A297891 Numbers that divide exactly two Euclid numbers.

277, 1051, 1381, 1657, 1867, 3001, 3373, 3499, 4637, 4877, 5147, 6673, 7547, 10859, 10987, 14797, 17291, 18749, 19531, 25939, 27337, 27953, 31013, 32203, 32983, 33547, 34123, 34591, 35747, 38047, 38197, 38711, 44293, 44357, 47059, 47569, 48809, 51151, 51437

Offset

1,1

Comments

The k-th Euclid number, A006862(k), is 1 plus the product of the first k primes, i.e., 1 + A002110(k). A113165 lists the numbers (> 1) that divide at least one Euclid number; a(1) = 277 = A113165(19); a(2) = 1051 = A113165(41); a(53) = 92143 = A113165(995).

Up to N = 10^5, roughly 5% of the terms in A113165 are also in this sequence. Does that ratio continue to hold as N increases?

It appears that the vast majority of terms in A113165 are prime, but that sequence contains a number of composite numbers as well, beginning with A113165(59) = 1843 = 19*97, A113165(125) = 5263 = 19*277, A113165(195) = 10147 = 73*139, and A113165(231) = 12629 = 73*173. But do any composites divide more than one Euclid number?

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000

Example

a(1) = 277 because 277 is the smallest number that divides exactly two Euclid numbers: 1 + 2*3*5*7*11*13*17 = 510511 and 1 + 2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59 = 1922760350154212639071.

Crossrefs

Cf. A002110 (primorials), A006862 (Euclid numbers), A113165 (numbers > 1 that divide Euclid numbers).

Sequence in context: A142127 A142831 A105977 * A236803 A048525 A189609

Adjacent sequences: A297888 A297889 A297890 * A297892 A297893 A297894

Keyword

nonn

Author

Jon E. Schoenfield, Jan 07 2018

Status

approved