A363332 -id:A363332 - cp's OEIS frontend

A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 64.

Differs from A363332 at n = 1, 216, 432, 648, 864, 1000, ... .

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

Cf. A051903, A109613, A350390, A368711.
Cf. A331273, A363332.
Similar sequences: A007424, A368781, A372602, A372603, A372604.

f[n_] := n - If[EvenQ[n], 1, 0]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = (n+1) \ 2 * 2 - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A350390(n)).
a(n) = A109613(A051903(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{i>=1} (1 - (1/zeta(2*i+1))) = 1.42929441950714075659... .

A363334 a(n) is the sum of divisors of n that are both coreful and bi-unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 26, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 62, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 78, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 118, 65, 66, 67

Offset: 1

Amiram Eldar, May 28 2023

First differs from A363331 at n = 16.

The number of these divisors is A363332(n).

			a(8) = 14 since 8 has 3 divisors that are both bi-unitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.

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

Cf. A004709, A057723, A188999, A222266, A362852, A363331, A363332.

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - 1 - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1 - if(f[i, 2]%2, 0, f[i, 1]^(f[i, 2]/2)));}

Multiplicative with a(p^e) = (p^(e+1) - 1)/(p - 1) - 1, if e is odd, and (p^(e+1) - 1)/(p - 1) - p^(e/2) - 1 if e is even.
a(n) >= n, with equality if and only if n is cubefree (A004709).
a(n) >= A362852(n), with equality if and only if n = 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (p/(p+1))*(1+1/p-1/p^3+2/p^5) = 0.557782322450569540209... .
Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(3*s-2) + 2/p^(3*s-1) - 2/p^(4*s-2)). - Amiram Eldar, Oct 01 2023

A363333 Numbers with a record number of divisors that are both coreful and bi-unitary.

Original entry on oeis.org

1, 8, 32, 128, 216, 864, 3456, 7776, 13824, 31104, 108000, 279936, 432000, 972000, 1728000, 3888000, 15552000, 34992000, 62208000, 97200000, 139968000, 248832000, 333396000, 559872000, 592704000, 874800000, 1333584000, 5334336000, 12002256000, 21337344000, 33339600000

Offset: 1

Amiram Eldar, May 28 2023

nonn

Indices of records in A363332.

The corresponding record values are 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..261
Amiram Eldar, Table of n, a(n), A363332(a(n)) for n = 1..261

Cf. A363332.
Subsequence of A025487.
Similar sequences: A005934, A293185.

f[p_, e_] := If[OddQ[e], e, e - 1]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120]
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

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A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

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A363334 a(n) is the sum of divisors of n that are both coreful and bi-unitary.

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Formula

A363333 Numbers with a record number of divisors that are both coreful and bi-unitary.

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Author

Keywords

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Mathematica