A033150 -id:A033150 - cp's OEIS frontend

A382219 Product of the largest and smallest exponents in the prime factorization of n.

1, 1, 1, 4, 1, 1, 1, 9, 4, 1, 1, 2, 1, 1, 1, 16, 1, 2, 1, 2, 1, 1, 1, 3, 4, 1, 9, 2, 1, 1, 1, 25, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 4, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 36, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 16, 1, 1, 2, 1, 1, 1, 3, 1, 2

Offset: 1

Ilya Gutkovskiy, Mar 19 2025

nonn

The asymptotic density of the occurrences of k = 1, 2, ... in this sequence is 1/zeta(2) for k = 1 and 1/zeta(k+1) - 1/zeta(k) for k >= 2, and the asymptotic mean of this sequence is A033150, the same densities and mean as in A051903, since a(n) = A051903(n) for nonpowerful numbers n (A052485) whose asymptotic density is 1. - Amiram Eldar, Mar 28 2025

Cf. A005361, A033150, A051903, A051904, A052485, A062977, A066048, A304233, A333352.

Table[Max @@ (#[[2]] & /@ FactorInteger[n]) Min @@ (#[[2]] & /@ FactorInteger[n]), {n, 90}]

a(n) = if(n == 1, 1, my(e = factor(n)[,2]); vecmin(e) * vecmax(e)); \\ Amiram Eldar, Mar 28 2025

If n = Product (p_j^k_j) then a(n) = min{k_j} * max{k_j}.

a(n) = A051903(n) * A051904(n) for n > 1.

A384659 Numbers k such that A384655(k)/k > A384655(m)/m for all m < k.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 48, 72, 96, 144, 288, 432, 576, 720, 1440, 2160, 2880, 4320, 8640, 17280, 21600, 25920, 30240, 43200, 60480, 120960, 151200, 181440, 241920, 302400, 604800, 907200, 1209600, 1814400, 3326400, 3628800, 5443200, 6350400, 6652800, 9979200

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

All the terms above 2 are nonsquarefree (A013929).

			The first values of A384655(k)/k, for k = 1..8, are {0, 1/2, 1/3, 3/4, 1/5, 2/3, 1/7, 7/8}. The record values, 0, 1/2, 3/4 and 7/8, occur at k = 1, 2, 4 and 8, the first 4 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..355

Subsequence of A025487.

Cf. A013929, A033150, A384655.

f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; r[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s/n]; r[1] = 0;
seq[lim_] := Module[{s = {}, rm = -1, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]

r(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, p[i]^(e[i]-k)))); s/n);
list(lim) = {my(rm = -1, r1); for(k = 1, lim, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", ")));}

Limit_{n->oo} A384655(a(n))/a(n) = c, where c is Niven's constant (A033150).

A242972 Decimal expansion of a constant related to Niven's constant.

Original entry on oeis.org

8, 9, 2, 8, 9, 4, 5, 7, 1, 4, 5, 1, 2, 6, 6, 0, 9, 0, 4, 5, 7, 0, 0, 9, 4, 3, 0, 0, 2, 2, 4, 2, 7, 0, 9, 3, 3, 6, 0, 5, 0, 4, 0, 8, 5, 9, 4, 4, 5, 6, 8, 4, 3, 2, 6, 4, 7, 4, 9, 5, 6, 7, 9, 0, 7, 4, 3, 7, 2, 7, 3, 4, 3, 8, 7, 2, 7, 6, 5, 6, 4, 9, 4, 9, 0, 6, 6, 9, 6, 8, 8, 7, 3, 6, 9, 4, 1, 7, 8, 3

Offset: 0

Jean-François Alcover, May 28 2014

			0.89289457145126609045700943002242709336...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, p. 113.

Eric Weisstein's MathWorld, Niven's Constant

Cf. A033150, A072102, A090699.

digits = 100; k0 = 100; dk = 50; $MaxExtraPrecision = 12*digits; z[n_?NumericQ] := Zeta[Prime[n // Floor]]; Clear[s]; s[k_] := s[k] = NSum[z[n] - 1, {n, 1, k}, WorkingPrecision -> digits + 10, NSumTerms -> 10*digits]*(1 + NSum[Zeta[n] - 1, {n, k + 1, Infinity}, WorkingPrecision -> digits + 10]); s[k0] ; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits] != RealDigits[s[k - dk], 10, digits], Print["k = ", k]; k = k + dk]; RealDigits[s[k], 10, digits] // First

Equals Sum_(p prime) (zeta(p)-1).

Equals Sum_{k>=2} Sum_{p prime} 1/k^p. - Amiram Eldar, Aug 21 2020

A245055 Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.

Original entry on oeis.org

1, 7, 4, 2, 6, 5, 2, 3, 1, 1, 0, 3, 3, 5, 1, 5, 4, 3, 5, 2, 4, 8, 9, 0, 4, 8, 0, 6, 9, 4, 1, 2, 9, 8, 6, 4, 1, 1, 5, 4, 4, 3, 7, 9, 8, 9, 8, 3, 8, 1, 0, 4, 6, 2, 8, 1, 4, 2, 9, 0, 4, 7, 9, 5, 7, 4, 6, 5, 5, 5, 0, 3, 8, 7, 0, 0, 8, 1, 3, 5, 0, 8, 6, 8, 0, 5, 8, 1, 4, 7, 4, 1, 7, 5, 2, 4, 7, 8, 8, 1, 2

Offset: 1

Jean-François Alcover, Oct 22 2014

			1.7426523110335154352489048069412986411544379898381...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 273.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 33.
Carl Pomerance, The expected number of random elements to generate a finite abelian group, Periodica Mathematica Hungarica 43 (2001), 191-198.

Cf. A021002, A033150, A084892, A084893, A249141.

digits = 101; max = 400; c = 1/Product[N[Zeta[k], digits + 100], {k, 2, max}]; p[j_] := Product[N[Zeta[k], digits + 100], {k, 2, j}]; tau = Sum[1 - (1 - 2^-j)*c*p[j], {j, 1, max}]; RealDigits[tau, 10, digits ] // First

default(realprecision,120); suminf(j=1, 1-(1-2^(-j))*prodinf(k=j+1, 1/zeta(k))) \\ Vaclav Kotesovec, Oct 22 2014

tau = sum_{j >= 1} (1-(1-2^(-j))*prod_{k >= j+1} zeta(k)^(-1)).

tau = sum_{j >= 1} (1-(1-2^(-j))*c*prod_{k = 2..j} zeta(k)), where c is A068982.

A335532 Decimal expansion of the asymptotic value of the second raw moment of the maximal exponent in the prime factorizations of n (A051903).

Original entry on oeis.org

4, 3, 0, 1, 3, 0, 2, 4, 0, 0, 3, 1, 3, 3, 6, 6, 5, 9, 9, 9, 8, 0, 6, 8, 9, 3, 4, 0, 4, 1, 8, 7, 7, 5, 7, 9, 9, 2, 2, 9, 8, 9, 1, 2, 9, 7, 6, 3, 4, 7, 7, 4, 3, 1, 6, 4, 7, 3, 8, 6, 9, 9, 1, 7, 2, 7, 2, 4, 8, 1, 5, 9, 3, 0, 3, 2, 5, 0, 3, 8, 7, 7, 0, 0, 3, 4, 1

Offset: 1

Amiram Eldar, Oct 18 2020

Let H(n) = A051903(n) be the maximal exponent in the prime factorizations of n. The asymptotic density of the numbers whose maximal exponent is k is d(k) = 1/zeta(k+1) - 1/z(k). For example, k=1 corresponds to the squarefree numbers (A005117), and k=2 corresponds to the cubefree numbers which are not squarefree (A067259). The asymptotic mean of H is = Sum_{k>=1} k*d(k) = 1 + Sum_{j>=2} (1 - 1/zeta(j)) = 1.705211... which is Niven's constant (A033150). The second raw moment of the distribution of maximal exponents is = Sum_{k>=1} k^2*d(k), whose simplified formula in terms of zeta functions is given in the FORMULA section.

The second central moment, or variance, of H is - ^2 = 4.3013024003... - 1.7052111401...^2 = 1.3935573679... and the standard deviation is sqrt( - ^2) = 1.1804903082...

			4.30130240031336659998068934041877579922989129763477...
For the numbers n=1..2^20, the values of H(n) = A051903(n) are in the range [0..20]. Their mean value is 894015/524288 = 1.705198..., their second raw moment is 140939/32768 = 4.301116..., and their standard deviation is sqrt(383019202687/274877906944) = 1.180430...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, pp. 112-113.

Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
D. Suryanarayana and R. Sita Rama Chandra Rao, On the maximum and minimum exponents in factoring integers, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
Eric Weisstein's World of Mathematics, Niven's Constant.
Wikipedia, Niven's constant.

Cf. A005117, A033150, A051903, A067259, A242977.

RealDigits[1 + Sum[(2*j - 1)*(1 - 1/Zeta[j]), {j, 2, 400}], 10, 100][[1]]

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A051903(k)^2.

Equals 1 + Sum_{j>=2} (2*j-1) * (1 - 1/zeta(j)).

cp's OEIS Frontend

A382219 Product of the largest and smallest exponents in the prime factorization of n.

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A384659 Numbers k such that A384655(k)/k > A384655(m)/m for all m < k.

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A242972 Decimal expansion of a constant related to Niven's constant.

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A245055 Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.

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A335532 Decimal expansion of the asymptotic value of the second raw moment of the maximal exponent in the prime factorizations of n (A051903).

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