A360721 -id:A360721 - cp's OEIS frontend

A368105 The number of bi-unitary divisors of n that are powerful (A001694).

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

Offset: 1

Amiram Eldar, Dec 12 2023

First differs from A095691 and A365552 at n = 32.

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

Cf. A013661, A001694, A005117, A062503, A286324,
Similar sequences: A005361, A323308, A360721.
Cf. A095691, A365552.

f[p_, e_] := If[e == 2 || OddQ[e], e, e -1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2 || x == 2, x, x-1), factor(n)[, 2]));

Multiplicative with a(p^e) = e if e = 2 or e is odd, and e-1 otherwise.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A286324(n), with equality if and only if n equals the square of a squarefree number (A062503).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4 + 1/p^5) = 1.87133814920590891161... .

A360723 Numbers that have at least one exponent in their canonical prime factorization that is neither 2 nor of the form 2^k-1, k>=1.

Original entry on oeis.org

16, 32, 48, 64, 80, 81, 96, 112, 144, 160, 162, 176, 192, 208, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 400, 405, 416, 432, 448, 464, 480, 486, 496, 512, 528, 544, 560, 567, 576, 592, 608, 624, 625, 648, 656, 672, 688, 704, 720, 729, 736, 752

Offset: 1

Amiram Eldar, Feb 18 2023

Numbers that have at least one powerful divisor that is not infinitary divisor, i.e., numbers k such that A360721(k) < A005361(k).
The complement of this sequence is the sequence of numbers all of whose powerful divisors are also infinitary divisors. The related sequence of numbers all of whose infinitary divisors are powerful is the sequence of squares (A000290).
The asymptotic density of this sequence is 1 - Product_{p prime} ((1 - 1/p) * (1 + 1/p^2 + Sum_{i>=1} 1/p^(2^i-1))) = 0.071899867098952952524... .

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

Cf. A000225, A000290, A001694, A005361, A077609, A360721.

q[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # != 2 && # + 1 != 2^IntegerExponent[# + 1, 2] &]; Select[Range[1000], q]

is(n) = {my(e = factor(n)[, 2]); for(i = 1, #e, if(e[i] != 2 && (e[i]+1)>>valuation(e[i]+1, 2) != 1, return(1))); 0;}

A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 49, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 17

Offset: 1

Amiram Eldar, Feb 18 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers.

Cf. A001694, A049417, A077609, A077610, A360723.

Similar sequences: A183097, A360721.

f[p_, e_] := Times @@ (p^(2^(-1 + Flatten @ Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])) + 1) - If[OddQ[e], p, 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = f(p, e) if e is even, and f(p, e) - p is e is odd, where f(p, e) = Product{k>=1, e_k=1} (p^(2^k) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
a(n) <= A049417(n), with equality if and only if n is a square.
a(n) <= A183097(n), with equality if and only if n is not in A360723.

A377709 Numbers that have a record number of infinitary divisors that are powerful (A001694).

Original entry on oeis.org

1, 4, 8, 36, 72, 128, 216, 1152, 3456, 27000, 28800, 86400, 279936, 432000, 4233600, 6998400, 21168000, 34992000, 148176000, 342921600, 1714608000, 8957952000, 12002256000, 197222256000, 207467568000, 438939648000, 1071630000000, 1452272976000, 3072577536000, 7501410000000

Offset: 1

Amiram Eldar, Nov 04 2024

nonn

Indices of records in A360721.

The corresponding record values are 1, 2, 3, 4, 6, 7, 9, 14, 21, 27, 28, 42, 49, 63, 84, 98, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..117
Amiram Eldar, Table of n, a(n), A360721(a(n)) for n = 1..117

Cf. A001694, A037445, A360721.

Subsequence of A025487.

f[p_, e_] := 2^DigitCount[e, 2, 1] - Mod[e, 2]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; 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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A368105 The number of bi-unitary divisors of n that are powerful (A001694).

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A360723 Numbers that have at least one exponent in their canonical prime factorization that is neither 2 nor of the form 2^k-1, k>=1.

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A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).

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A377709 Numbers that have a record number of infinitary divisors that are powerful (A001694).

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