A377515 - OEIS

%I #10 Oct 31 2024 01:10:58

%S 1,2,3,2,5,6,7,2,9,10,11,6,13,14,15,2,17,18,19,10,21,22,23,6,25,26,9,

%T 14,29,30,31,2,33,34,35,18,37,38,39,10,41,42,43,22,45,46,47,6,49,50,

%U 51,26,53,18,55,14,57,58,59,30,61,62,63,2,65,66,67,34,69,70

%N The largest divisor of n that is a term in A276078.

%C First differs from A327937 at n = 625 = 5^4: a(625) = 125, while A327937(625) = 625.

%C The number of these divisors is A377516(n), and their sum is A377517(n).

%H Amiram Eldar, <a href="/A377515/b377515.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^min(pi(p), e), where pi(n) = A000720(n).

%F a(n) = n if and only if n is in A276078.

%F Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((pi(p)+1)*s) - p^(pi(p)+1) - p^(pi(p)*s) + p^pi(p))/p^((pi(p)+1)*s).

%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^pi(p) * (p+1))) = 0.80906238421914194523... .

%t f[p_, e_] := p^Min[PrimePi[p], e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(primepi(f[i,1]), f[i,2]));}

%Y Cf. A000720, A276078, A327937, A377516, A377517, A377518.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Oct 30 2024