A129132 -id:A129132 - 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

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);}

A116546 Erroneous version of A129132.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 15, 16, 17, 18, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, 37, 40, 42, 43, 44, 45, 50, 51, 52, 53, 55, 56, 57, 58, 61, 62, 63, 64, 66, 68, 69, 70, 74, 76, 78, 79, 81, 82, 85, 86, 89, 90, 91, 92, 94, 95, 96, 98, 104, 105, 106, 107, 109, 110, 111

Offset: 1

Giovanni Teofilatto, Mar 28 2006

dead

Cf. A051903.

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller, Mar 15 1996

Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]
a(n) = A128651(A129132(n-1) + 2) for n > 1. - Reinhard Zumkeller, Mar 30 2007
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length.  For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017

			G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandache-type multiplicative sequences
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
Neville Holmes, Integer Sequences [Broken link]
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
Wikipedia, Radical of an integer.

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
Range of values is A005117.
Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
Bisections: A099984, A099985.
Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
A003961, A059896 are used to express relationship between terms of this sequence.
Cf. A001615, A008683, A076479, A078615.

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
seq(A007947(i),i=1..69); # Peter Luschny, Mar 22 2011
A:= n -> convert(numtheory:-factorset(n),`*`):
seq(A(n),n=1..100); # Robert Israel, Aug 10 2014
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
From Antti Karttunen, Jun 18 2017: (Start)
a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
a(n) = A019565(A087207(n)). (End)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
From Peter Munn, Jan 01 2020: (Start)
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A003961(n)) = A003961(a(n)).
a(n^2) = a(n).
a(A225546(n)) = A019565(A267116(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

A051903 Maximum exponent in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Labos Elemer, Dec 16 1999

Smallest number of factors of all factorizations of n into squarefree numbers, see also A128651, A001055. - Reinhard Zumkeller, Mar 30 2007
Maximum number of invariant factors among abelian groups of order n. - Álvar Ibeas, Nov 01 2014
a(n) is the highest of the frequencies of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1..r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24) = 3; indeed, the partition having Heinz number 24 = 2*2*2*3 is [1,1,1,2], where the distinct parts 1 and 2 have frequencies 3 and 1, respectively. - Emeric Deutsch, Jun 04 2015
From Thomas Ordowski, Dec 02 2019: (Start)
a(n) is the smallest k such that b^(phi(n)+k) == b^k (mod n) for all b.
The Euler phi function can be replaced by the Carmichael lambda function.
Problems:
(*) Are there composite numbers n > 4 such that n == a(n) (mod phi(n))? By Lehmer's totient conjecture, there are no such squarefree numbers.
(**) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod lambda(n))? These are odd numbers n such that a(n) > 1 and b^n == b^a(n) (mod n) for all b.
(***) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod ord_{n}(2))? These are odd numbers n such that a(n) > 1 and 2^n == 2^a(n) (mod n).
Note: if (***) do not exist, then (**) do not exist. (End)
Niven (1969) proved that the asymptotic mean of this sequence is 1 + Sum_{j>=2} 1 - (1/zeta(j)) (A033150). - Amiram Eldar, Jul 10 2020

			For n = 72 = 2^3*3^2, a(72) = max(exponents) = max(3,2) = 3.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Benjamin Merlin Bumpus and Zoltan A. Kocsis, Spined categories: generalizing tree-width beyond graphs, arXiv:2104.01841 [math.CO], 2021.
Cao Hui-Zhong, The Asymptotic Formulas Related to Exponents in Factoring Integers, Math. Balkanica, Vol. 5 (1991), Fasc. 2.
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.
Index entries for sequences computed from exponents in factorization of n

Average value is A033150 = 1.7052....

Cf. A002322, A005361, A008479, A028234, A051904, A052409, A067029, A091050, A129132, A327295, A328310, A329885.

a051903 1 = 0
a051903 n = maximum $ a124010_row n -- Reinhard Zumkeller, May 27 2012

A051903 := proc(n)
        a := 0 ;
        for f in ifactors(n)[2] do
                a := max(a,op(2,f)) ;
        end do:
        a ;
end proc: # R. J. Mathar, Apr 03 2012
# second Maple program:
a:= n-> max(0, seq(i[2], i=ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, May 09 2020

Table[If[n == 1, 0, Max @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=if(n>1,vecmax(factor(n)[,2]),0) \\ Charles R Greathouse IV, Oct 30 2012

from sympy import factorint
def A051903(n):
    return max(factorint(n).values()) if n > 1 else 0
# Chai Wah Wu, Jan 03 2015

;; With memoization-macro definec.
(definec (A051903 n) (if (= 1 n) 0 (max (A067029 n) (A051903 (A028234 n))))) ;; Antti Karttunen, Aug 08 2016

a(n) = max_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(1) = 0; for n > 1, a(n) = max(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 08 2016
Conjecture: a(n) = a(A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - Velin Yanev, Sep 02 2017
Comment from David J. Seal, Sep 18 2017: (Start)
This conjecture seems very easily provable to me: if the factorization of n is p1^k1 * p2^k2 * ... * pm^km, then the factorization of the largest squarefree divisor of n is p1 * p2 * ... * pm. So the factorization of A003557(n) is p1^(k1-1) * p2^(k2-1) * ... * pm^(km-1) if exponents of zero are allowed, or with the product terms that have an exponent of zero removed if they're not (if that results in an empty product, consider it to be 1 as usual).
The formula then follows from the fact that provided all ki >= 1, Max(k1, k2, ..., km) = Max(k1-1, k2-1, ..., km-1) + 1, and Max(k1-1, k2-1, ..., km-1) is not altered by removing the ki-1 values that are 0, provided we treat the empty Max() as being 0. That proves the formula and the provisos about empty products and Max() correspond to a(1) = 0.
Also, for any n, applying the formula Max(k1, k2, ..., km) times to n = p1^k1 * p2^k2 * ... * pm^km reduces all the exponents to zero, i.e., to the case a(1) = 0, so that case and the formula generate the sequence. (End)
Sum_{k=1..n} (-1)^k * a(k) ~ c * n, where c = Sum_{k>=2} 1/((2^k-1)*zeta(k)) = 0.44541445377638761933... . - Amiram Eldar, Jul 28 2024
a(n) <= log(n)/log(2). - Hal M. Switkay, Jul 03 2025

A380264 a(n) is the numerator of the mean value of A051903(k) at the range k = 1..n.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 5, 4, 13, 14, 4, 17, 9, 19, 23, 24, 13, 27, 29, 10, 31, 32, 35, 37, 19, 41, 43, 44, 3, 46, 51, 52, 53, 54, 14, 57, 29, 59, 31, 63, 32, 65, 67, 23, 35, 71, 25, 11, 79, 80, 41, 83, 43, 87, 45, 91, 46, 93, 19, 96, 97, 11, 105, 106, 107, 108, 55

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, 4/3, ...

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. A033150, A051903, A129132, A359071, A359072, A380265 (denominators).

f[n_] := Max[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 += vecmax(factor(n)[,2]);  print1(numerator(s/n), ", "));}

a(n) = numerator((Sum_{k=1..n} A051903(k))/n).
a(n)/A380265(n) = A129132(n)/n.
Limit_{n->oo} a(n)/A380265(n) = c, where c is Niven's constant (A033150).
abs(a(n)/A380265(n) - c) <= 3*log_2(n)/sqrt(n).

A128651 In the sequence of natural numbers, replace the nonsquarefree numbers with their minimal factorization into squarefree numbers.

Original entry on oeis.org

1, 2, 3, 2, 2, 5, 6, 7, 2, 2, 2, 3, 3, 10, 11, 6, 2, 13, 14, 15, 2, 2, 2, 2, 17, 6, 3, 19, 10, 2, 21, 22, 23, 6, 2, 2, 5, 5, 26, 3, 3, 3, 14, 2, 29, 30, 31, 2, 2, 2, 2, 2, 33, 34, 35, 6, 6, 37, 38, 39, 10, 2, 2, 41, 42, 43, 22, 2, 15, 3, 46, 47, 6, 2, 2, 2, 7, 7, 10, 5, 51, 26, 2, 53, 6, 3, 3, 55

Offset: 1

Reinhard Zumkeller, Mar 18 2007

nonn

a(A129132(n-1) + 2) = A007947(n) for n > 1. - Reinhard Zumkeller, Mar 30 2007

			1 2 3 _ 4 5 6 7 _ _ 8 _ 9 10 11 _12 13 14 15 _ _ _16 ....
1 2 3 2 2 5 6 7 2 2 2 3 3 10 11 6 2 13 14 15 2 2 2 2 ....

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree

Cf. A013929, A007947, A003557.

a(n) = f(n,1,2,2,1) with f(n,m,x,y,z) = if n=m then z else if rad(y)=y then f(n,m+1,x+1,x+1,y) else f(n,m+1,x,y/rad(y),rad(y)), rad = squarefree kernel (A007947).

A359071 Numerators of the partial sums of the reciprocals of the maximal exponent in prime factorization of the positive integers (A051903).

Original entry on oeis.org

1, 2, 5, 7, 9, 11, 35, 19, 22, 25, 53, 59, 65, 71, 145, 157, 163, 175, 181, 193, 205, 217, 221, 227, 239, 81, 83, 87, 91, 95, 479, 499, 519, 539, 549, 569, 589, 609, 1847, 1907, 1967, 2027, 2057, 2087, 2147, 2207, 1111, 563, 1141, 1171, 593, 608, 613, 628, 211

Offset: 2

Amiram Eldar, Dec 15 2022

			Fractions begin with 1, 2, 5/2, 7/2, 9/2, 11/2, 35/6, 19/3, 22/3, 25/3, 53/6, 59/6, ...

Amiram Eldar, Table of n, a(n) for n = 2..10000
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. Chandra Rao, On the maximum and minimum exponents in factoring integers, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
Index entries for sequences computed from exponents in factorization of n.

Cf. A051903, A129132, A242977, A359072 (denominators).

f[n_] := Max[FactorInteger[n][[;; , 2]]]; f[1] = 0; Numerator[Accumulate[Table[1/f[n], {n, 2, 100}]]]

a(n) = numerator(Sum_{k=2..n} 1/A051903(k)).

a(n)/A359072(n) = c_1 * n + O(n^(1/2)*exp(-c_2*log(n)^(3/5)/log(log(n))^(1/5))), where c_1 = A242977 and c_2 is a constant, 0 < c_2 < 1/2^(8/5) (Suryanarayana and R. Chandra Rao, 1977).

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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A116546 Erroneous version of A129132.

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A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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A051903 Maximum exponent in the prime factorization of n.

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A380264 a(n) is the numerator of the mean value of A051903(k) at the range k = 1..n.

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A128651 In the sequence of natural numbers, replace the nonsquarefree numbers with their minimal factorization into squarefree numbers.

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