1,2

Sequence includes all even, square-free "idoneal" or "convenient" numbers (A000926); all members are even and square-free 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 (noe(AT)sspectra.com), Jun 06 2006, Nov 26 2007

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.

G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), p. 56.

T. D. Noe, Table of n, a(n) for n = 1..1000

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].

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.

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 (noe(AT)sspectra.com), Jun 06 2006

Sequence in context: A112861 A140775 A077064 this_sequence A034168 A055745 A001172

Adjacent sequences: A080712 A080713 A080714 this_sequence A080716 A080717 A080718

nonn,nice

Matthew Vandermast (ghodges14(AT)comcast.net), Mar 23 2003