A380264 -id:A380264 - cp's OEIS frontend

A382475 Numbers k where record values occur for A129132(k)/k = A380264(k)/A380265(k), the mean value of the maximum exponent in the prime factorization of the numbers {1, 2, ..., k}.

1, 2, 3, 4, 8, 9, 16, 18, 20, 24, 25, 27, 28, 32, 56, 64, 81, 128, 162, 176, 192, 256, 352, 384, 736, 768, 896, 1026, 1029, 1056, 1280, 1792, 1863, 1864, 1928, 2052, 2058, 2064, 2080, 2304, 2432, 2560, 2944, 3776, 4376, 4384, 4480, 4482, 5104, 5120, 5121, 5125

Offset: 1

Amiram Eldar, Mar 28 2025

First differs from A382476 at n = 72: a(72) = 39936 while A382476(72) = 39937.

Niven (1969) proved that abs(A129132(k)/k - c) < f(k) = (3/k) * Sum_{i=2 .. floor(log_2(k))} k^(1/i), where c = A033150 is Niven's constant. For k = 81984 we have A129132(k)/k - c = 2.40277...*10^(-5). There are no other terms in this sequence that are larger than 81984 up to 16500000000, and for k = 16500000000 we have abs(A129132(k)/k - c) < f(k) = 2.39403...*10^(-5). Therefore, this sequence is finite and a(73) = 81984 is the last term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..73
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. A033150, A051903, A129132, A380264, A380265, A382476.

f[k_] := Max[FactorInteger[k][[;; , 2]]]; f[1] = 0; seq[lim_] := Module[{v = {}, s = 0, rm = -1, r}, Do[s += f[k]; r = s/k; If[r > rm, rm = r; AppendTo[v, k]], {k, 1, lim}]; v]; seq[10^5]

f(k) = if(k == 1, 0, vecmax(factor(k)[, 2]));
list(lim) = {my(v = List(), s = 0, rm = -1, r); for(k = 1, lim, s += f(k); r = s/k; if(r > rm, rm = r; listput(v, k))); Vec(v);}

A382476 Numbers k where record low values occur for abs(A129132(k)/k - c) = abs(A380264(k)/A380265(k) - c), where c = A033150 is Niven's constant.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 16, 18, 20, 24, 25, 27, 28, 32, 56, 64, 81, 128, 162, 176, 192, 256, 352, 384, 736, 768, 896, 1026, 1029, 1056, 1280, 1792, 1863, 1864, 1928, 2052, 2058, 2064, 2080, 2304, 2432, 2560, 2944, 3776, 4376, 4384, 4480, 4482, 5104, 5120, 5121, 5125

Offset: 1

Amiram Eldar, Mar 28 2025

nonn

First differs from A382475 at n = 72: a(72) = 39937 while A382475(72) = 39936.

Since lim_{k->oo} A129132(k)/k = c, this sequence is infinite if Niven's constant is irrational.

Amiram Eldar, Table of n, a(n) for n = 1..103

Cf. A033150, A051903, A129132, A380264, A380265, A382475.

f[k_] := Max[FactorInteger[k][[;; , 2]]]; f[1] = 0; seq[lim_] := Module[{Niven = 1 + NSum[1 - 1/Zeta[k], {k, 2, Infinity}, NSumTerms -> 100, WorkingPrecision -> 50], dm = 2, d, s = {}, sum = 0}, Do[sum += f[k]; d = Abs[sum/k - Niven]; If[d < dm, dm = d; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^4]

default(realprecision, 120);
f(k) = if(k == 1, 0, vecmax(factor(k)[, 2]));
list(lim) = {my(niven = 1 + suminf(k = 2, 1-1/zeta(k)), dm = 2, d, s = List(), sm = 0); for(k = 1, lim, sm += f(k); d = abs(sm/k - niven); if(d < dm, dm = d; listput(s, k))); Vec(s);}

A086197 Incorrect version of A380264.

Original entry on oeis.org

1, 2, 3, 1, 1, 1, 1, 11, 13, 14, 5, 17, 9, 19, 5, 24, 25, 27, 7, 10, 31, 32, 11, 36, 19, 13, 3, 44, 3, 46, 47, 52, 53, 54, 55, 57, 29, 59, 3, 63, 32, 65, 3, 68, 35, 71, 3, 76, 39, 80, 81, 83, 14, 87, 11, 91, 46, 93, 47, 96, 97, 14, 25, 106, 107, 108, 109, 37, 8, 113, 19, 117

Offset: 1

Eric W. Weisstein, Jul 11 2003

Previous name was: Numerators of running averages of A051903.

Due to a bug in the Mathematica code in the World of Mathematics page, the sequence was calculated with running averages over {A051903(1), 1, A051903(2), A051903(3), ... } with the first term removed. See A380264 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. A051903, A086196, A086198 (denominators), A380264 .

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

a(n) = numerator((1 + Sum_{k=1..n} A051903(k))/(n+1)). - Amiram Eldar, Jan 18 2025

A380265 a(n) is the denominator of the mean value of A051903(k) at the range k = 1..n.

Original entry on oeis.org

1, 2, 3, 1, 1, 1, 1, 4, 3, 10, 11, 3, 13, 7, 15, 16, 17, 9, 19, 20, 7, 22, 23, 24, 25, 13, 27, 28, 29, 2, 31, 32, 33, 34, 35, 9, 37, 19, 39, 20, 41, 21, 43, 44, 15, 23, 47, 16, 7, 50, 51, 26, 53, 27, 55, 28, 57, 29, 59, 12, 61, 62, 7, 64, 65, 66, 67, 34, 23, 5, 71

Offset: 1

Amiram Eldar, Jan 18 2025

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. A051903, A380264 (numerators).

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

lista(nmax) = {my(s = 0); print1(1, ", "); for(n = 2, nmax, s += vecmax(factor(n)[,2]);  print1(denominator(s/n), ", "));}

a(n) = denominator((Sum_{k=1..n} A051903(k))/n).

cp's OEIS Frontend

A382475 Numbers k where record values occur for A129132(k)/k = A380264(k)/A380265(k), the mean value of the maximum exponent in the prime factorization of the numbers {1, 2, ..., k}.

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A382476 Numbers k where record low values occur for abs(A129132(k)/k - c) = abs(A380264(k)/A380265(k) - c), where c = A033150 is Niven's constant.

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A086197 Incorrect version of A380264.

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A380265 a(n) is the denominator of the mean value of A051903(k) at the range k = 1..n.

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