A282446 -id:A282446 - cp's OEIS frontend

A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

1, 2, 4, 6, 12, 30, 36, 60, 180, 420, 900, 1260, 4620, 6300, 13860, 44100, 55440, 69300, 180180, 485100, 720720, 900900, 2882880, 3063060, 6306300, 12252240, 15315300, 49008960, 58198140, 107207100, 232792560, 290990700, 931170240, 1163962800, 2036934900, 4655851200

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, ...

Amiram Eldar, Table of n, a(n) for n = 1..200

Subsequence of A025487.
Cf. A282446.
Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A309141 (nonunitary), A318278 (exponential).

recDivNum[1] = 1; recDivNum[n_] := recDivNum[n] = Times @@ (1 + recDivNum/@ (Last /@ FactorInteger[n])); rm = 0; s = {}; Do[r = recDivNum[n]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^4}]; s

A333926 The sum of recursive divisors of n.

Original entry on oeis.org

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).

A353898 a(n) is the number of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 10 2022

First differs from A049599 and A282446 at n=32.

			The divisors of 8 are 1, 2 = 2^1, 4 = 2^2 and 8 = 2^3. 3 of these divisors, 1, 2 and 4, are in A138302. Therefore, a(8) = 3.

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

Cf. A000005, A004709, A049599, A138302, A282446, A353897.

Similar sequences: A034444, A048105, A286324, A049419.

f[p_, e_] := Floor[Log2[e]] + 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = floor(log_2(e)) + 2.
a(n) > 1 for n > 1 and a(n) = 2 if and only if n is a prime.
a(n) = A000005(n) if and only if n is cubefree (A004709).

A366901 The number of exponentially odious divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 27 2023

First differs from A049599 and A282446 at n = 32, from A365551 at n = 64, and from A353898 at n = 128.
The number of divisors of n that are exponentially odious numbers (A270428), i.e., numbers having only odious (A000069) exponents in their canonical prime factorization.
The sum of these divisors is A366903(n) and the largest of them is A366905(n).

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

Cf. A000005, A000069, A004709, A115384, A270428, A366903, A366905.
Cf. A049599, A282446, A365551, A353898.
Similar sequences: A325837, A353898, A365680, A366902.

f[p_, e_] := Floor[e/2] + If[OddQ[e] || EvenQ[DigitCount[e + 1, 2, 1]], 1, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = 1 + n\2 + (n%2 || hammingweight(n+1)%2==0); \\ after Charles R Greathouse IV at A115384
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A115384(e) + 1.

a(n) <= A000005(n), with equality if and only if n is a cubefree number (A004709).

A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Sep 03 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (possibly of A282446 and A318469).

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

Cf. A049599, A318671 (numerators), A318673.

Cf. also A046644, A299150, A317932, A317934, A318498.

up_to = (2^16)+1;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA049599(n) = factorback(apply(e -> (1+numdiv(e)),factor(n)[,2]));
v318671_62 = DirSqrt(vector(up_to, n, A049599(n)));
A318671(n) = numerator(v318671_62[n]);
A318672(n) = denominator(v318671_62[n]);
A318673(n) = valuation(A318672(n),2);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A049599(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318673(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]))))

A287958 Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 4, 3, 4, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 12, 5, 12, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 9, 64, 35, 6, 35, 64, 9, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40

Offset: 1

Rémy Sigrist, Jun 03 2017

We say that m is a recursive multiple of d iff d is a recursive divisor of m (as described in A282446).
More informally, the prime tower factorization of T(n, k) is the union 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 LCM (A003990).
For any i > 0, j > 0 and k > 0:
- A007947(T(i, j)) = A007947(lcm(i, j)),
- T(i, j) >= 1,
- T(i, j) >= max(i, j),
- T(i, j) >= lcm(i, j),
- T(i, 1) = i,
- 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*j,
- if gcd(i, j) = 1 then T(i, j) = i*j.
See also A287957 for the GCD equivalent.

			Table starts:
n\k|     1   2   3   4   5   6   7   8   9  10
---+-----------------------------------------------
1  |     1   2   3   4   5   6   7   8   9  10  ...
2  |     2   2   6   4  10   6  14   8  18  10  ...
3  |     3   6   3  12  15   6  21  24   9  30  ...
4  |     4   4  12   4  20  12  28  64  36  20  ...
5  |     5  10  15  20   5  30  35  40  45  10  ...
6  |     6   6   6  12  30   6  42  24  18  30  ...
7  |     7  14  21  28  35  42   7  56  63  70  ...
8  |     8   8  24  64  40  24  56   8  72  40  ...
9  |     9  18   9  36  45  18  63  72   9  90  ...
10 |    10  10  30  20  10  30  70  40  90  10  ...
...
T(4, 8) = T(2^2, 2^3) = 2^(2*3) = 2^6 = 64.

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

Cf. A003990, A007947, A182318, A282446, A287957.

T(n,k) = if (n*k==0, return (max(n,k))); my (g=factor(lcm(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))

A328880 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) + 1), where pi = A000720, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 3, 2, 3, 8, 5, 6, 7, 6, 12, 2, 4, 6, 3, 8, 9, 10, 4, 6, 4, 14, 3, 6, 9, 24, 6, 2, 15, 8, 12, 6, 7, 6, 21, 8, 8, 18, 7, 10, 12, 8, 13, 6, 3, 8, 12, 14, 3, 6, 20, 6, 9, 18, 5, 24, 7, 12, 9, 2, 28, 30, 4, 8, 12, 24, 9, 6, 10, 14, 12, 6, 15, 42, 11, 8

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

			a(36) = 6 because 36 = 2^2 * 3^2 = prime(1)^2 * prime(2)^2 and (a(1) + 1) * (a(2) + 1) = (1 + 1) * (2 + 1) = 6.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A048250, A156061, A184160, A225395, A282446, A328879.

a[1] = 1; a[n_] := Times @@ (a[PrimePi[#[[1]]]] + 1 & /@ FactorInteger[n]); Table[a[n], {n, 1, 80}]

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + a(primepi(f[i])))} \\ Andrew Howroyd, Oct 29 2019

a(n) = a(prime(n)) - 1.

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

Original entry on oeis.org

6, 28, 264, 1104, 3360, 75840, 151062912, 606557952, 2171581440

Offset: 1

Amiram Eldar, Apr 10 2020

Since a recursive divisor is also a (1+e)-divisor (see A049599), then the first 6 terms and other terms of this sequence coincide with those of A049603.

			264 is a term since the sum of its recursive divisors is 1 + 2 + 3 + 6 + 8 + 11 + 22 + 24 + 33 + 66 + 88 + 264 = 528 = 2 * 264.

Cf. A049603, A282446, A333926.

Analogous sequences: A000396, A002827 (unitary), A007357 (infinitary), A054979 (exponential), A064591 (nonunitary).

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]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == 2*# &]

A349284 Numbers k such that A051378(k) > 2k and A333926(k) <= 2k.

Original entry on oeis.org

126720, 134400, 149760, 188160, 195840, 456960, 510720, 549120, 618240, 718080, 748800, 779520, 802560, 833280, 940800, 979200, 994560, 1094400, 1102080, 1155840, 1263360, 1324800, 1393920, 1424640, 1585920, 1639680, 1670400, 1785600, 1800960, 1908480, 1946880

Offset: 1

Amiram Eldar, Nov 13 2021

nonn

(1+e)-abundant numbers are numbers k such that A051378(k) > 2*k, i.e., numbers k whose sum of (1+e)-divisors exceeds 2*k.

Since all the recursive divisors (see A282446) of a number are also its (1+e)-divisors, the sequence of (1+e)-abundant numbers includes all the recursive abundant numbers (A333928). The first 21387 (1+e)-abundant numbers are also recursive abundant numbers. Therefore, this sequence includes only the (1+e)-abundant numbers that are not recursive abundant numbers.

			126720 is a term since A051378(126720) = 261144 > 2*126720 = 253440 and A333926(126720) = 246168 < 253440.

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

Cf. A049599, A049603, A051378, A282446, A333926, A333928.

oesigma[1] = 1; oesigma[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; 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]]; recsigma[1] = 1; recsigma[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], oesigma[#] > 2*# && recsigma[#] <= 2*# &]

cp's OEIS Frontend

A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

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A333926 The sum of recursive divisors of n.

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A353898 a(n) is the number of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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A366901 The number of exponentially odious divisors of n.

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A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

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

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A287958 Table read by antidiagonals: T(n, k) = least recursive multiple of n and k; n > 0 and k > 0.

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A328880 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) + 1), where pi = A000720, with a(1) = 1.

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A349284 Numbers k such that A051378(k) > 2k and A333926(k) <= 2k.