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A080765
Integers m such that m+1 divides lcm(1 through m).
10
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
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
KEYWORD
nonn,easy
AUTHOR
Lekraj Beedassy, Mar 10 2003
EXTENSIONS
More terms from Klaus Brockhaus, Jun 11 2004
STATUS
approved