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

1,1

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

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

Formula

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

Example

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.

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 *)

Prog

(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;
(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

Crossrefs

Cf. A115186, A154598.

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

Sequence in context: A136211 A041036 A041699 * A047023 A032319 A346060

Adjacent sequences: A154701 A154702 A154703 * A154705 A154706 A154707

Keyword

nonn,changed

Author

Klaus Brockhaus, Jan 14 2009, Jan 15 2009

Extensions

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

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

Status

approved