A154704 a(n) = smallest number k such that k-1 and k+1 both have exactly n prime divisors (counted with multiplicity).

4, 5, 19, 55, 271, 1889, 10529, 59777, 101249, 406783, 6581249, 12164095, 65071999, 652963841, 6548416001, 13858918399, 145046192129, 75389157377, 943344975871, 23114453401601, 108772434771967, 101249475018751, 551785225781249, 9740041658826751, 136182187711004671, 4560483868737535

Offset: 1

Author

Klaus Brockhaus, Jan 14 2009, Jan 15 2009

Keywords

• nonn

Comments

Similar to A154598, where k is restricted to primes.

m=2*a(n) is the least number m such that m-2 and m+2 have exactly n+1 prime factors, counted with multiplicity. - Hugo Pfoertner, Apr 02 2024

Examples

			For k = 4, k-1 = 3 and k+1 = 5 (twin primes) both have one factor and 4 is the smallest such number.
For k = 55, k-1 = 54 = 2*3*3*3 and k+1 = 56 = 2*2*2*7 both have four factors and 55 is the smallest such number.
For k = 59777, k-1 = 59776 = 2*2*2*2*2*2*2*467 and k+1 = 59778 = 2*3*3*3*3*3*3*41 both have eight factors and 59777 is the smallest such number.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..36
• David A. Corneth, Upper bounds on a(n) for n = 1..89

Crossrefs

Cf. A115186, A154598.

Cf. A001222 (number of prime divisors of n).

Programs

Magma

S:=[]; for n:=1 to 10 do k:=3; while not &+[ f[2]: f in Factorization(k-1) ] eq n or not &+[ f[2]: f in Factorization(k+1) ] eq n do k+:=1; end while; Append(~S, k); end for; S;

Mathematica

a[n_]:=Module[{k=2}, While[PrimeOmega[k-1]!=n || PrimeOmega[k+1]!=n, k++]; k]; Array[a,26] (* Stefano Spezia, Apr 02 2024 *)
Flatten[Table[Position[Partition[PrimeOmega[Range[410000]],3,1],?(#[[1]]==#[[3]]==n&),1,1],{n,10}]]+1//Quiet (* The program generates the first ten terms of the sequence. *) (* _Harvey P. Dale, Jul 21 2024 *)
Table[SequencePosition[PrimeOmega[Range[410000]],{n,,n},1],{n,10}][[;;,1,1]]+1 (* The program generates the first ten terms of the sequence. *) (* _Harvey P. Dale, Aug 29 2024 *)

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+2) == 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, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Aug 12 2023

Formula

a(n) = 2*A115186(n-1) + 1 for n > 1. - Hugo Pfoertner, Apr 02 2024

Extensions

a(15)-a(23) from Donovan Johnson, Jan 21 2009

a(24)-a(26) from Daniel Suteu, Aug 12 2023