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

Author

Gene Ward Smith, Apr 02 2006

Keywords

• nonn
• hard
• more

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

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A002071, A117583.

Extensions

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

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