| Displaying 1-4 of 4 results found.
page
1
Refactorable multiply-perfect numbers.
+10
8
1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 518666803200, 13661860101120, 740344994887680, 796928461056000, 212517062615531520, 87934476737668055040, 154345556085770649600, 170206605192656148480, 1161492388333469337600, 1802582780370364661760
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.
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).
Denominator of (n / tau(n) - sigma(n) / n).
+10
6
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
COMMENTS
See A245776 - numerator of (n / tau(n) - sigma(n) / n). First deviation from A245785 (denominator of (n/tau(n) + sigma(n)/n)) is at a(300); a(300) = 75, A245785(300) = 25. Sequence of numbers n such that A245785(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] - Jaroslav Krizek, Aug 15 2014
EXAMPLE
For n = 9; a(9) = denominator(9/tau(9) - sigma(9)/9) = denominator(9/3 - 13/9) = denominator(14/9) = 9.
MATHEMATICA
Table[Denominator[n/DivisorSigma[0, n] - DivisorSigma[1, n]/n], {n, 70}] (* Alonso del Arte, Aug 15 2014 *)
PROG
(Magma) [Denominator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n)): n in [1..100]]
(PARI) vector(150, n, denominator(n/numdiv(n) - sigma(n)/n)) \\ Derek Orr, Aug 01 2014
Numerator of (n/tau(n) - sigma(n)/n).
+10
5
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
COMMENTS
See A245777 - denominator of (n/tau(n) - sigma(n)/n).
FORMULA
a(n) = 1 for n = 3, 8 and 24.
a(n) < 0 for n = 2, 4, 6 and 12.
EXAMPLE
For n = 9; a(9) = numerator(9/tau(9) - sigma(9)/9) = numerator(9/3 - 13/9) = numerator(14/9) = 14.
MATHEMATICA
a245776[n_Integer] :=
Map[Numerator[#/DivisorSigma[0, #] - DivisorSigma[1, #]/#] &,
PROG
(Magma) [Numerator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n)): n in [1..1000]]
(PARI) vector(150, n, numerator(n/numdiv(n) - sigma(n)/n)) \\ Derek Orr, Aug 01 2014
Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.
+10
4
1, 672, 4680, 30240, 435708, 23569920, 45532800, 4138364160, 14182439040, 53798734080, 153003540480, 403031236608, 518666803200
COMMENTS
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. The numbers 13661860101120 and 740344994887680 are also terms. - Giovanni Resta, Nov 14 2019
EXAMPLE
672 is in sequence because 672 / tau(672) - sigma(672) / 672 = 672 / 24 - 2016 / 672 = 25 (integer).
MAPLE
select(n -> type(n/numtheory:-tau(n) - numtheory:-sigma(n)/n, integer), [$1..10^8]); # Robert Israel, Aug 03 2014
PROG
(Magma) [n: n in [1..100000] | (Denominator((n/(#[d: d in Divisors(n)])) - (SumOfDivisors(n)/n)) eq 1)]
(PARI)
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
Search completed in 0.005 seconds
|