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
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.
Programs
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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..] -- Reinhard Zumkeller, Apr 12 2012
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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
Comments