cp's OEIS Frontend

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.

%I A117581 #34 Apr 13 2025 14:55:57
%S A117581 2,9,81,4375,9801,123201,336141,11859211,11859211,177182721,
%T A117581 1611308700,3463200000,63927525376,421138799640,1109496723126,
%U A117581 1453579866025,20628591204481,31887350832897,31887350832897,119089041053697,2286831727304145,9591468737351909376,9591468737351909376,9591468737351909376,9591468737351909376,9591468737351909376,19316158377073923834001
%N 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.
%C A117581 By a theorem of Størmer, the number of such integers is finite; moreover he provides an algorithm for finding the complete list.
%C A117581 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.
%C A117581 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
%H A117581 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/stormer.py">stormer.py</a>.
%H A117581 D. H. Lehmer, <a href="http://projecteuclid.org/euclid.ijm/1256067456">On a problem of Størmer</a>, Ill. J. Math., 8 (1964), 57-79.
%H A117581 Wikipedia, <a href="http://en.wikipedia.org/wiki/Stormer%27s_theorem">Størmer's theorem</a>.
%F A117581 a(n) = A002072 + 1.
%Y A117581 Cf. A002071, A002072, A116486, A117582, A117583.
%K A117581 nonn,hard
%O A117581 1,1
%A A117581 _Gene Ward Smith_, Mar 29 2006
%E A117581 Entry edited by _N. J. A. Sloane_, Apr 01 2006
%E A117581 Corrected and extended by _Don Reble_, Nov 21 2006
%E A117581 More terms from A002072 added by _Amiram Eldar_, Apr 13 2025