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 *)
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 *)
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
KEYWORD
nonn
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