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A080715
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Numbers k such that for any positive integers (a, b), if a * b = k then a + b is prime.
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14
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1, 2, 6, 10, 22, 30, 42, 58, 70, 78, 82, 102, 130, 190, 210, 310, 330, 358, 382, 442, 462, 478, 562, 658, 742, 838, 862, 970, 1038, 1222, 1282, 1318, 1618, 1810, 1870, 1978, 2038, 2062, 2098, 2242, 2398, 2458, 2578, 2902, 2938, 2962, 3018, 3082, 3322, 3642, 3862, 4218, 4258, 4282, 4678, 5098, 5590, 5938, 6042, 6078
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OFFSET
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1,2
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COMMENTS
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Sequence includes all even, squarefree "idoneal" or "convenient" numbers (A000926); all members are even and squarefree except 1 (which is also idoneal).
Is it known, or can it be proved, that this sequence is infinite?
Let p and p+2 be twin primes. If 2p+1 is also prime, 2p is in this sequence. - T. D. Noe, Jun 06 2006, Nov 26 2007
2*A045536 are the n with two prime factors. 2*A128279 are the n with three prime factors. 2*A128278 are the n with four prime factors. 2*A128277 are the n with five prime factors. 2*A128276 lists the least n having k prime factors. - T. D. Noe, Nov 14 2010
Numbers n such that d + n/d is prime for every d|n. Then n+1 is a prime p = 2 or p == 3 (mod 4). - Thomas Ordowski, Apr 12 2013
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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EXAMPLE
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1 is the product of two positive integers in one way: 1 * 1. The sum of the multiplicands is 2, which is prime.
310 (2*5*31) is the product of two positive integers in 4 ways: 1 * 310, 2 * 155, 5 * 62 and 10 * 31. The sums of the pairs of multiplicands are 311, 157, 67 and 41, respectively; all are primes.
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MATHEMATICA
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t={}; Do[ds=Divisors[n]; If[EvenQ[Length[ds]], ok=True; k=1; While[k<=Length[ds]/2 && (ok=PrimeQ[ds[[k]]+ds[[ -k]]]), k++ ]; If[ok, AppendTo[t, n]]], {n, 2, 4000}]; t (* T. D. Noe, Jun 06 2006 *)
Select[Range[10^4], (d=Divisors[#]; And@@PrimeQ[d + # / d])&] (* Vincenzo Librandi, Jul 14 2017 *)
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PROG
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(Haskell)
a080715 n = a080715_list !! (n-1)
a080715_list = 1 : filter (\x -> all ((== 1) . a010051) $
zipWith (+) (a027750_row x) (reverse $ a027750_row x)) [2, 4..]
(PARI) is_ok(n)=fordiv(n, d, if(!isprime(d+n/d), return(0))); return(1);
for(n=1, 10^4, if(is_ok(n), print1(n, ", "))); \\ Joerg Arndt, Jul 10 2014
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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