A380267 -id:A380267 - cp's OEIS frontend

A086196 Incorrect version of A380267.

2, 3, 4, 1, 1, 1, 1, 9, 10, 11, 4, 13, 14, 5, 16, 17, 3, 19, 10, 7, 11, 23, 4, 25, 26, 27, 28, 29, 10, 31, 32, 33, 34, 35, 36, 37, 19, 39, 20, 41, 3, 43, 22, 45, 23, 47, 24, 7, 10, 17, 52, 53, 18, 11, 56, 19, 58, 59, 4, 61, 62, 21, 64, 13, 33, 67, 17, 69, 7, 71, 18, 73, 74

Offset: 1

Eric W. Weisstein, Jul 11 2003

Previous name was: Denominators of running averages of A051904.

Due to a bug in the Mathematica code in the World of Mathematics page, the sequence was calculated with running averages over {A051904(1), 1, A051904(2), A051904(3), ... } with the first term removed. See A380267 for the correct sequence. - Amiram Eldar, Jan 18 2025

			1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, ...

Eric Weisstein's World of Mathematics, Niven's Constant.

Cf. A051904, A086195 (numerators), A380267.

f[n_] := Min[FactorInteger[n][[;; , 2]]]; f[1] = 0; With[{m = 100}, Denominator[(1 + Accumulate[Array[f, m]]) / Range[2, m + 1]]] (* Amiram Eldar, Jan 18 2025 *)

a(n) = denominator((1 + Sum_{k=1..n} A051904(k))/(n+1)). - Amiram Eldar, Jan 18 2025

A380266 a(n) is the numerator of the mean value of A051904(k) at the range k = 1..n.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 5, 4, 13, 14, 5, 16, 17, 6, 11, 23, 4, 25, 13, 9, 14, 29, 5, 32, 33, 4, 37, 38, 13, 40, 45, 46, 47, 48, 25, 51, 26, 53, 27, 55, 4, 57, 29, 59, 30, 61, 31, 64, 13, 22, 67, 68, 23, 14, 71, 24, 73, 74, 5, 76, 77, 26, 21, 17, 43, 87, 22, 89, 9

Offset: 1

Amiram Eldar, Jan 18 2025

			Fractions begin with 0, 1/2, 2/3, 1, 1, 1, 1, 5/4, 4/3, 13/10, 14/11, 5/4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant.
Wikipedia, Niven's constant.

Cf. A051904, A090699, A380267 (denominators).

f[n_] := Min[FactorInteger[n][[;;, 2]]]; f[1] = 0; With[{m = 100}, Numerator[Accumulate[Array[f, m]] / Range[m]]]

lista(nmax) = {my(s = 0); print1(0, ", "); for(n = 2, nmax, s += vecmin(factor(n)[,2]);  print1(numerator(s/n), ", "));}

a(n) = numerator((Sum_{k=1..n} A051904(k))/n).

a(n)/A380267(n) = 1 + c/sqrt(n) + o(1/sqrt(n)), where c = zeta(3/2)/zeta(3) (A090699).

cp's OEIS Frontend

A086196 Incorrect version of A380267.

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