A396829 - OEIS

%I #15 Jun 09 2026 10:14:17

%S 3,5,9,18,31,46,67,91,122,158,194,238,284,334,392,456,522,591,668,749,

%T 835,929,1028,1133,1242,1352,1469,1594,1727,1869,2019,2163,2315,2471,

%U 2636,2802,2977,3157,3342,3534,3731,3933,4145,4358,4581,4811,5053,5293,5531

%N 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.

%C 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.

%H Amiram Eldar, <a href="/A396829/b396829.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%t 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]

%o (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

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

%K nonn

%O 1,1

%A _Peter Munn_, _M. F. Hasler_, and _Amiram Eldar_, Jun 07 2026