A091051 -id:A091051 - cp's OEIS frontend

A091050 Number of divisors of n that are perfect powers.

1, 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, 4, 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, 5, 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, 4, 1, 1

Offset: 1

Reinhard Zumkeller, Dec 15 2003

nonn

Not the same as A005361: a(72)=5 <> A005361(72)=6.

			Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108},
a(108) = #{1^2, 2^2, 3^2, 3^3, 6^2} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A000005, A001597, A005361, A072102, A091051.

a091050 = sum . map a075802 . a027750_row
-- Reinhard Zumkeller, Dec 13 2012

ppQ[n_] := GCD @@ Last /@ FactorInteger@ n > 1; ppQ[1] = True; f[n_] := Length@ Select[ Divisors@ n, ppQ]; Array[f, 105] (* Robert G. Wilson v, Dec 12 2012 *)

a(n) = 1+ sumdiv(n, d, ispower(d)>1); \\ Michel Marcus, Sep 21 2014

a(n)={my(f=factor(n)[,2]); 1 + if(#f, sum(k=2, vecmax(f), moebius(k)*(1 - prod(i=1, #f, 1 + f[i]\k))))} \\ Andrew Howroyd, Aug 30 2020

a(n) = 1 iff n is squarefree: a(A005117(n)) = 1, a(A013929(n)) > 1.
a(p^k) = k for p prime, k>0: a(A000961(n)) = A025474(n).
a(n) = Sum_{k=1..A000005(n)} A075802(A027750(n,k)). - Reinhard Zumkeller, Dec 13 2012
G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 20 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + A072102 = 1.874464... . - Amiram Eldar, Dec 31 2023

Wrong formula deleted by Amiram Eldar, Apr 29 2020

A183097 a(n) = sum of powerful divisors d (including 1) of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 29, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 61, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 125, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 29, 118, 1, 1, 5, 1, 1, 1, 13, 1, 10, 1, 5, 1, 1, 1, 61, 1, 50, 10, 130

Offset: 1

Jaroslav Krizek, Dec 25 2010

Sequence is not the same as A091051(n); a(72) = 130, A091051(72) = 58.

a(n) = sum of divisors d of n from set A001694 - powerful numbers.

			For n = 12, set of such divisors is {1, 4}; a(12) = 1+4 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences related to sums of divisors.

Cf. A001694, A091051, A183102.

Cf. A002117, A078434.

A183097 := proc(n)
    local a,pe,p,e ;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if e > 1 then
            a := a* ( (p^(e+1)-1)/(p-1)-p) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 02 2020

fun[p_,e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)

A183097(n) = sumdiv(n, d, ispowerful(d)*d); \\ Antti Karttunen, Oct 07 2017

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1) - f[i,1]);} \\ Amiram Eldar, Dec 24 2023

a(n) = A000203(n) - A183098(n) = A183100(n) + 1.
a(1) = 1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = ((p^(k+1)-1) / (p-1))-p, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
From Amiram Eldar, Dec 24 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)^2/(3*zeta(3)) = 1.892451... . (End)

A183104 a(n) = product of divisors of n that are perfect powers.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 32, 9, 1, 1, 4, 1, 1, 1, 512, 1, 9, 1, 4, 1, 1, 1, 32, 25, 1, 243, 4, 1, 1, 1, 16384, 1, 1, 1, 1296, 1, 1, 1, 32, 1, 1, 1, 4, 9, 1, 1, 512, 49, 25, 1, 4, 1, 243, 1, 32, 1, 1, 1, 4, 1, 1, 9, 1048576, 1, 1, 1, 4, 1, 1, 1, 10368

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

Sequence is not the same as A183102: a(72) = 10368, A183102(72) = 746496.

Not multiplicative, as a(4)*a(9) <> a(36). - R. J. Mathar, Jun 07 2011

			For n = 12, set of such divisors is {1, 4}; a(12) = 1*4 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A001597, A007955, A091051, A183102, A183105.

isA001597 := proc(n) local e ; e := seq(op(2,p),p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:
A183104 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if isA001597(d) then a := a*d; end if; end do; a ; end proc:
seq(A183104(n),n=1..72) ; # R. J. Mathar, Jun 07 2011

perfPQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1; Table[Times@@Select[Divisors[n],perfPQ[#]&],{n,120}] (* Harvey P. Dale, Mar 07 2024 *)

A183104(n) = { my(m=1); fordiv(n, d, if(ispower(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183105(n).

a(1) = 1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = p^((1/2*k*(k+1))-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

A183101 a(n) = sum of divisors of n that are not perfect powers.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 2, 3, 17, 11, 23, 13, 23, 23, 2, 17, 29, 19, 37, 31, 35, 23, 47, 5, 41, 3, 51, 29, 71, 31, 2, 47, 53, 47, 41, 37, 59, 55, 77, 41, 95, 43, 79, 68, 71, 47, 95, 7, 67, 71, 93, 53, 83, 71, 107, 79, 89, 59, 163, 61, 95, 94, 2, 83, 143, 67, 121, 95, 143, 71, 137, 73, 113, 98, 135, 95, 167, 79, 157, 3, 125, 83, 219, 107, 131, 119, 167, 89, 224, 111, 163, 127, 143, 119, 191, 97, 121, 146, 87

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

Sequence is not the same as A183098(n): a(72) = 137, A183098(72) = 65.

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 2+3+6+12=23.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences related to sums of divisors

Cf. A000203, A091051, A183098, A183105.

N:= 1000: # to get a(1) to a(N)
S:=  {seq(seq(i^k, i=1..floor(N^(1/k))), k=2..ilog2(N))}:
seq(convert(numtheory:-divisors(t) minus S,`+`), t=1..N); # Robert Israel, Oct 02 2014

Table[Total[DeleteCases[Divisors[n],?(GCD@@FactorInteger[#][[All,2]]>1&)]],{n,100}]-1 (* _Harvey P. Dale, May 30 2021 *)

a(n) = sumdiv(n, d, d*((d!=1) && !ispower(d))); \\ Michel Marcus, Oct 02 2014

a(n) = A000203(n) - A091051(n).

a(1) = 0, a(p) = p, a(pq) = p+q+pq, a(pq...z) = [(p+1)*(q+1)*…*(z+1)]-1, a(p^k) = p, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

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A091050 Number of divisors of n that are perfect powers.

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A183097 a(n) = sum of powerful divisors d (including 1) of n.

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A183104 a(n) = product of divisors of n that are perfect powers.

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A183101 a(n) = sum of divisors of n that are not perfect powers.

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