A120301 -id:A120301 - cp's OEIS frontend

A123944 Numbers k such that A120301(k) differs from A058313(k).

19, 28, 87, 99, 104, 196, 203, 210, 222, 228, 231, 238, 281, 328, 367, 499, 579, 620, 888, 967, 1036, 1147, 1204, 1352, 1372, 1403, 1419, 1430, 1470, 1481, 1498, 1503, 1666, 1693, 1907, 2211, 2359, 2440, 2499, 2521, 2556, 2678, 2948, 3407, 3467, 3504, 3537, 3892, 4046, 4079, 4108

Offset: 1

Alexander Adamchuk, Nov 22 2006

nonn

The ratio A120301(n)/A058313(n) = 1 for most n.
The ratio A120301(a(n))/A058313(a(n)) = {5, 7, 11, 5, 13, 7, 17, 7, 37, 10, 29, 119, 47, 41, 23, 5, 29, 31, 37, 11, 37, 41, 43, 13, 7, 13, 71, 13, 49, 13, 7,...} is prime for the most a(n).
The first composite ratio A120301(a(n))/A058313(a(n)) corresponds to a(n) = a(29) = 1470 because A120301(1470)/A058313(1470) = 49 = 7^2. [Edited by Petros Hadjicostas, May 09 2020]

Cf. A058313, A120301.

f=0; Do[f=f+(-1)^(n+1)*1/n; g=Abs[(2(-1)^n*n+(-1)^n-1)/4]*f; rfg=Numerator[g]/Numerator[f]; If[(rfg==1)==False, Print[{n,rfg}]], {n,1,15000}]

isok(n) = my(sn = sum(k=1, n, (-1)^(k+1)/k)); numerator(sn) != abs(numerator((-1/4) * (2*(-1)^n*n + (-1)^n - 1)*sn));
for (n=1, 4200, if (isok(n), print1(n, ", "))); \\ Michel Marcus, May 10 2020

a(47)-a(51) from Petros Hadjicostas, May 09 2020

A167372 a(n) = A120301(A123944(n))/A058313(A123944(n)).

Original entry on oeis.org

5, 7, 11, 5, 13, 7, 17, 7, 37, 19, 29, 119, 47, 41, 23, 5, 29, 31, 37, 11, 37, 41, 43, 13, 7, 13, 71, 13, 49, 13, 7, 47, 7, 7, 53, 79, 59, 61, 5, 97, 71, 103, 67, 71, 17, 73, 61, 139, 17, 17, 79, 19, 19, 19, 19, 83, 151, 89, 29, 97, 263, 29, 101, 103, 223, 107, 109, 271, 37, 23, 113, 359

Offset: 1

Alexander Adamchuk, Nov 02 2009

nonn

The ratio A120301(n)/A058313(n) = 1 for most n.
a(n) is prime for most n.
The first composite ratio a(12) = 119 = 7*17 corresponds to A123944(12) = 238.
The next two composite ratios a(29) = a(76) = 49 = 7^2 correspond to A123944(29) = 1470 and A123944(76) = 10290. [Edited by Petros Hadjicostas, May 09 2020]

Cf. A058313, A120301, A123944.

f = 0; Do[f = f + (-1)^(n + 1) * 1/n; g = Abs[(2(-1)^n * n + (-1)^n - 1)/4] * f; rfg = Numerator[g]/Numerator[f]; If[(rfg == 1) == False, Print[rfg]], {n, 1500}]

lista(nn) = {for (n=1, nn, my(sn = sum(k=1, n, (-1)^(k+1)/k)); if ((s=numerator(sn)) != (ss=abs(numerator((-1/4) * (2*(-1)^n*n + (-1)^n - 1) * sn))), print1(ss/s, ", ")););} \\ Michel Marcus, May 10 2020

a(32)-a(46) from Petros Hadjicostas, May 09 2020, using Michel Marcus's program and the data from A123944

a(47)-a(72) from Petros Hadjicostas, May 09 2020, using the Mathematica program

A334724 Denominator of the sum of all elements of the n X n matrix M with M[i,j] = (-1)^(i+j)*i/j for i,j = 1..n.

Original entry on oeis.org

1, 2, 3, 6, 20, 20, 105, 210, 504, 504, 4620, 4620, 51480, 51480, 9009, 18018, 272272, 272272, 23279256, 23279256, 21162960, 21162960, 446185740, 446185740, 2059318800, 2059318800, 5736673800, 5736673800, 22181805360, 22181805360, 644658718275, 1289317436550, 1213475234400

Offset: 1

Petros Hadjicostas, May 09 2020

			The absolute values of the first few fractions are 1, 1/2, 5/3, 7/6, 47/20, 37/20, 319/105, 533/210, 1879/504, ... = A120301/A334724.

Cf. A120301 (absolute values of numerators).

a(n) = denominator(sum(j=1, n, sum(i=1, n, (-1)^(i+j)*i/j))); \\ Michel Marcus, May 09 2020

More terms from Michel Marcus, May 09 2020

cp's OEIS Frontend

A123944 Numbers k such that A120301(k) differs from A058313(k).

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A167372 a(n) = A120301(A123944(n))/A058313(A123944(n)).

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A334724 Denominator of the sum of all elements of the n X n matrix M with M[i,j] = (-1)^(i+j)*i/j for i,j = 1..n.

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