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

1,2

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

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.

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A010051, A027750.

Sequence in context: A186296 A140775 A077064 * A034168 A055745 A371283

Adjacent sequences: A080712 A080713 A080714 * A080716 A080717 A080718

Keyword

nonn,nice

Author

Matthew Vandermast, Mar 23 2003

Status

approved