, If[! PrimeQ[m], i = m; While[! PrimeQ[i], --i]; j = m; While[! PrimeQ[j], ++j]; dist = Min[1/i - 1/m, 1/m - 1/j]; If[dist > epsilon, table = Append[table, {dist, m}]]]]; init = Reverse[Sort[table]]; Transpose[init][[2]]

a382529(nterms) = {my(m=nterms+nterms/log(nterms)+3*nterms/log(nterms)^2, mc=floor(m*if(nterms<337963, 4, log(m)/2-1)), C=vectorsmall(mc), L=List(), nc=0); forcomposite(n=4, mc, C[nc++]=n; my(d=min(1/precprime(n)-1/n,1/n-1/nextprime(n))); listput(L,d)); my(P=vecsort(L,,5)); vecextract(Vec(C),P)[1..nterms]}; \\ Hugo Pfoertner, Apr 22 2025

More terms from Hugo Pfoertner, Mar 30 2025

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A380139 Prime gaps between 10^m and 10^(m+1), m>=0, sorted first by falling number of occurrences and then by rising gap size, written as an irregular triangle.

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A382529 The composite numbers ordered by decreasing reciprocal-distance from the primes (reciprocal distance defined in comments).

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