A054372 -id:A054372 - cp's OEIS frontend

A073093 Number of prime power divisors of n.

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

Offset: 1

Reinhard Zumkeller, Aug 24 2002

Also, number of prime divisors of 2n (counted with multiplicity).
A001221(n) < a(n) <= A000005(n) for all n; a(n)=A001221(n)+1 iff n is squarefree (A005117); a(n)=A000005(n) iff n is a prime power (A000961).
a(n) is also the number of kBenoit Cloitre, Oct 13 2002
a(n) is also 1 + the number of divisors of n with omega(d)=1, where omega is A001221. - Enrique Pérez Herrero, Nov 05 2009
Length of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012
a(n) depends only on the prime signature of n with a(A025487(n)) = 1, 2, 3, 3, 4, 4, 5, 5, 4, 6, 5, 6, 5, 7, 6, 7 ,.. = A036041(n)+1; (n>=1). - R. J. Mathar, May 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970.
T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A000961, A023888, A054372. Bisection of A001222.

a073093 = length . a210208_row  -- Reinhard Zumkeller, Mar 18 2012

[n eq 1 select 1 else &+[p[2]: p in Factorization(n)]+1: n in [1..100]]; // Vincenzo Librandi, Jan 06 2017

seq(numtheory:-bigomega(n)+1, n=1..1000); # Robert Israel, Sep 06 2015

f[n_] := Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; Table[ f[2n], {n, 105}] (* Robert G. Wilson v, Dec 23 2004 *)
A001221[n_] := (Length[ FactorInteger[n]]); SetAttributes[A001221, Listable]; A073093[n_]:=Length[Select[A001221[Divisors[n]], # == 1 &]]; (* Enrique Pérez Herrero, Nov 05 2009 *)
PrimeOmega[Range[100]] + 1 (* Paolo Xausa, Nov 23 2024 *)

numlib::Omega (2*n)$ n=1..105 // Zerinvary Lajos, May 13 2008

a(n)=sum(k=1,n,if(1-polresultant(polcyclo(n),polcyclo(k)),1,0))

A073093(n)=bigomega(n)+1   \\ M. F. Hasler, Dec 08 2010

If n = Product (p_j^k_j), a(n) = 1 + Sum (k_j).
a(n) = bigomega(n)+1 = A001222(n)+1 = A001222(2*n).
a(n) = if n=1 then 1 else a(A032742(n)) + 1. - Reinhard Zumkeller, Sep 24 2009
a(n) = max { a(d) ; d 1. - David W. Wilson, Dec 08 2010
a(n) = Sum_{k = 1 .. A001221(n)} A010055(A027750(n,k)). - Reinhard Zumkeller, Mar 18 2012
G.f.: x/(1 - x) + Sum_{k>=2} floor(1/omega(k))*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017

A052128 a(1) = 1; for n > 1, a(n) is the largest divisor of n that is coprime to a larger divisor of n.

Original entry on oeis.org

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

Offset: 1

James Sellers, Jan 21 2000

nonn

Least k > 0 such that the resultant of the k-th cyclotomic polynomial and the n-th cyclotomic polynomial is not 1. - Benoit Cloitre, Oct 13 2002
From Jianing Song, Sep 28 2022: (Start)
a(n) is the largest divisor d of n such that d <= sqrt(n) and that gcd(d,n/d) = 1.
Proof: write n = Product_{1<=i<=r} (p_i)^(e_i), let d be the largest divisor of n such that d <= sqrt(n) and that gcd(d,n/d) = 1. Obviously we have a(n) >= d. Suppose that a(n) = Product_{1<=i<=s} (p_i)^(m_i) for s <= r, 1 <= m_i <= e_i, then the larger divisor to which a(n) is coprime is a divisor of Product_{s+1<=i<=r} (p_i)^(e_i), so by definition we have a(n) <= min{Product_{1<=i<=s} (p_i)^(e_i), Product_{s+1<=i<=r} (p_i)^(e_i)} <= d. Thus a(n) = d. (End)

			a(6) = 6 / 3^1 = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A054372, A076388, A354933.

Table[best = 1; d = Divisors[n]; While[Length[d] > 1, e = d[[1]]; d = Rest[d]; If[Min[GCD[e, d]] == 1, best = e]]; best, {n, 102}] (* T. D. Noe, Aug 23 2013 *)

a(n) = my(i, j, d = divisors(n)); forstep (i = #d-1, 1, -1, for (j = i+1, #d, if (gcd(d[i], d[j]) == 1, return (d[i])))); 1 \\ Michel Marcus, Aug 22 2013

a(n)=my(f=factor(n),v=[1]); for(i=1,#f~,v=concat(v, f[i,1]^f[i,2] *v)); v=vecsort(v); forstep(i=#v\2,2,-1,for(j=i+1,#v-1, if(gcd(v[i],v[j])==1,return(v[i])))); 1 \\ Charles R Greathouse IV, Aug 22 2013

A052128(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(n/d))); \\ Antti Karttunen, Jun 16 2022

a(n) = n / A354933(n) = A354933(n) - A076388(n). - Antti Karttunen, Jun 16 2022

Terms corrected by Charles R Greathouse IV, Aug 22 2013

Definition rewritten by Jianing Song, Sep 28 2022

A075795 Number of k, 0

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 5, 4, 6, 7, 9, 8, 11, 11, 12, 11, 15, 14, 17, 16, 18, 19, 21, 19, 22, 23, 23, 24, 27, 26, 29, 26, 30, 31, 32, 31, 35, 35, 36, 35, 39, 38, 41, 40, 41, 43, 45, 42, 46, 46, 48, 48, 51, 49, 52, 51, 54, 55, 57, 55, 59, 59, 59, 57, 62, 62, 65, 64, 66, 66, 69, 66, 71

Offset: 1

Benoit Cloitre, Oct 13 2002

nonn

a(n) >= A000010(n)-1 since if 2<=kRobert Israel, Jul 24 2016

For n>1 a(n) = number of roots of the n-th polynomial in A275345, equal to 1. - Mats Granvik, Jul 24 2016

Robert Israel, Table of n, a(n) for n = 1..10000
T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970.
T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975.
Eric Weisstein's World of Mathematics, Cyclotomic polynomials.

Cf. A001222, A054372, A073093.

seq(n -numtheory:-bigomega(n)-1, n=1..1000); # Robert Israel, Jul 25 2016

Table[n - PrimeOmega@ n - 1, {n, 73}] (* Michael De Vlieger, Jul 26 2016 *)

a(n)=sum(k=1,n,if(1-polresultant(polcyclo(n),polcyclo(k)),0,1))

a(n) = n - A073093(n).

a(n) = n - A001222(n) - 1. - Michel Marcus, Jul 24 2016

a(30)=2 and a(31)=6 merged into a(30)=26 by Mats Granvik, Jul 24 2016

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A073093 Number of prime power divisors of n.

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A052128 a(1) = 1; for n > 1, a(n) is the largest divisor of n that is coprime to a larger divisor of n.

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A075795 Number of k, 0

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