A080715 Numbers k such that for any positive integers (a, b), if a * b = k then a + b is prime.

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

Offset: 1

Author

Matthew Vandermast, Mar 23 2003

Keywords

• nonn
• nice

Comments

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

Examples

			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.
		

References

• 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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• 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].
• Mircea Becheanu, Florian Luca and Igor E. Shparlinski, On the sums of complementary divisors, Int. J. Number Theory, Vol. 3, No. 4 (2007), pp. 635-648.
• Günther Frei, Euler's convenient numbers, Math. Intell., Vol. 7, No. 3 (1985), p. 56.

Crossrefs

Cf. A010051, A027750.

Programs

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..]
-- Reinhard Zumkeller, Apr 12 2012

Mathematica

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 *)

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