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A384248 The sum of the integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

%I A384248 #6 May 27 2025 01:17:11
%S A384248 1,1,3,6,10,6,21,16,36,20,55,36,78,42,60,120,136,72,171,120,126,110,
%T A384248 253,96,300,156,243,252,406,120,465,256,330,272,420,432,666,342,468,
%U A384248 320,820,252,903,660,720,506,1081,720,1176,600,816,936,1378,486,1100,672
%N A384248 The sum of the integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.
%H A384248 Amiram Eldar, <a href="/A384248/b384248.txt">Table of n, a(n) for n = 1..10000</a>
%F A384248 a(n) = n * A384247(n) / 2, for n >= 2.
%F A384248 a(n) <= A333576(n), with equality if and only if n is in A138302.
%F A384248 a(n) >= A023896(n), with equality if and only if n is an exponentially odd number (A268335).
%F A384248 Sum_{k=1..n} a(k) ~ c * n^2 / 6, where c = Product_{p prime} f(1/p) = 0.66718130416373472394..., and f(x) = 1 - (1-x)*Sum_{k>=1} x^(2^k)/(1-x^(2^k)).
%t A384248 f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); a[1] = 1; a[n_] := (n/2) * Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384248 (PARI) a(n) = if(n == 1, 1, my(f = factor(n)); n^2 * prod(i = 1, #f~, (1 - 1/f[i,1]^(1 << valuation(f[i,2], 2)))) / 2);
%Y A384248 Row sums of A384246.
%Y A384248 Cf. A138302, A268335, A384247.
%Y A384248 Analogous sequences: A023896, A200723, A333576.
%K A384248 nonn,easy
%O A384248 1,3
%A A384248 _Amiram Eldar_, May 23 2025