A117581 For each successive prime p, the largest integer n such that both n and n-1 factor into primes less than or equal to p.

2, 9, 81, 4375, 9801, 123201, 336141, 11859211, 11859211, 177182721, 1611308700, 3463200000, 63927525376, 421138799640, 1109496723126, 1453579866025, 20628591204481, 31887350832897, 31887350832897, 119089041053697

Offset

1,1

Comments

By a theorem of Størmer, the number of such integers is finite; moreover he provides an algorithm for finding the complete list.

Størmer came to this problem from music theory. Another way to formulate the statement of the theorem is that for any prime p, there are only a finite number of superparticular ratios R = n/(n-1) such that R factors into primes less than or equal to p. The numerator of the smallest such R for the i-th prime is the i-th element of the above sequence. For instance, 81/80, the syntonic comma, is the smallest 5-limit superparticular "comma", i.e., small ratio greater than one.

An effective abc conjecture (c < rad(abc)^2) would imply that a(21) = 2286831727304145 and a(22) = ... = a(26) = 9591468737351909376 and a(27) = ... = a(32) = 19316158377073923834001 and a(33) = 124225935845233319439174. - Lucas A. Brown, Oct 16 2022

Links

Table of n, a(n) for n=1..20.

Lucas A. Brown, stormer.py.

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

Wikipedia, Størmer's theorem.

Crossrefs

Cf. A002071, A116486, A117582, A117583. Equals A002072(n) + 1.

Sequence in context: A147302 A301861 A112670 * A110567 A123570 A006040

Adjacent sequences: A117578 A117579 A117580 * A117582 A117583 A117584

Keyword

nonn,hard

Author

Gene Ward Smith, Mar 29 2006

Extensions

Entry edited by N. J. A. Sloane, Apr 01 2006

Corrected and extended by Don Reble, Nov 21 2006

Status

approved