A245786 -id:A245786 - cp's OEIS frontend

A245782 Refactorable multiply-perfect numbers.

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 518666803200, 13661860101120, 740344994887680, 796928461056000, 212517062615531520, 87934476737668055040, 154345556085770649600, 170206605192656148480, 1161492388333469337600, 1802582780370364661760

Offset: 1

Jaroslav Krizek, Aug 01 2014

nonn

Multiply-perfect numbers k (A007691) such that k / tau(k) is an integer.
Also multiply-perfect numbers k (A007691) such that (k / tau(k) - sigma(k) / k) = (k / A000005(k) - A000203(k) / k) is an integer.
Also multiply-perfect numbers k (A007691) such that (k / tau(k) + sigma(k) / k) = (k / A000005(k) + A000203(k) / k) is an integer.

			Multiply-perfect number 672 is in sequence because 672 / tau(672) = 28 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..407

Intersection of A033950 (refactorable numbers) and A007691 (multiply-perfect numbers).
Subsequence of A245778 and A245786.
Supersequence of A047728.
Cf. A000005, A000203.
Cf. A245776, A245777, A245779.

[n:n in [A007691(n)] | (Denominator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n))) eq 1];

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

isok(n) = !(n % numdiv(n)) && !(sigma(n) % n); \\ Michel Marcus, Aug 11 2014

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !(k % d);} \\ Amiram Eldar, May 09 2024

a(14)-a(18) from Amiram Eldar, May 09 2024

A245785 Denominator of (n/tau(n) + sigma(n)/n).

Original entry on oeis.org

1, 2, 6, 12, 10, 2, 14, 8, 9, 10, 22, 3, 26, 14, 20, 80, 34, 6, 38, 30, 84, 22, 46, 2, 75, 26, 108, 3, 58, 20, 62, 96, 44, 34, 140, 36, 74, 38, 156, 4, 82, 28, 86, 33, 30, 46, 94, 60, 147, 150, 68, 78, 106, 36, 220, 7, 228, 58, 118, 5, 122, 62, 126, 448, 260

Offset: 1

Jaroslav Krizek, Aug 15 2014

Denominator of (n/A000005(n) + A000203(n)/n).
See A245784 - numerator of (n/tau(n) + sigma(n)/n).
A245784(n) / a(n) = integer for numbers n in A245786; a(n) = 1.
First deviation from A245777 (denominator of (n/tau(n) - sigma(n)/n)) is at a(300); a(300) = 25, A245777(300) = 75. Sequence of numbers n such that A245777(n) is not equal to a(n): 300, 768, 1452, 1764, 2100, 3468, 3900, 5376, 5700, 6084, 6348, 9075, 9300, ... See (Magma) [n: n in [1..10000] | (Denominator((n/(#[d: d in Divisors(n)]))+(SumOfDivisors(n)/n))) - (Denominator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n))) ne 0]

			For n = 9; a(9) = denominator(9/tau(9) + sigma(9)/9) = denominator(9/3 + 13/9) = denominator(40/9) = 9.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A245784, A245786.

[Denominator((n/(#[d: d in Divisors(n)]))+(SumOfDivisors(n)/n)): n in [1..1000]]

for(n=1, 100, s=n/numdiv(n); t=sigma(n)/n; print1(denominator(s+t),", ")) \\ Derek Orr, Aug 15 2014

A245784 Numerator of (n/tau(n) + sigma(n)/n).

Original entry on oeis.org

2, 5, 17, 37, 37, 7, 65, 31, 40, 43, 145, 13, 197, 73, 107, 411, 325, 31, 401, 163, 569, 157, 577, 11, 718, 211, 889, 20, 901, 123, 1025, 701, 427, 343, 1417, 235, 1445, 421, 1745, 29, 1765, 211, 1937, 305, 277, 601, 2305, 443, 2572, 1529, 963, 823, 2917, 323

Offset: 1

Jaroslav Krizek, Aug 15 2014

Numerator of (n/A000005(n) + A000203(n)/n).

See A245785 - denominator of (n/tau(n) + sigma(n)/n).

			For n = 9; a(9) = numerator(9/tau(9) + sigma(9)/9) = numerator(9/3 + 13/9) = numerator(40/9) = 40.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A245785, A245786.

[Numerator((n/(#[d: d in Divisors(n)]))+(SumOfDivisors(n)/n)): n in [1..1000]]

for(n=1, 100, s=n/numdiv(n); t=sigma(n)/n; print1(numerator(s+t),", ")) \\ Derek Orr, Aug 15 2014

cp's OEIS Frontend

A245782 Refactorable multiply-perfect numbers.

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A245785 Denominator of (n/tau(n) + sigma(n)/n).

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A245784 Numerator of (n/tau(n) + sigma(n)/n).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI