A183098 -id:A183098 - cp's OEIS frontend

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

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)

A183100 a(n) is the sum of divisors d of n which are either 1 or of the form Product_{i} (p_i^e_i) where at least one e_i = 1.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 24, 14, 24, 24, 3, 18, 30, 20, 38, 32, 36, 24, 48, 6, 42, 4, 52, 30, 72, 32, 3, 48, 54, 48, 42, 38, 60, 56, 78, 42, 96, 44, 80, 69, 72, 48, 96, 8, 68, 72, 94, 54, 84, 72, 108, 80, 90, 60, 164, 62, 96, 95, 3, 84, 144, 68, 122, 96, 144, 72, 66, 74, 114, 99, 136, 96, 168, 80, 158, 4, 126, 84, 220, 108, 132, 120, 168, 90, 225, 112, 164, 128, 144, 120, 192, 98, 122, 147, 88

Offset: 1

Jaroslav Krizek, Dec 25 2010

a(n) = sum of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >= 1.

			For n = 12, the set of such divisors is {1, 2, 3, 6, 12}; a(12) = 1+2+3+6+12 = 24.

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

Cf. A000203, A001694, A183098, A183099.

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := f1[p, e] - p; a[1] = 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f + 1; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

A183100(n) = (1 + sumdiv(n, d, d*(!ispowerful(d)))); \\ Antti Karttunen, Oct 07 2017

a(n) = A000203(n) - A183099(n) = A183098(n) + 1.

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

A183103 a(n) = product of non-powerful divisors d of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 36, 7, 2, 3, 100, 11, 432, 13, 196, 225, 2, 17, 648, 19, 2000, 441, 484, 23, 10368, 5, 676, 3, 5488, 29, 810000, 31, 2, 1089, 1156, 1225, 7776, 37, 1444, 1521, 80000, 41, 3111696, 43, 21296, 10125, 2116, 47, 497664, 7, 5000, 2601, 35152, 53, 34992, 3025, 307328, 3249, 3364, 59, 11664000000, 61, 3844, 27783, 2, 4225, 18974736, 67, 78608, 4761, 24010000, 71, 186624, 73

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

Sequence is not the same as A183105: a(72) = 186624, A183105(72) = 13436928.

Not multiplicative, for example a(2)*a(3) <> a(6). - R. J. Mathar, Jun 07 2011

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 1*2*3*6*12 = 432.

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

Cf. A001694, A007955, A183098, A183102, A183105.

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A183103 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if not isA001694(d) then a := a*d; end if; end do; a ; end proc:
seq(A183103(n),n=1..73) ; # R. J. Mathar, Jun 07 2011

powerfulQ[n_] := Min[FactorInteger[n][[All, 2]]] > 1;
a[n_] := Times @@ Select[Divisors[n], !powerfulQ[#]&];
Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jun 01 2024 *)

A183103(n) = { my(m=1); fordiv(n, d, if(!ispowerful(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183102(n).

a(1) = 1, a(p) = p, a(pq) = (pq)^2, a(pq...z) = (pq...z)^(2^(k-1)), a(p^k) = p, 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.

cp's OEIS Frontend

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

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A183100 a(n) is the sum of divisors d of n which are either 1 or of the form Product_{i} (p_i^e_i) where at least one e_i = 1.

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A183103 a(n) = product of non-powerful divisors d of n.

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

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