A033150 -id:A033150 - cp's OEIS frontend

A062977 Difference between largest and smallest positive exponent in prime factorization of n; a(1) = 0 by convention.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 0, 0, 0, 0, 2, 0

Offset: 1

Henry Bottomley, Jul 24 2001

			a(24) = 2 since 24 = 2^3*3^1 and max(3,1) - min(3,1) = 3 - 1 = 2;
a(25) = 0 since 25 = 5^2 and max(2) - min(2) = 2 - 2 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 4000 terms from Harry J. Smith)
Index entries for sequences computed from exponents in factorization of n.

Cf. A072774 (positions of zeros), A059404 (of nonzeros).

Cf. A001222, A033150, A051903, A051904, A071178, A108951, A325226.

dlsp[n_]:=Module[{xp=FactorInteger[n][[All,2]]},Max[xp]-Min[xp]]; Join[ {0},Array[ dlsp,120]] (* Harvey P. Dale, Jan 28 2021 *)

{ for (n=1, 4000, if (n<2, M=m=0, f=factor(n)~; M=m=f[2, 1]; for (i=2, length(f), M=max(M, f[2, i]); m=min(m, f[2, i]))); write("b062977.txt", n, " ", M - m) ) } \\ Harry J. Smith, Aug 14 2009

A062977(n) = if((1==n),0,n=(factor(n)[, 2]); vecmax(n)-vecmin(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A051903(n) - A051904(n).
a(A108951(n)) = A325226(n) = A001222(n) - A071178(n). - Antti Karttunen, Nov 17 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150 - 1 = 0.705211... . - Amiram Eldar, Jan 05 2024

A368713 The maximal exponent in the prime factorization of the nonsquarefree numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 04 2024

The terms of A051903 that are larger than 1.

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

Cf. A013661, A013929, A033150, A051903.

Similar sequences: A368710, A368711, A368712.

s[n_] := Max @@ Last /@ FactorInteger[n]; s /@ Select[Range[250], !SquareFreeQ[#] &]
(* or *)
f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e > 1, e, Nothing]]; Array[f, 250]

lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(e > 1, print1(e, ", ")));}

a(n) = A051903(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (c * zeta(2) - 1)/(zeta(2) - 1) = 2.798673520766..., where c = 1.705211... is Niven's constant (A033150).

A372603 The maximal exponent in the prime factorization of the powerful part of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 07 2024

First differs from A275812 at n = 36, and from A212172 at n = 37.

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

Cf. A051903, A057521, A087156, A368710.
Cf. A013661, A033150, A059956.
Cf. A212172, A275812.
Similar sequences: A007424, A368781, A372601, A372602, A372604.

f[n_] := If[n == 1, 0, n]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = if(n == 1, 0, n);
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A057521(n)).
a(n) = A087156(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - 1/zeta(2) + Sum_{i>=2} (1 - 1/zeta(i)) = A033150 - A059956 = 1.09728403825134113562... .

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).

A033151 Continued fraction for Niven's constant.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, 1, 2, 1, 1, 11, 1, 4, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 7, 14, 2, 1, 13, 2, 1, 6, 2, 3, 1, 1, 1, 5, 2, 2, 8, 29, 1, 6, 1, 18, 1, 3, 2, 1, 5, 1, 1, 1, 18, 1, 3, 1, 372, 3, 3, 1, 47, 2, 1, 6, 1, 5, 1, 4, 1, 2, 4, 2, 2, 1, 1, 2, 7, 1, 7, 1, 14, 6, 5, 1, 559, 1

Offset: 0

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
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A033150 (decimal expansion).

rd[n_] := rd[n] = RealDigits[ N[1 + Sum[1 - 1/Zeta[j], {j, 2, 2^n}], 105]][[1]]; rd[n = 4]; While[rd[n] =!= rd[n - 1], n++]; Niven = FromDigits[{rd[n], 1}]; ContinuedFraction[Niven, 100] (* Jean-François Alcover, Oct 30 2012 *)

Offset changed by Andrew Howroyd, Jul 04 2024

A066301 a(n) = 0 if n is squarefree, otherwise 1 + a(n/rad(n)) where rad = A007947 (squarefree kernel).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 01 2002

This sequence is not the same as A046660.

			a(24) = 1 + a(24/rad(24)) = 1 + a(24/6) = 1 + a(4) = 1 + (1+a(4/rad(4))) = 1 + (1+a(4/2)) = 2 + a(2) = 2 + 0 = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005117, A007947, A003557, A008966, A033150, A046660, A051903.

a066301 1 = 0
a066301 n = a051903 n - 1  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Max[FactorInteger[n][[;;, 2]]] - 1; Array[a, 100] (* Amiram Eldar, Jan 05 2024 *)

a(n)=if(n>1, vecmax(factor(n)[,2])-1, 0) \\ Charles R Greathouse IV, Jul 15 2013

a(n) = A051903(n)-1 for n > 1, a(1) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150 - 1 = 0.705211... . - Amiram Eldar, Jan 05 2024

A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

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, 6, 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

Offset: 1

Jaroslav Krizek, Mar 05 2009

nonn

a(n) for n >= 2 equals LCM of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 deviates from A072411, first different term is a(360), a(360) = 3, A072411(360) = 6.

			For n = 12 = 2^2 * 3^1 we have a(12) = lcm(2,1) = 2.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(4,2) = 4.

Cf. A000040, A003990, A006881, A033150, A120944, A000961, A000027, A072411, A051904, A051903, A158378.

Table[LCM @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); lcm(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150. - Amiram Eldar, Sep 11 2024

A244000 Decimal expansion of the Bateman-Grosswald constant zeta(2/3)/zeta(2), a constant (negated) arising in the asymptotic evaluation of the number of square-full numbers (also called "powerful" numbers).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 17 2014

			-1.4879506635322726315987491125787134987961...

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

Wikipedia, Powerful number

Cf. A001694, A033150, A059956 (analog for squarefree integers), A090699.

RealDigits[ Zeta[2/3] / Zeta[2], 10, 101] // First

A375538 Numerator of the asymptotic mean over the positive integers of the maximum exponent in the prime factorization of the largest prime(n)-smooth divisor function.

Original entry on oeis.org

1, 13, 51227, 926908275845, 548123689541583443758024333411, 629375533747930240763697631488051776709110194920714685268467462860005271344878614119

Offset: 1

Amiram Eldar, Aug 19 2024

The numbers of digits of the terms are 1, 2, 5, 12, 30, 84, 215, 537, 1237, 2930, 6775, 15484, 35185, ... .

			Fractions begins: 1, 13/10, 51227/36540, 926908275845/636617813832, 548123689541583443758024333411/369693143251781030056182487680, ...
For n = 1, prime(1) = 2, the "2-smooth numbers" are the powers of 2 (A000079), and the sequence that gives the exponent of the largest power of 2 that divides n is A007814, whose asymptotic mean is 1.
For n = 2, prime(2) = 3, the 3-smooth numbers are in A003586, and the sequence that gives the maximum exponent in the prime factorization of the largest 3-smooth divisor of n is A244417, whose asymptotic mean is 13/10.

Cf. A033150, A375537, A375539 (denominators).
Cf. A375538 (numerators).
Cf. A000079, A003586, A007814, A244417, A375536.

d[k_, n_] := Product[1 - 1/Prime[i]^k, {i, 1, n}]; f[n_] := Sum[k * (d[k+1, n] - d[k, n]), {k, 1, Infinity}]; Numerator[Array[f, 6]]

Let f(n) = a(n)/A375539(n). Then:
f(n) = lim_{m->oo} (1/m) * Sum_{i=1..m} A375537(n, i).
f(n) = Sum_{k>=1} k * (d(k+1, prime(n)) - d(k, prime(n))), where d(k, p) = Product_{q prime <= p} (1 - 1/q^k).
Limit_{n->oo} f(n) = A033150.

A033152 Position of first occurrence of n in the continued fraction for Niven's constant.

Original entry on oeis.org

0, 2, 9, 6, 46, 40, 33, 12, 139, 251, 19, 334, 37, 34, 326, 184, 199, 54

Offset: 1

Eric W. Weisstein

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

Next term > 500. - Michel ten Voorde Jun 14 2003

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, A000009, A000010.

rd[n_] := rd[n] = RealDigits[ N[1 + Sum[1 - 1/Zeta[j], {j, 2, 2^n}], 400]][[1]]; rd[n = 4]; While[rd[n] =!= rd[n-1], n++]; Niven = FromDigits[{rd[n], 1}]; A033151 = ContinuedFraction[Niven]; a[n_] := Position[A033151, n][[1, 1]]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Oct 31 2012 *)

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

More terms from Michel ten Voorde, Jun 14 2003

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

cp's OEIS Frontend

A062977 Difference between largest and smallest positive exponent in prime factorization of n; a(1) = 0 by convention.

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A368713 The maximal exponent in the prime factorization of the nonsquarefree numbers.

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A372603 The maximal exponent in the prime factorization of the powerful part 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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A033151 Continued fraction for Niven's constant.

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A066301 a(n) = 0 if n is squarefree, otherwise 1 + a(n/rad(n)) where rad = A007947 (squarefree kernel).

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A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

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