A204142 -id:A204142 - cp's OEIS frontend

A204580 Number of primes of the form A204142[i]*A204142[j]+2 (i,j <= n), larger than A204142[n].

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 15, 17, 20, 24, 27, 30, 32, 32, 34, 36, 39, 41, 43, 46, 50, 51, 54, 60, 64, 68, 71, 76, 78, 80, 83, 86, 92, 93, 98, 101, 104, 106, 109, 113, 114, 117, 126, 128, 133, 135, 139, 143, 147, 150, 160, 166, 173, 181

Offset: 1

M. F. Hasler, Feb 11 2012

nonn

It is not yet known whether there is a finite number of "good numbers" A204142. This is equivalent to the question whether this sequence will eventually decrease to zero.

One has a(100)=408, a(316)=2179, a(1000)=13136, a(3162)=86603, a(10000)=609377.

M. F. Hasler, Table of n, a(n) for n = 1..3000

A204580(n)={ my(r=1,s); while( A204142[r]^2+2<=A204142[n], r++); s=r; sum(i=r,n, while(s>1 & A204142[s-1]*A204142[i]+2 > A204142[n],s--); sum(j=s,i, is/*pseudo*/prime(A204142[j]*A204142[i]+2)))} /* From n~3000 on, use of ispseudoprime is significantly faster. The vector A204142 must be already computed. */

A063638 Primes p such that p-2 is a semiprime.

Original entry on oeis.org

11, 17, 23, 37, 41, 53, 59, 67, 71, 79, 89, 97, 113, 131, 157, 163, 179, 211, 223, 239, 251, 269, 293, 307, 311, 331, 337, 367, 373, 379, 383, 397, 409, 419, 439, 449, 487, 491, 499, 503, 521, 547, 593, 599, 613, 631, 673, 683, 691, 701, 709, 719, 733, 739

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Primes of form p*q + 2, where p and q are primes.
11 is the only prime of this form where p=q. For prime p>3, 3 divides p^2+2. - T. D. Noe, Mar 01 2006
The asymptotic growth of this sequence is relevant for A204142. We have a(10^k) = (11, 79, 1571, 27961, 407741, 5647823, ...). - M. F. Hasler, Feb 13 2012

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A005385, A001358, A063637, A109611 (Chen primes), A204142, A064911, A040976, A241809.

a063638 n = a063638_list !! (n-1)
a063638_list = map (+ 2) $ filter ((== 1) . a064911) a040976_list
-- Reinhard Zumkeller, Feb 22 2012

Take[Select[ # + 2 & /@ Union[Flatten[Outer[Times, Prime[Range[100]], Prime[Range[100]]]]], PrimeQ], 60]
Select[Prime[Range[200]],PrimeOmega[#-2]==2&] (* Paolo Xausa, Oct 30 2023 *)

n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p - 2) == 2, write("b063638.txt", n++, " ", p); if (n==1000, break))) \\ Harry J. Smith, Aug 26 2009

forprime(p=3,9999, bigomega(p-2)==2 & print1(p","))

p=2; for(n=1,1e4, until(bigomega(-2+p=nextprime(p+1))==2,); write("b063638.txt", n" "p)) \\ M. F. Hasler, Feb 13 2012

list(lim)=my(v=List(), t); forprime(p=3, (lim-2)\3, forprime(q=3, min((lim-2)\p, p), t=p*q+2; if(isprime(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 05 2016

a(n) = A241809(n) + 2. - Hugo Pfoertner, Oct 30 2023

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A204580 Number of primes of the form A204142[i]*A204142[j]+2 (i,j <= n), larger than A204142[n].

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A063638 Primes p such that p-2 is a semiprime.

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