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).

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

Author

Amiram Eldar, Jun 24 2023

Keywords

• nonn

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.

Crossrefs

Cf. A002378, A001222, A059958, A075088, A076550.

Programs

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]

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); }

Extensions

a(29)-a(32) from Martin Ehrenstein, Jul 08 2023