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

1,1

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)

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.

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A003418.

Sequence in context: A049049 A118358 A101731 * A226039 A257292 A234285

Adjacent sequences: A080762 A080763 A080764 * A080766 A080767 A080768

Keyword

nonn,easy

Author

Lekraj Beedassy, Mar 10 2003

Extensions

More terms from Klaus Brockhaus, Jun 11 2004

Status

approved