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A185718
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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).
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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)
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]
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PROG
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(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", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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