A183101 -id:A183101 - cp's OEIS frontend

A183098 a(1) = 0, a(n) = sum of divisors d of n such that if d = Product_{i} (p_i^e_i) then not all e_i are > 1.

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, 65, 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

a(n) = sum of non-powerful divisors d of n where powerful numbers are numbers from A001694(m) for m >= 2.

Sequence is not the same as A183101(n): a(72) = 65, A183101(72) = 137.

			For n = 12, the 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, A001694, A183097, A183100, A183101, A183103.

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := f1[p, e] - p; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

A183098(n) = sumdiv(n, d, d*(!ispowerful(d))); \\ Antti Karttunen, Oct 07 2017

a(n) = A000203(n) - A183097(n) = A183100(n) - 1.

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

Name corrected by Jon E. Schoenfield, Aug 29 2023

A183105 a(n) = product of divisors of n that are not perfect powers.

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, 13436928

Offset: 1

Jaroslav Krizek, Dec 25 2010

Sequence is not the same as A183103: a(72) = 13436928, A183103(72) = 186624.

Not multiplicative since 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. A001597, A007955, A183101, A183103, A183104.

isA001597 := proc(n) local e ; e := seq(op(2,p),p=ifactors(n)[2]) ; return ( igcd(e) >=2 ) ; end proc:
A183105 := proc(n) local a,d; a := 1 ; for d in numtheory[divisors](n) do if not isA001597(d) then a := a*d; end if; end do; a ; end proc:
seq(A183105(n),n=1..72) ; # R. J. Mathar, Jun 07 2011

perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1;
a[n_] := Times @@ Select[Divisors[n], !perfectPowerQ[#]&];
Table[a[n], {n, 1, 72}] (* Jean-François Alcover, May 31 2024 *)

A183105(n) = { my(m=1); fordiv(n, d, if(!ispower(d), m *= d)); m; }; \\ Antti Karttunen, Oct 07 2017

a(n) = A007955(n) / A183104(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.

cp's OEIS Frontend

A183098 a(1) = 0, a(n) = sum of divisors d of n such that if d = Product_{i} (p_i^e_i) then not all e_i are > 1.

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