A117582 The number of ratios t/(t-1), where t is a square number, which factor into primes less than or equal to prime(n).

0, 2, 5, 10, 15, 24, 34, 46, 57, 74, 90, 114, 141, 174, 208, 244, 287, 334, 387

Offset

1,2

Comments

By a theorem of Størmer, the number of ratios m/(m-1) factoring into primes only up to p is finite. Some of these have square numerators.

Equivalently, a(n) is the number of triples of consecutive prime(n)-smooth numbers. - Lucas A. Brown, Oct 04 2022

Links

Table of n, a(n) for n=1..19.

Lucas A. Brown, stormer.py.

E. F. Ecklund and R. B. Eggleton, Prime factors of consecutive integers, Amer. Math. Monthly, 79 (1972), 1082-1089.

D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79.

Wikipedia, Størmer's theorem

Example

The ratios counted by a(3) are 4/3, 9/8, 16/15, 25/24, and 81/80.
The ratios counted by a(4) are 4/3, 9/8, 16/15, 25/24, 36/35, 49/48, 64/63, 81/80, 225/224, and 2401/2400.

Crossrefs

Cf. A002071, A117583.

Sequence in context: A163059 A099738 A064513 * A002134 A243971 A062472

Adjacent sequences: A117579 A117580 A117581 * A117583 A117584 A117585

Keyword

nonn,hard,more

Author

Gene Ward Smith, Apr 02 2006

Extensions

Offset 1 and a(14)-a(18) by Lucas A. Brown, Oct 04 2022

a(19) from Lucas A. Brown, Oct 16 2022

Status

approved