cp's OEIS Frontend

A280050 a(n) = Sum_{k=2..n} k/lpf(k), where lpf(k) is the least prime dividing k (A020639).

%I A280050 #36 Jul 03 2025 08:40:23
%S A280050 0,1,2,4,5,8,9,13,16,21,22,28,29,36,41,49,50,59,60,70,77,88,89,101,
%T A280050 106,119,128,142,143,158,159,175,186,203,210,228,229,248,261,281,282,
%U A280050 303,304,326,341,364,365,389,396,421,438,464,465,492,503,531,550,579,580,610,611,642,663,695,708,741,742,776,799,834,835
%N A280050 a(n) = Sum_{k=2..n} k/lpf(k), where lpf(k) is the least prime dividing k (A020639).
%C A280050 Sum of the largest proper divisors of all positive integers <= n.
%H A280050 Amiram Eldar, <a href="/A280050/b280050.txt">Table of n, a(n) for n = 1..10000</a>
%H A280050 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeastPrimeFactor.html">Least Prime Factor</a>.
%H A280050 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ProperDivisor.html">Proper Divisor</a>.
%F A280050 a(n) = Sum_{k=2..n} k/A020639(k).
%F A280050 a(n) + 1 = Sum_{k=1..n} A032742(k).
%F A280050 a(p^k) = a(p^k-1) + p^(k-1), when p is prime.
%F A280050 a(n) ~ c * n^2, where c = (1/2) * Sum_{k>=1} A005867(k-1)/(prime(k)*A002110(k)) = 0.165049... . - _Amiram Eldar_, Jul 03 2025
%e A280050 For n = 8 the divisors of the first eight positive integers are {1}, {1, 2}, {1, 3}, {1, 2, 4}, {1, 5}, {1, 2, 3, 6}, {1, 7}, {1, 2, 4, 8}, so a(8) = 1 + 1 + 2 + 1 + 3 + 1 + 4 = 13.
%t A280050 Table[Sum[k/FactorInteger[k][[1, 1]], {k, 2, n}], {n, 71}]
%t A280050 Join[{0}, Accumulate[Table[k/FactorInteger[k][[1, 1]], {k, 2, 71}]]] (* _Amiram Eldar_, Jul 03 2025 *)
%o A280050 (PARI) list(kmax) = {my(s = 0); print1(s, ", "); for(k = 2, kmax, s += k/factor(k)[1,1]; print1(s, ", "));} \\ _Amiram Eldar_, Jul 03 2025
%Y A280050 Cf. A001065, A002110, A005867, A020639, A032742, A046669, A046670, A153485.
%K A280050 nonn,easy
%O A280050 1,3
%A A280050 _Ilya Gutkovskiy_, Jan 02 2017