A397212 - OEIS

%I #17 Jun 19 2026 13:01:24

%S 10,21,15,35,143,91,187,133,299,319,155,629,287,559,611,583,413,671,

%T 871,497,949,1501,1411,1691,2231,1313,1133,1819,1853,1243,2413,1441,

%U 3151,3197,4321,1963,2983,3749,1837,3979,3401,3439,2483,3281,3743,4577,4009,5129,3859

%N 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).

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

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

%H David A. Corneth, <a href="/A397212/b397212.txt">Table of n, a(n) for n = 1..10000</a>

%H David A. Corneth, <a href="/A397212/a397212.gp.txt">PARI program</a>

%e 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.

%e 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.

%o (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)))

%Y Cf. A006881.

%K nonn

%O 1,1

%A _David A. Corneth_, Jun 18 2026