A185718 For positive n with prime decomposition n = Product_{j=1..m} (p_j^k_j) define A_n = Sum_{j=1..m} (p_j*k_j) and B_n = Sum_{j=1..m} (p_j^k_j). This sequence gives those n for which A_n and B_n are both prime and B_n = A_n + 2 (i.e., form a twin prime pair).

40, 88, 184, 424, 808, 1048, 1384, 1528, 1864, 2104, 2184, 3080, 4504, 4744, 5224, 5928, 6440, 6568, 7224, 8104, 8360, 8840, 9784, 10264, 10472, 11480, 11544, 11848, 12808, 12904, 14136, 14840, 14968, 16280, 16648, 18664, 19608, 20344, 21080, 22040, 23240, 23704, 24440, 24648, 24920, 26008, 26584, 27384, 27608, 27688, 28264, 28952, 29240

Offset

1,1

Comments

Assuming the twin prime conjecture, my advisor and I are able to prove there are infinitely many of these pairs. In other words, there are infinitely many n such that A_n and B_n are prime and B_n = A_n + 2.

From Bobby Browning, Feb 14 2011: (Start)

8*A046138 is a subsequence of A185718 for the following reasons:

i) the n in A185718 for which A_n and B_n form a twin prime pair are of the form n=2^3*p_1*p_2*...p_k.

ii) the A046138 sequence consists of primes p such that p+6 and p+8 form a twin prime pair.

iii) so if p is a prime such that p+6 and p+8 form a twin prime pair and n = 2^3*p then A_n = p+6 and B_n = p+8. Thus, the integers such that n = 2^3*p are a subsequence of A185718. (End)

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Example

a(1) = 40  = 2^3*5^1, with a = 11 and b = 13.
a(2) = 88  = 2^3*11^1 with a = 17 and b = 19.
a(3) = 184 = 2^3*23^1 with a = 29 and b = 31.

Mathematica

okQ[n_] := Module[{p, e, a, b}, {p, e} = Transpose[FactorInteger[n]]; a = Plus @@ (p*e); b = Plus @@ (p^e); b == a + 2 && PrimeQ[a] && PrimeQ[b]]; Select[Range[30000], okQ]

Prog

(PARI) sopfr(n)=my(f=factor(n)); sum(i=1, #f[, 1], f[i, 1]*f[i, 2]);
forstep(n=8, 1e5, 16, if(issquarefree(n/8)&&isprime(k=sopfr(n))&isprime(k+2), print1(n", ")))

Crossrefs

Cf. A001414 (A_n), A008475 (B_n).

Sequence in context: A243803 A203855 A188335 * A043414 A044178 A044559

Adjacent sequences: A185715 A185716 A185717 * A185719 A185720 A185721

Keyword

nonn

Author

Bobby Browning, Feb 10 2011

Status

approved