A080765 Integers m such that m+1 divides lcm(1 through m).

5, 9, 11, 13, 14, 17, 19, 20, 21, 23, 25, 27, 29, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 104, 105, 107

Offset: 1

Author

Lekraj Beedassy, Mar 10 2003

Keywords

• nonn
• easy

Comments

Integers m for which A003418(m) = A003418(m+1).

a(n) = A024619(n) - 1. Proof:

If N+1 is a power of a prime (N+1=P^K), then only smaller powers of that prime divide numbers up to N and so lcm(1..N) doesn't have K powers of P; that is, N+1=P^K doesn't divide lcm(1..N).

From Don Reble, Mar 12 2003: (Start)

If N+1 is not a power of a prime, then it has at least two prime factors. Call one of them P, let K be such that P^K divides N+1, but P^(K+1) doesn't, and let N+1=P^K*R. Then

- R is greater than 1 because it is divisible by another prime factor of N+1;

- P^K and R are each less than N+1 because the other is greater than one;

- lcm(P^K,R) divides lcm(1..N) because 1..N includes both numbers;

- lcm(P^K,R)=N+1 because P doesn't divide R;

- N+1 divides lcm(1..N). (End)

Examples

			17 is the sequence because lcm(1,2,...,17)=12252240 and 17+1=18 divides 12252240.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Andrei Asinowski, Cyril Banderier, Benjamin Hackl, Flip-sort and combinatorial aspects of pop-stack sorting, arXiv:2003.04912 [math.CO], 2020.

Crossrefs

Cf. A003418.

Programs

Mathematica

Select[Range[120], Divisible[LCM @@ Range[#], #+1]&] (* Jean-François Alcover, Jun 21 2018 *)

PARI

a=1;for(n=1,108,a=lcm(a,n);if(a%(n+1)==0,print1(n,","))) \\ Klaus Brockhaus, Jun 11 2004

PARI

first(n) = {my(u = max(2*n, 50), charact = vector(u, i, 1), res = List()); forprime(p = 2, 2*n, for(t = 1, logint(u, p), charact[p^t - 1] = 0)); for(i = 1, u, if(charact[i] == 1, listput(res, i); if(#res >= n, return(res)))); res } \\ David A. Corneth, Aug 30 2019

Sage

[x - 1 for x in (1..108) if not is_prime_power(n)]  # Peter Luschny, May 23 2013

Formula

a(n) ~ n. - David A. Corneth, Aug 30 2019

Extensions

More terms from Klaus Brockhaus, Jun 11 2004