A397212 a(n) is the smallest semiprime prime(n)*q such that some subset of {2*prime(n), 3*prime(n),...,q*prime(n)} has the denominator of reciprocals not divisible by prime(n).

10, 21, 15, 35, 143, 91, 187, 133, 299, 319, 155, 629, 287, 559, 611, 583, 413, 671, 871, 497, 949, 1501, 1411, 1691, 2231, 1313, 1133, 1819, 1853, 1243, 2413, 1441, 3151, 3197, 4321, 1963, 2983, 3749, 1837, 3979, 3401, 3439, 2483, 3281, 3743, 4577, 4009, 5129, 3859

Offset

1,1

Comments

a(n) is never a square and no square semiprime is part of such subset.

If a sum of reciprocals leading to A397192(k) some semiprime is divisible by prime(n) then A397192(k) >= a(n).

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

David A. Corneth, PARI program

Example

a(1) = 10 as the sum of reciprocals of {6, 10}, which is a subset of {3*prime(1), 5*prime(1)} = {6, 10} is 1/6 + 1/10 = 4/15 and the denominator is not divisible by prime(1) = 2. No such subset exists without a semiprime < 10.
a(5) = 143 as the sum of reciprocals of {22, 33, 55, 77, 143}, which is a subset of {2*prime(5), 3*prime(5), ..., 13*prime(5)} = {22, 33, 55, 77, 121, 143}, is 1/22 + 1/33 + 1/55 + 1/77 + 1/143 = 311/2730 the denominator 2730 is not divisible by prime(5) = 11.

Prog

(PARI) a(n)=my(p = prime(n), l = List()); forprime(q = 2, oo, if(q != p, listput(l, p*q), next); for(i = 1, #l, forvec(x = vector(i, j, [1, #l]), c = sum(j = 1, i, 1/l[x[j]]); if(denominator(c) % p != 0, return(p*q)); , 2)))

Crossrefs

Cf. A006881.

Sequence in context: A085222 A085221 A341111 * A128536 A202318 A251128

Adjacent sequences: A397205 A397208 A397209 * A397215 A397216 A397217

Keyword

nonn

Author

David A. Corneth, Jun 18 2026

Status

approved