A322300 a(n) is the least k such that A001222(k)=n and A001222(k+1)=n+1.

1, 3, 26, 99, 495, 728, 1215, 6560, 309824, 1896128, 1043199, 15752960, 178149375, 399112191, 4226550272, 7219625984, 45990608895, 558743781375, 1565795778560, 28996228218879, 63685431525375, 45922887663615, 1956754664980479, 30987856352641023

Offset: 0

Author

Robert Israel and J. M. Bergot, Dec 02 2018

Keywords

• nonn
• more

Comments

Dickson's conjecture implies that a(n) exists for each n.

Examples

			a(3)= 99 because 99=3^2*11 has three prime factors (counted with multiplicity) and 99+1=2^2*5^2 has four, and 99 is the least number with those properties.
		

Crossrefs

Cf. A001222, A076156, A079482.

Programs

Maple

b:= 0:
for n from 2 do
  a:= b;
  b:= numtheory:-bigomega(n);
  if b = a+1 and not assigned(A[a]) then
     A[a]:= n-1;
     if a = 9 then break fi
  fi
od:
seq(A[i],i=0..9);

Mathematica

a[n_] := Module[{k = 1}, While[PrimeOmega[k] != n || PrimeOmega[k + 1] != n + 1, k++]; k]; Array[a, 10, 0] (* Amiram Eldar, Dec 03 2018 *)

PARI

isok(n,k) = bigomega(k) == n && bigomega(k+1) == n+1;
a(n) = for(k=1, oo, if(isok(n,k), return(k))); \\ Daniel Suteu, May 05 2022

PARI

generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(bigomega(m*q-1) == k, listput(list, m*q-1))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n+1, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jul 09 2023

Extensions

a(9)-a(13) from Rémy Sigrist, Dec 03 2018

a(14)-a(18) from Giovanni Resta, Jun 11 2020

a(19)-a(21) from Daniel Suteu, May 05 2022

a(22)-a(23) from Daniel Suteu, Jul 09 2023