A335667 a(n) is the smallest even number k such that k-1 and k+1 are both n-almost primes.

4, 34, 274, 2276, 8126, 184876, 446876, 18671876, 95234374, 1144976876, 6018359374, 281025390626, 2068291015624, 6254345703124, 181171630859374, 337813720703126, 31079046044921876, 205337027587890626

Offset: 1

Author

Zak Seidov and Amiram Eldar, Jun 17 2020

Keywords

• nonn
• more

Comments

10^13 < a(15) <= 181171630859374. - Giovanni Resta, Jun 21 2020

Examples

			a(1) = 4 since 4 - 1 and 4 + 1 are both primes.
a(2) = 34 since 34 - 1 = 33 = 3*11 and 34 + 1 = 35 = 5*7 are both semiprimes.
a(3) = 274 since 274 - 1 = 273 = 3*7*13 and 274 + 1 = 275 = 5^2 * 11 are both 3-almost primes.
		

Links

• David A. Corneth, Upper bounds on a(n) for n = 1..37 where values with * are proven values

Crossrefs

Cf. A001222, A088077, A115186, A154704.

Programs

Mathematica

m = 8; v = Table[0, {m}]; c = 0; o1 = 1; n = 4; While[c < m, o2 = PrimeOmega[n + 1]; If[o1 == o2 && v[[o1]] == 0, c++; v[[o1]] = n]; o1 = o2; n += 2]; v

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, 3, 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, Jul 10 2023

Extensions

a(12)-a(14) from Giovanni Resta, Jun 21 2020

a(15)-a(18) from Daniel Suteu, Jul 10 2023