

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
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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, Flipsort and combinatorial aspects of popstack 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]&] (* JeanFrançois 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



