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
Keywords
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)
Programs
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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
Comments