login
A363847
Numbers k such that Omega(m*(m+1)) < Omega(k*(k+1)) for all m < k, where Omega(k) is the number of prime divisors of k counted with multiplicity (A001222).
0
1, 2, 3, 7, 8, 15, 32, 63, 224, 255, 512, 3968, 4095, 14336, 32768, 65535, 180224, 262143, 1048575, 14680064, 16777215, 134217728, 268435455, 1073741823, 8589934592, 12884901887, 34359738368, 68719476735, 1099511627775, 4398046511103, 17592186044415, 35184372088832
OFFSET
1,2
COMMENTS
Terms a(2)-a(18) were found by Erdős and Nicolas (1978-1979).
Equivalently, numbers k such that Omega(m) + Omega(m+1) < Omega(k) + Omega(k+1), for all m < k.
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 20, 22, 24, 26, 27, 31, 33, 34, 37, 38, 39, 40, 46, 48, 50, 51, 52, ... .
LINKS
Paul Erdős and Jean-Louis Nicolas, Sur la fonction "nombre de facteurs premiers de n", Séminaire Delange-Pisot-Poitou, Théorie des nombres, Vol. 20, No. 2 (1978-1979), Talk no. 32, pp. 1-19. See p. 10.
MATHEMATICA
seq[kmax_] := Module[{o1 = 0, o2, om = 0, s = {}}, Do[o2 = PrimeOmega[k]; o = o1 + o2; If[o > om, om = o; AppendTo[s, k - 1]]; o1 = o2, {k, 2, kmax}]; s]; seq[10^5]
PROG
(PARI) lista(kmax) = {my(o1 = 0, o2, om = 0); for(k = 2, kmax, o2 = bigomega(k); o = o1 + o2; if(o > om, om = o; print1(k-1, ", ")); o1 = o2); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 24 2023
EXTENSIONS
a(29)-a(32) from Martin Ehrenstein, Jul 08 2023
STATUS
approved