A105418 Smallest prime p such that the sum of it and the following prime has n prime factors including multiplicity, or 0 if no such prime exists.

2, 0, 3, 11, 53, 71, 61, 191, 953, 1151, 3833, 7159, 4093, 30713, 36857, 110587, 360439, 663547, 2064379, 786431, 3932153, 5242877, 9437179, 63700991, 138412031, 169869311, 436207613, 3875536883, 1358954453, 1879048183, 10066329587, 8053063661, 14495514619

Offset: 1

Author

Giovanni Teofilatto and Robert G. Wilson v, Apr 06 2005

Keywords

• nonn

Comments

a(2) = 0 since it is impossible.

Examples

			a(5) = 53 because (53 + 59) = 112 = 2^4*7.
a(24) = 63700991 because (63700991 + 63700993) = 127401984 = 2^19*3^5.
a(28) = 3875536883 because (3875536883 + 3875536909) = 7751073792 = 2^25*3*7*11.
a(29) = 1358954453 because (1358954453 + 1358954539) = 2717908992 = 2^25*3^4.
a(30) = 1879048183 because (1879048183 + 1879048201) = 3758096384 = 2^29*7.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 1..500 (terms 1..38 from Amiram Eldar)

Crossrefs

Cf. A001043, A001222, A071215, A098037, A098048.

Programs

Mathematica

f[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]]; t = Table[0, {40}]; Do[a = f[Prime[n] + Prime[n + 1]]; If[a < 41 && t[[a]] == 0, t[[a]] = Prime[n]; Print[{a, Prime[n]}]], {n, 111500000}]; t

PARI

almost_primes(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, listput(list, m*q)), 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) = if(n==2, return(0)); my(x=2^n, y=2*x); while(1, my(v=almost_primes(x, y, n)); for(k=1, #v, my(p=precprime(max(v[k]>>1, 2)), q=nextprime(p+1)); if(p+q == v[k], return(p))); x=y+1; y=2*x); \\ Daniel Suteu, Aug 06 2024

Extensions

a(28)=3875536883 from Ray Chandler and Robert G. Wilson v, Apr 10 2005

Edited by Ray Chandler, Apr 10 2005

a(31)-a(33) from Daniel Suteu, Nov 18 2018

Definition slightly modified by Harvey P. Dale, Jul 17 2024