A287958 -id:A287958 - cp's OEIS frontend

A333926 The sum of recursive divisors of n.

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, Apr 10 2020

The definition of recursive divisors and the number of recursive divisors of n are in A282446.

First differs from A051378 at n = 256.

			The recursive divisors of 8 are 1, 2 and 8, therefore a(8) = 1 + 2 + 8 = 11.

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

Cf. A000203, A051378, A282446, A287957, A287958.

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]

Multiplicative with a(p^k) = 1 + Sum_{d recursive divisor of k} p^d.

a(n) <= A051378(n) <= A000203(n).

A287957 Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Jun 03 2017

We use the definition of recursive divisor given in A282446.
More informally, the prime tower factorization of T(n, k) is the intersection of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).
This sequence has connections with the classical GCD (A003989).
For any i > 0, j > 0 and k > 0:
- T(i, j) = 1 iff gcd(i, j) = 1,
- A007947(T(i, j)) = A007947(gcd(i, j)),
- T(i, j) >= 1,
- T(i, j) <= min(i, j),
- T(i, j) <= gcd(i, j),
- T(i, 1) = 1,
- T(i, i) = i,
- T(i, j) = T(j, i) (the sequence is commutative),
- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),
- T(i, i*j) <= i,
- if gcd(i, j) = 1 then T(i*j, k) = T(i, k) * T(j, k) (the sequence is multiplicative),
- T(i, 2*i) = A259445(i).
See also A287958 for the LCM equivalent.

			Table starts:
n\k|    1   2   3   4   5   6   7   8   9   10
---+-----------------------------------------------
1  |    1   1   1   1   1   1   1   1   1    1  ...
2  |    1   2   1   2   1   2   1   2   1    2  ...
3  |    1   1   3   1   1   3   1   1   3    1  ...
4  |    1   2   1   4   1   2   1   2   1    2  ...
5  |    1   1   1   1   5   1   1   1   1    5  ...
6  |    1   2   3   2   1   6   1   2   3    2  ...
7  |    1   1   1   1   1   1   7   1   1    1  ...
8  |    1   2   1   2   1   2   1   8   1    2  ...
9  |    1   1   3   1   1   3   1   1   9    1  ...
10 |    1   2   1   2   5   2   1   2   1   10  ...
...
T(4, 8) = T(2^2, 2^3) = 2.

Rémy Sigrist, First 100 antidiagonals of array, flattened
Rémy Sigrist, Illustration of the first terms

Cf. A003989, A007947, A182318, A259445, A282446, A287958.

T(n,k) = my (g=factor(gcd(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))

cp's OEIS Frontend

A333926 The sum of recursive divisors of n.

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