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

1,2

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

Table of n, a(n) for n=1..32.

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

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

Sequence in context: A307328 A097095 A101755 * A076550 A062269 A244508

Adjacent sequences: A363844 A363845 A363846 * A363848 A363849 A363850

Keyword

nonn

Author

Amiram Eldar, Jun 24 2023

Extensions

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

Status

approved