A374231 a(n) is the minimum number of distinct numbers with exactly n prime factors (counted with multiplicity) whose sum of reciprocals exceeds 1.

3, 13, 96, 1772, 108336, 35181993

Offset

1,1

Links

Table of n, a(n) for n=1..6.

Example

a(1) = 3 since Sum_{k=1..2} 1/prime(k) = 1/2 + 1/3 = 5/6 < 1 and Sum_{k=1..3} 1/prime(k) = 1/2 + 1/3 + 1/5 = 31/30 > 1.
a(2) = 13 since Sum_{k=1..12} 1/A001358(k) = 1/4 + 1/6 + 1/9 + 1/10 + 1/14 + 1/15 + 1/21 + 1/22 + 1/25 + 1/26 + 1/33 + 1/34 = 15271237/15315300 < 1 and Sum_{k=1..13} 1/A001358(k) = 1/4 + 1/6 + ... + 1/35 = 15708817/15315300 > 1.

Mathematica

next[p_, n_] := Module[{k = p + 1}, While[PrimeOmega[k] != n, k++]; k]; a[n_] := Module[{k = 0, sum = 0, p = 0}, While[sum <= 1, p = next[p, n]; sum += 1/p; k++]; k]; Array[a, 5]

Prog

(PARI) nextnum(p, n) = {my(k = p + 1); while(bigomega(k) != n, k++); k; }
a(n) = {my(k = 0, sum = 0, p = 0); while(sum <= 1, p = nextnum(p, n); sum += 1/p; k++); k; }

Crossrefs

Cf. A078840.

Cf. A000040, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A046314.

Sequence in context: A183283 A006898 A277492 * A342305 A275528 A129375

Adjacent sequences: A374228 A374229 A374230 * A374232 A374233 A374234

Keyword

nonn,hard,more

Author

Amiram Eldar, Jul 01 2024

Status

approved