A033150 -id:A033150 - cp's OEIS frontend

A033153 Incrementally largest terms in the continued fraction for Niven's constant.

Original entry on oeis.org

1, 2, 4, 8, 11, 14, 29, 372, 559, 1671

Offset: 1

Eric W. Weisstein

I. Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc. 22, 356-360, 1969.
Eric Weisstein's World of Mathematics, Niven's Constant.

Cf. A033150, A033151, A033152, A000010.

One more term from Michel ten Voorde, Jun 14 2003

A188385 Highest exponent in the prime factorization of n^n.

Original entry on oeis.org

0, 2, 3, 8, 5, 6, 7, 24, 18, 10, 11, 24, 13, 14, 15, 64, 17, 36, 19, 40, 21, 22, 23, 72, 50, 26, 81, 56, 29, 30, 31, 160, 33, 34, 35, 72, 37, 38, 39, 120, 41, 42, 43, 88, 90, 46, 47, 192, 98, 100, 51, 104, 53, 162, 55, 168, 57, 58, 59, 120, 61, 62, 126, 384

Offset: 1

A. Timothy Royappa, Mar 29 2011

			For n = 1, 1^1 = 1, giving a(1) = 0.
For n = 12, 12^12 = 8916100448256 = (2^24)(3^12), giving a(12) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000312, A033150, A051903.

Join[{0}, Table[n*Max[Last /@ FactorInteger[n]], {n, 2, 100}]] (* T. D. Noe, Mar 30 2011 *)

a(n) = if (n==1, 0, n*vecmax(factor(n)[,2])); \\ Michel Marcus, Dec 08 2020

a(n) = n * A051903(n). - Franklin T. Adams-Watters, Mar 29 2011

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A033150 = 1.705211... . - Amiram Eldar, Jan 05 2024

A242977 Decimal expansion of Sum_{k>1} 1/(k(k-1)zeta(k)), a constant related to Niven's constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 28 2014

The asymptotic mean of the reciprocals of the maximal exponent in prime factorization of the positive integers. - Amiram Eldar, Dec 15 2022

			0.766944490521088241652417923...

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

Wolfgang Schwarz and Jürgen Spilker, A remark on some special arithmetical functions, in: E. Laurincikas , E. Manstavicius and V. Stakenas (eds.), Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996, New Trends in Probability and Statistics, Vol. 4, VSP BV & TEV Ltd. (1997), pp. 221-245.
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.

Cf. A033150, A051903, A072102, A090699.

digits = 98; m0 = 100; dm = 100; Clear[f]; f[m_] := f[m] = NSum[1/(k*(k - 1)*Zeta[k]), {k, 2, m}, WorkingPrecision -> digits + 10, NSumTerms -> m] + 1/m; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], Print["m = ", m ]; m = m + dm]; RealDigits[f[m], 10, digits] // First

sumpos(k = 2, 1/(k*(k-1)*zeta(k))) \\ Amiram Eldar, Dec 15 2022

Equals lim_{n->oo} (1/n) * Sum_{k=2..n} 1/A051903(k). - Amiram Eldar, Oct 16 2020

Equals 1 + Sum_{k>=2} (1/zeta(k)-1)/(k*(k-1)). - Amiram Eldar, Dec 15 2022

A370494 Oblong numbers of the form (k-1)*k where k is the product of an odd number of distinct primes.

Original entry on oeis.org

2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 870, 930, 1332, 1640, 1722, 1806, 2162, 2756, 3422, 3660, 4290, 4422, 4830, 4970, 5256, 6006, 6162, 6806, 7832, 9312, 10100, 10302, 10506, 10920, 11342, 11772, 11990, 12656, 12882, 16002, 16770, 17030, 18632, 18906, 19182

Offset: 1

Amiram Eldar, Feb 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A370495 within A368249.

Cf. A002378, A030059, A033150, A368250.

Table[n*(n - 1), {n, Select[Range[150], MoebiusMu[#] == -1 &]}]

lista(kmax) = forsquarefree(k=1, kmax, if(moebius(k) == -1, print1(k[1]*(k[1]-1), ", ")));

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A370494(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-primepi(x)-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length(),2)))
    return (k:=bisection(f,n,n))*(k-1) # Chai Wah Wu, Jan 28 2025

a(n) = A002378(A030059(n)-1).

Sum_{n>=1} 1/a(n) = (A368250 + A033150 - 1)/2 = 0.776922504035... .

A370495 Oblong numbers of the form (k-1)*k where k is the product of an even number of distinct primes.

Original entry on oeis.org

0, 30, 90, 182, 210, 420, 462, 650, 1056, 1122, 1190, 1406, 1482, 2070, 2550, 2970, 3192, 3306, 3782, 4160, 4692, 5402, 5852, 6642, 7140, 7310, 7482, 8190, 8556, 8742, 8930, 11130, 12210, 13110, 13806, 14042, 14762, 15006, 16512, 17556, 17822, 19740, 20022, 20306

Offset: 1

Amiram Eldar, Feb 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A370494 within A368249.

Cf. A002378, A030059, A033150, A368250.

Table[n*(n - 1), {n, Select[Range[150], MoebiusMu[#] == 1 &]}]

lista(kmax) = forsquarefree(k=1, kmax, if(moebius(k) == 1, print1(k[1]*(k[1]-1), ", ")));

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A370495(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n-1+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,x.bit_length(),2)))
    return (k:=bisection(f,n,n))*(k-1) # Chai Wah Wu, Jan 28 2025

a(n) = A002378(A030229(n)-1).

Sum_{n>=2} 1/a(n) = (A368250 - A033150 + 1)/2 = 0.071711363929... .

A375040 The maximum exponent in the prime factorization of 2*n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 3, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 2, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 3, 1, 2, 2, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 2, 1, 5, 4, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 1, 2, 1, 4, 1

Offset: 1

Amiram Eldar, Jul 28 2024

Bisection of A051903.

Cf. A033150, A375039.

a[n_] := Max[FactorInteger[2*n][[;; , 2]]]; Array[a, 100]

PARI
```
a(n) = vecmax(factor(2*n)[,2]);
```

a(n) = A051903(2*n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=2} (1 - (2^k-2)/((2^k-1)*zeta(k))) = 2.15062559388175538361... .

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}.

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

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);}

A033154 Position of incrementally largest terms in continued fraction for Niven's constant.

Original entry on oeis.org

0, 2, 6, 12, 19, 34, 50, 67, 95, 461

Offset: 1

Eric W. Weisstein

The continued fraction expansion is indexed [a_0; a_1, a_2, a_3, ...];

I. Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc. 22, 356-360, 1969.
Eric Weisstein's World of Mathematics, Niven's Constant

Cf. A033150, A033151, A033152, A033153.

A033151(a(n)) = A033153(n). - Andrew Howroyd, Sep 11 2024

One more term from Michel ten Voorde, Jun 14 2003

Terms decreased by 1 for consistency with offset change in A033151 by Andrew Howroyd, Sep 11 2024

A249141 Decimal expansion of 'sigma', a constant associated with the expected number of random elements to generate a finite abelian group.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 22 2014

			2.11845656347016353238252776910236476428859...

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.

digits = 102; jmax = 400; P[j_] := 1/Product[N[Zeta[k], digits+100], {k, j, jmax}]; sigma = 1+Sum[1 - P[j], {j, 2, jmax}]; RealDigits[sigma, 10, digits] // First

default(realprecision,120); 1 + suminf(j=2, 1 - prodinf(k=j, 1/zeta(k))) \\ Michel Marcus, Oct 22 2014

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

A272531 Decimal expansion of C_2 (so named by S. Finch), a constant which is an analog of Niven's constant when mean of exponents is considered instead of maximum.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 02 2016

			0.1187309349576408430176668841155338623125786669962548878395960878789...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95) and Section 2.6 Niven's constant p.112.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 15.
Eric Weisstein's MathWorld, Niven's Constant

Cf. A033150, A077761, A085548, A136141.

digits = 100;
C1 = NSum[PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits] ;
M = EulerGamma - NSum[PrimeZetaP[n]/n, {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits] ;
N0 = PrimeZetaP[2];
C2 = C1 (1 - M) - N0;
RealDigits[C2, 10, digits][[1]]

C_2 = C_1 (1 - M) - N, using Finch's notation, where C_1 is A136141, M A077761 and N A085548.

cp's OEIS Frontend

A033153 Incrementally largest terms in the continued fraction for Niven's constant.

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A188385 Highest exponent in the prime factorization of n^n.

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A242977 Decimal expansion of Sum_{k>1} 1/(k*(k-1)*zeta(k)), a constant related to Niven's constant.

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A370494 Oblong numbers of the form (k-1)*k where k is the product of an odd number of distinct primes.

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A370495 Oblong numbers of the form (k-1)*k where k is the product of an even number of distinct primes.

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A375040 The maximum exponent in the prime factorization of 2*n.

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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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A033154 Position of incrementally largest terms in continued fraction for Niven's constant.

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A242977 Decimal expansion of Sum_{k>1} 1/(k(k-1)zeta(k)), a constant related to Niven's constant.