A321169 a(n) is the smallest prime p such that p + 2 is a product of n primes (counted with multiplicity).

3, 2, 43, 79, 241, 727, 3643, 15307, 19681, 164023, 1673053, 885733, 2657203, 18600433, 23914843, 100442347, 358722673, 645700813, 4519905703, 18983603959, 48427561123, 31381059607, 261508830073, 1307544150373, 3295011258943, 24006510600883, 12709329141643, 53379182394907, 190639937124673, 2615579937350539

Offset

1,1

Comments

a(n) ~ c * 3^n. - David A. Corneth, Jan 11 2019

Links

David A. Corneth, Table of n, a(n) for n = 1..801

Example

a(1) = 3 as 3 + 2 = 5 (prime),
a(2) = 2 as 2 + 2 = 4 = 2*2 (semiprime),
a(3) = 43 as 43 + 2 = 45 = 3*3*5  (3-almost prime),
a(4) = 79 as 79 + 2 = 81 = 3*3*3*3 (4-almost prime).

Mathematica

ptns[n_, 0] := If[n == 0, {{}}, {}]; ptns[n_, k_] := Module[{r}, If[n < k, Return[{}]]; ptns[n, k] = 1 + Union @@ Table[PadRight[#, k] & /@ ptns[n - k, r], {r, 0, k}]]; a[n_] := Module[{i, l, v}, v = Infinity; For[i = n, True, i++, l = (Times @@ Prime /@ # &) /@ ptns[i, n]; If[Min @@ l > v, Return[v]]; minp = Min @@ Select[l - 2, PrimeQ]; If[minp < v, v = minp]]] ; Array[a, 10] (* after Amarnath Murthy at A073919 *)

Prog

(PARI) a(n) = forprime(p=2, , if (bigomega(p+2) == n, return (p))); \\ Michel Marcus, Jan 10 2019
(PARI) a(n) = {my(p3 = 3^n, u, c); if(n <= 2, return(4 - n)); if(isprime(p3 - 2), return(p3 - 2)); forprime(p = 5, oo, if(isprime(p3 / 3 * p - 2), u = p3 / 3 * p - 2; break ) ); for(i = 2, n, if(p3 * (5/3)^i > u, return(u)); for(j = 1, oo, if(p3 * j \ 3^i > u, next(2)); if(bigomega(j) == i, if(isprime(p3 / 3^(i) * j - 2), u = p3 / 3^(i) * j - 2; next(2) ) ) ) ); return(u) } \\ David A. Corneth, Jan 11 2019

Crossrefs

Cf. A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A046314, A073919, A118883.

Sequence in context: A179984 A260888 A065085 * A093398 A002680 A009395

Adjacent sequences: A321166 A321167 A321168 * A321170 A321171 A321172

Keyword

nonn

Author

Amiram Eldar and Zak Seidov, Jan 10 2019

Status

approved