A245777 -id:A245777 - 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

A245776 Numerator of (n/tau(n) - sigma(n)/n).

Original entry on oeis.org

0, -1, 1, -5, 13, -1, 33, 1, 14, 7, 97, -1, 141, 25, 43, 101, 253, 5, 321, 37, 313, 85, 481, 1, 532, 127, 569, 8, 781, 27, 897, 323, 299, 235, 1033, 53, 1293, 301, 1297, 11, 1597, 83, 1761, 179, 173, 457, 2113, 133, 2230, 971, 771, 529, 2701, 163, 2737, 34, 2929, 751, 3361, 11, 3597, 865, 1115

Offset: 1

Jaroslav Krizek, Aug 01 2014

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

See A245777 - 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(14/9) = 14.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A245777, A245778, A245779, A245782.

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

a245776[n_Integer] :=
Map[Numerator[#/DivisorSigma[0, #] - DivisorSigma[1, #]/#] &,
  Range[n]]; a245776[63] (* Michael De Vlieger, Aug 07 2014 *)

vector(150, n, numerator(n/numdiv(n) - sigma(n)/n)) \\ Derek Orr, Aug 01 2014

a(n)/A245777(n) < 1 for numbers n in A245779.
a(n)/A245777(n) is an integer for numbers n in A245778.
a(n) = 1 for n = 3, 8 and 24.
a(n) < 0 for n = 2, 4, 6 and 12.

A245779 Numbers n such that (n/tau(n) - sigma(n)/n) < 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 18, 24

Offset: 1

Jaroslav Krizek, Aug 02 2014

Numbers n such that A245776(n)/A245777(n) = n/A000005(n) - A000203(n)/n < 1.

Finite sequence with 10 terms.

			24 is in sequence because 24/tau(24) - sigma(24)/24 = 24/8 - 60/24 = 1/2.

Cf. A000005, A000203, A245776, A245777, A245778, A245782.

[n:n in [1..1000000] | (Numerator((n /(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n))) / (Denominator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n))) lt 1]

a245779[n_Integer] :=
Select[Range[n],
  If[#/DivisorSigma[0, #] - DivisorSigma[1, #]/# < 1, True, False] &]; a245779[1000] (* Michael De Vlieger, Aug 07 2014 *)
Select[Range[25],#/DivisorSigma[0,#]-DivisorSigma[1,#]/#<1&] (* Harvey P. Dale, Nov 21 2023 *)

for(n=1,10^3, if(n/numdiv(n) - sigma(n)/n < 1, print1(n,", "))) \\ Derek Orr, Aug 02 2014

A245778 Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.

Original entry on oeis.org

1, 672, 4680, 30240, 435708, 23569920, 45532800, 4138364160, 14182439040, 53798734080, 153003540480, 403031236608, 518666803200

Offset: 1

Jaroslav Krizek, Aug 01 2014

Numbers n such that A245776(n) / A245777(n) =  (n / A000005(n) - A000203(n) / n) is an integer.
Sequence of integers k(n): 0, 25, 94, 311, 4031, 73652, 118571, …
Conjecture: subsequence of A216793.
Refactorable multiply-perfect numbers (A245782) are members of this sequence.
a(14) > 10^13. - Giovanni Resta, Jul 13 2015
The numbers 13661860101120 and 740344994887680 are also terms. - Giovanni Resta, Nov 14 2019

			672 is in sequence because 672 / tau(672) - sigma(672) / 672 = 672 / 24 - 2016 / 672 = 25 (integer).

Cf. A000005, A000203, A216793, A245776, A245777, A245779, A245782.

[n: n in [1..100000] | (Denominator((n/(#[d: d in Divisors(n)])) - (SumOfDivisors(n)/n)) eq 1)]

select(n -> type(n/numtheory:-tau(n) - numtheory:-sigma(n)/n,integer), [$1..10^8]); # Robert Israel, Aug 03 2014

for(n=1,10^8,s=n/numdiv(n);t=sigma(n)/n;if(floor(s-t)==s-t,print1(n,", "))) \\ Derek Orr, Aug 01 2014

A245777(a(n)) = 1.

a(8)-a(13) from Giovanni Resta, Jul 13 2015

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

cp's OEIS Frontend

A245782 Refactorable multiply-perfect numbers.

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A245776 Numerator of (n/tau(n) - sigma(n)/n).

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A245779 Numbers n such that (n/tau(n) - sigma(n)/n) < 1.

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A245778 Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.

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

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