%I #58 Dec 09 2021 11:16:36
%S 1,2,6,10,22,30,42,58,70,78,82,102,130,190,210,310,330,358,382,442,
%T 462,478,562,658,742,838,862,970,1038,1222,1282,1318,1618,1810,1870,
%U 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
%N Numbers k such that for any positive integers (a, b), if a * b = k then a + b is prime.
%C Sequence includes all even, squarefree "idoneal" or "convenient" numbers (A000926); all members are even and squarefree except 1 (which is also idoneal).
%C Is it known, or can it be proved, that this sequence is infinite?
%C 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
%C 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
%C 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
%D 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.
%H Amiram Eldar, <a href="/A080715/b080715.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Mircea Becheanu, Florian Luca and Igor E. Shparlinski, <a href="http://dx.doi.org/10.1142/S1793042107001152">On the sums of complementary divisors</a>, Int. J. Number Theory, Vol. 3, No. 4 (2007), pp. 635-648.
%H Günther Frei, <a href="http://dx.doi.org/10.1007/BF03025809">Euler's convenient numbers</a>, Math. Intell., Vol. 7, No. 3 (1985), p. 56.
%e 1 is the product of two positive integers in one way: 1 * 1. The sum of the multiplicands is 2, which is prime.
%e 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.
%t 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 *)
%t Select[Range[10^4], (d=Divisors[#]; And@@PrimeQ[d + # / d])&] (* _Vincenzo Librandi_, Jul 14 2017 *)
%o (Haskell)
%o a080715 n = a080715_list !! (n-1)
%o a080715_list = 1 : filter (\x -> all ((== 1) . a010051) $
%o zipWith (+) (a027750_row x) (reverse $ a027750_row x)) [2,4..]
%o -- _Reinhard Zumkeller_, Apr 12 2012
%o (PARI) is_ok(n)=fordiv(n,d,if(!isprime(d+n/d),return(0)));return(1);
%o for(n=1,10^4,if(is_ok(n),print1(n,", "))); \\ _Joerg Arndt_, Jul 10 2014
%Y Cf. A010051, A027750.
%K nonn,nice
%O 1,2
%A _Matthew Vandermast_, Mar 23 2003