A254860 Sorted integers m = (prime(n+1)^2 - prime(n)^2)/24, where prime(n) is A000040(n), with duplicates removed.

1, 2, 3, 5, 7, 10, 12, 13, 15, 17, 18, 23, 25, 27, 28, 30, 32, 33, 35, 37, 38, 40, 43, 45, 47, 52, 55, 58, 62, 65, 67, 70, 72, 75, 77, 80, 85, 87, 88, 93, 95, 100, 103, 105, 107, 110, 117, 118, 120, 127, 130, 133, 135, 137, 138, 140, 143, 147

Offset: 1

Author

Richard R. Forberg, Feb 19 2015

Keywords

• nonn

Comments

A069482 gives the values of (prime(n+1)^2 - prime(n)^2), in order, with duplicates.

For n>=3 (prime(n+1)^2 - prime(n)^2)/24 is an integer.

The list here is sorted with duplicates removed to examine the nature and scope coverage over the integers of these ratios.

a(n) values have increasing differences on average, and approximately fit a curve for the n-th distinct value, given by (1/3)*n*log(n) + (3/10)*n*log(log(n))^3 for the first 10,000 values.

The differences between adjacent a(n) values, examined over the first 100,000 values, indicates all integers are covered (i.e., for any integer k there is at least one n where k = a(n+1) - a(n)).

Prime factorization of a(n) indicates every prime will appear as a factor for at least one a(n) value.

Crossrefs

Cf. A069482, A000040.

Programs

Mathematica

DeleteDuplicates[Sort[Table[(Prime[n + 1]^2 - Prime[n]^2)/24, {n, 3, 300}]]]
Union[Differences[Prime[Range[3,300]]^2]/24] (* Harvey P. Dale, Jul 18 2025 *)