A245786 -id:A245786

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A245782

Refactorable multiply-perfect numbers.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Multiply-perfect numbers k (A007691) such that k / tau(k) is integer.

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

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

LINKS

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

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

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Aug 01 2014

EXTENSIONS

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

STATUS

approved

A245785

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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]

LINKS

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

EXAMPLE

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

PROG

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

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

CROSSREFS

Cf. A000005, A000203, A245784, A245786.

KEYWORD

nonn,frac

AUTHOR

Jaroslav Krizek, Aug 15 2014

STATUS

approved

A245784

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

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

LINKS

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

EXAMPLE

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

PROG

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

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

CROSSREFS

Cf. A000005, A000203, A245785, A245786.

KEYWORD

nonn,frac

AUTHOR

Jaroslav Krizek, Aug 15 2014

STATUS

approved

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