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

1,2

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.

Links

Table of n, a(n) for n=1..58.

Mathematica

DeleteDuplicates[Sort[Table[(Prime[n + 1]^2 - Prime[n]^2)/24, {n, 3, 300}]]]

Crossrefs

Cf. A069482, A000040.

Sequence in context: A191211 A263133 A346592 * A144726 A123885 A010062

Adjacent sequences: A254857 A254858 A254859 * A254861 A254862 A254863

Keyword

nonn

Author

Richard R. Forberg, Feb 19 2015

Status

approved