A198277 -id:A198277 - cp's OEIS frontend

A066938 Primes of the form p*q+p+q, where p and q are primes.

11, 17, 23, 31, 41, 47, 53, 59, 71, 79, 83, 89, 107, 113, 127, 131, 151, 167, 179, 191, 227, 239, 251, 263, 269, 271, 293, 311, 359, 383, 419, 431, 439, 443, 449, 479, 491, 503, 521, 587, 593, 599, 607, 631, 647, 659, 683, 701, 719, 727, 743, 773, 809, 827

Offset: 1

Reinhard Zumkeller, Jan 24 2002

nonn

For p not equal to q, either p*q or p+q is odd, so their sum is odd.
The representation is ambiguous, e.g. 2*7+2+7 = 23 = 3*5+3+5.
Complement of A198273 with respect to A000040. - Reinhard Zumkeller, Oct 23 2011
None of these primes are in A158913 since if p*q+p+q is a prime, then sigma(p*q+p+q) = sigma(p*q). - Amiram Eldar, Nov 15 2021

			59 is in the sequence because 59 = 2 * 19 + 2 + 19.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A054973, A067432, A072668, A072673, A088709, A158913, A198273, A198277.

a066938 n = a066938_list !! (n-1)
a066938_list = map a000040 $ filter ((> 0) . a067432) [1..]
-- Reinhard Zumkeller, Oct 23 2011

nn = 1000; n2 = PrimePi[nn/3]; Select[Union[Flatten[Table[(Prime[i] + 1) (Prime[j] + 1) - 1, {i, n2}, {j, n2}]]], # <= nn && PrimeQ[#] &]

is(n)=fordiv(n+1,d,my(p=d-1,q=(n+1)/d-1); if(isprime(p) && isprime(q), return(isprime(n)))); 0 \\ Charles R Greathouse IV, Jul 23 2013

A067432(A049084(a(n))) > 0. - Reinhard Zumkeller, Oct 23 2011

A054973(a(n)+1) >= 2. - Amiram Eldar, Nov 15 2021

Edited by Robert G. Wilson v, Feb 01 2002

A067432 Number of ways to represent the n-th prime in form p*q+p+q, where p and q are primes (see A066938).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 4, 0, 3, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 4, 0, 1, 1, 1, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Feb 15 2002

nonn

a(A049084(A066938(n))) > 0; a(A049084(A198273(n))) = 0; a(A049084(A198277(n))) = n and a(A049084(m)) <> n for m < A198277(n). [Reinhard Zumkeller, Oct 23 2011]

a(n) < A072670(n).

			a(15) = 2 as A000040(15) = 47 = 3*11+3+11 = 5*7+5+7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000196, A010051.

a067432 n = length [p | let prime_n = a000040 n,
   p <- takeWhile (< a000196 prime_n) a000040_list,
   let (q,m) = divMod (prime_n - p) (p + 1),
   m == 0, a010051 q == 1]
a067432_list = map a067432 [1..]
-- Reinhard Zumkeller, Oct 23 2011

A347008 Numbers that can be written in exactly two ways as p*q+p+q where p and q are primes with p < q.

Original entry on oeis.org

23, 47, 119, 167, 179, 323, 407, 419, 527, 587, 639, 647, 879, 935, 1043, 1103, 1119, 1139, 1215, 1223, 1247, 1271, 1331, 1367, 1403, 1455, 1595, 1599, 1631, 1691, 1775, 1791, 1859, 1895, 1931, 1943, 1959, 1967, 1979, 2099, 2111, 2175, 2183, 2219, 2231, 2435, 2471, 2483, 2495, 2543, 2559, 2603

Offset: 1

J. M. Bergot and Robert Israel, Aug 10 2021

nonn

			a(3) = 119 is a term because 119 = 5*19+5+19 = 3*29+3+29 are the two ways to produce 119 = p*q+p+q with primes p < q.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A198277.

N:= 10000: # to produce terms <= N
R:= Vector(N):
P:= select(isprime, [2,seq(i,i=3..N/3,2)]):
for i from 1 to nops(P) do
  for j from 1 to i-1 do
   v:=P[i]*P[j]+P[i]+P[j];
   if v <= N then R[v]:= R[v]+1 fi
od od:
select(t -> R[t]=2, [$1..N]);

from sympy import primerange
from collections import Counter
def aupto(limit):
    primes = list(primerange(2, limit//3+1))
    nums = [p*q+p+q for i, p in enumerate(primes) for q in primes[i+1:]]
    counts = Counter([k for k in nums if k <= limit])
    return sorted(k for k in counts if counts[k] == 2)
print(aupto(2604)) # Michael S. Branicky, Aug 10 2021

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A066938 Primes of the form p*q+p+q, where p and q are primes.

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A067432 Number of ways to represent the n-th prime in form p*q+p+q, where p and q are primes (see A066938).

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A347008 Numbers that can be written in exactly two ways as p*q+p+q where p and q are primes with p < q.

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