A372328 -id:A372328 - cp's OEIS frontend

A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 128, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 128, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 384, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Apr 28 2024

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

Cf. A036537, A372328.
Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193, A356194.
Differs from A102631 at n = 8, 24, 27, 32, 40, 54, 56, 64, ... .

f[p_, e_] := p^(2^Ceiling[Log2[e + 1]] - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e+1)) - 1).
a(n) = n * A372328(n).
a(n) = n if and only if n is in A036537.
a(n) <= n^2, with equality if and only if n = 1.

A372331 The number of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Apr 28 2024

First differs from A370077 and A370080 at n = 32.
The number of divisors d of n that are infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.
Equivalently, the number of divisors d of n such that for each prime divisor p of d, bitand(v_p(n), v_p(d)) = 0, where v_p(k) is the highest power of p that divides k. Note that for infinitary divisors d of n (A077609), bitand(v_p(n), v_p(d)) = v_p(d).

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

Cf. A036537, A037445, A064379, A077609, A080100, A372328.

Cf. A370077, A370080.

f[p_, e_] := 2^DigitCount[e, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 2^(logint(x, 2) + 1 - hammingweight(x)), factor(n)[, 2]));

a(n) = A037445(A372328(n)).
Multiplicative with a(p^e) = 2^A023416(e) = A080100(e).
a(n) = 1 if and only if n is in A036537.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A080100(k)/p^k) = 1.51209151045338102469... .

A372692 The sum of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 15, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 15, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 15, 40, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, May 10 2024

The sum of divisors d of n that are infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.

The numbers of these divisors is A372331(n) and the largest of them is A372328(n).

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

Cf. A001146, A036537, A049417, A064379, A372328, A372331.

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

s(n) = apply(x -> 1 - x, binary(n));
a(n) = {my(f = factor(n), k); prod(i = 1, #f~, k = s(f[i, 2]); prod(j = 1, #k, if(k[j], f[i, 1]^(2^(#k-j)) + 1, 1)));}

Multiplicative with a(p^e) = Product_{k >= 0, 2^k < e, bitand(e, 2^k) = 0} (p^(2^k) + 1).
a(n) >= 1, with equality if and only if n is in A036537.
a(n) <= n-1, with equality if and only if n = 2^(2^k) for k >= 0.

cp's OEIS Frontend

A372329 a(n) is the smallest multiple of n whose number of divisors is a power of 2 (A036537).

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A372331 The number of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

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A372692 The sum of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).

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Mathematica

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