A396829 a(n) is the least number of prime factors for any squarefree abundant number with prime(n) (the n-th prime) as its least prime factor.

3, 5, 9, 18, 31, 46, 67, 91, 122, 158, 194, 238, 284, 334, 392, 456, 522, 591, 668, 749, 835, 929, 1028, 1133, 1242, 1352, 1469, 1594, 1727, 1869, 2019, 2163, 2315, 2471, 2636, 2802, 2977, 3157, 3342, 3534, 3731, 3933, 4145, 4358, 4581, 4811, 5053, 5293, 5531

Offset

1,1

Comments

If we replace "abundant" in the definition with "nondeficient" so that perfect numbers qualify, we get the same sequence with an initial 2 instead of 3, as the perfect number 6 = 2*3 has only 2 prime factors but no other perfect numbers are squarefree.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Formula

a(n) = A007684(n) - n + 1, for n >= 2.

a(n) = A001222(A007741(n)).

Mathematica

a[1] = 3; a[n_] := a[n] = Module[{c = a[n-1] - 1, p, s}, p = Prime[n+c]; s = Product[1 + 1/p, {p, Prime[Range[n, n-1+c]]}]; While[s <= 2, s *= 1 + 1/p; p = NextPrime[p]; c++]; c]; Array[a, 50]

Prog

(PARI) apply( {A396829(n)=my(s=1+1/prime(n), a=n); until(2 < s*=1+1/prime(a++), ); a-n+1}, [1..44]) \\ M. F. Hasler, Jun 08 2026

Crossrefs

Cf. A001222, A007741, A007684, A007707, A087248, A108227.

Sequence in context: A281852 A120941 A108227 * A289912 A289914 A251704

Adjacent sequences: A396826 A396827 A396828 * A396831 A396833 A396834

Keyword

nonn

Author

Peter Munn, M. F. Hasler, and Amiram Eldar, Jun 07 2026

Status

approved