cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A184793 Numbers m such that prime(m) is of the form floor(k*r), where r=(1+sqrt(5))/2; complement of A180736.

Original entry on oeis.org

2, 5, 7, 8, 10, 12, 14, 16, 17, 18, 19, 20, 22, 25, 26, 27, 30, 31, 32, 33, 34, 38, 40, 41, 42, 45, 46, 47, 48, 50, 52, 53, 55, 56, 58, 60, 61, 63, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 81, 84, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 98, 103, 104, 105, 106, 107, 108, 110, 112, 114, 115, 117, 118, 119, 121, 122, 123, 131, 134, 137, 138, 139, 140, 142, 143, 146, 148, 149, 152, 153, 155, 157, 160, 162, 163
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2011

Keywords

Examples

			See A184792.

Crossrefs

Cf. A184774, A184792, A184795.

Programs

A317980 a(n) = Product_{i=1..n} floor(5*i/2).

Original entry on oeis.org

2, 10, 70, 700, 8400, 126000, 2142000, 42840000, 942480000, 23562000000, 636174000000, 19085220000000, 610727040000000, 21375446400000000, 790891516800000000, 31635660672000000000, 1328697748224000000000, 59791398670080000000000, 2810195737493760000000000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 02 2018

Keywords

Comments

If p > 2 and p is odd, then Product_{i=1..n} floor(p*i/2) ~ (p/2)^n * n! * 2^(1/(2*p)) * sqrt(Pi) / (Gamma(1/2 - 1/(2*p)) * n^(1/(2*p))).

Links

Crossrefs

Cf. A010786, A180736, A275062, A319948, A319950.

Programs

  • Mathematica
    Table[Product[Floor[i*5/2], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{4 a[n] - 10 a[n - 1] - 5 (n - 1) (5 n - 6) a[n - 2] == 0, a[1] == 2, a[2] == 10}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)
FoldList[Times,Floor[5*Range[20]/2]] (* Harvey P. Dale, Sep 17 2020 *)

Formula

a(n) ~ (5/2)^n * n! * 2^(1/10) * sqrt(Pi) / (Gamma(2/5) * n^(1/10)).
Recurrence: 4*a(n) - 10*a(n-1) - 5*(n - 1)*(5*n - 6)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319948 a(n) = Product_{i=1..n} floor(3*i/2).

Original entry on oeis.org

1, 3, 12, 72, 504, 4536, 45360, 544320, 7076160, 106142400, 1698278400, 30569011200, 580811212800, 12197035468800, 268334780313600, 6440034727526400, 161000868188160000, 4347023441080320000, 121716656350248960000, 3651499690507468800000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 02 2018

Keywords

Crossrefs

Cf. A010786, A180736, A275062, A319949, A319950, A317980.

Programs

  • Mathematica
    Table[Product[Floor[i*3/2], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{4 a[n] - 6 a[n - 1] - 3 (n - 1) (3 n - 4) a[n - 2] == 0, a[1] == 1, a[2] == 3}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)

Formula

a(n) ~ (3/2)^n * n! * 2^(1/6) * sqrt(Pi) / (Gamma(1/3) * n^(1/6)).
Recurrence: 4*a(n) - 6*a(n-1) - 3*(n - 1)*(3*n - 4)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319949 a(n) = Product_{i=1..n} floor(4*i/3).

Original entry on oeis.org

1, 2, 8, 40, 240, 1920, 17280, 172800, 2073600, 26956800, 377395200, 6038323200, 102651494400, 1847726899200, 36954537984000, 776045297664000, 17072996548608000, 409751917166592000, 10243797929164800000, 266338746158284800000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 02 2018

Keywords

Links

Crossrefs

Cf. A004773, A010786, A180736, A275062, A319948, A319950.

Programs

  • Mathematica
    Table[Product[Floor[i*4/3], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{27*(3*n - 7)*a[n] == 54*(2*n - 5)*a[n-1] + 12*(12*n^2 - 42*n + 35)*a[n-2] + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a[n-3], a[1]==1, a[2]==2, a[3]==8}, a, {n, 1, 20}]
FoldList[Times,Floor[4 Range[20]/3]] (* Harvey P. Dale, Mar 21 2024 *)
  • PARI
    a(n) = prod(i=1, n, (4*i)\3); \\ Michel Marcus, Oct 03 2018

Formula

a(n) ~ (4/3)^n * n! * 2*sqrt(Pi) / (3^(1/4) * Gamma(1/4) * n^(1/4)).
Recurrence: 27*(3*n - 7)*a(n) = 54*(2*n - 5)*a(n-1) + 12*(12*n^2 - 42*n + 35)*a(n-2) + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a(n-3).

A319950 a(n) = Product_{i=1..n} floor(5*i/3).

Original entry on oeis.org

1, 3, 15, 90, 720, 7200, 79200, 1029600, 15444000, 247104000, 4447872000, 88957440000, 1868106240000, 42966443520000, 1074161088000000, 27928188288000000, 781989272064000000, 23459678161920000000, 727250023019520000000, 23999250759644160000000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 02 2018

Keywords

Comments

If p > 3 and gcd(p,3)=1 then Product_{i=1..n} floor(i*p/3) ~ (p/3)^n * n! * 2*Pi * 3^(1/p - 1/2) / (c(p) * n^(1/p)), where
c(p) = Gamma(2/3 - 2/(3*p)) * Gamma(1/3 - 1/(3*p)) if mod(p, 3) = 1,
c(p) = Gamma(1/3 - 2/(3*p)) * Gamma(2/3 - 1/(3*p)) if mod(p, 3) = 2.
In general, if q > 1, p > q and gcd(p,q)=1, then Product_{i=1..n} floor(i*p/q) ~ c(p,q) * (p/q)^n * n! / n^((q-1)/(2*p)), where c(p,q) is a constant.

Crossrefs

Cf. A010786, A047220, A180736, A275062, A319948, A319949, A317980.

Programs

  • Mathematica
    Table[Product[Floor[i*5/3], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{27*(15*n - 32)*a[n] == 675*(n-2)*a[n-1] + 15*(75*n^2 - 255*n + 194)*a[n-2] + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a[n-3], a[1]==1, a[2]==3, a[3]==15}, a, {n, 1, 20}]
  • PARI
    a(n) = prod(i=1, n, (5*i)\3); \\ Michel Marcus, Oct 03 2018

Formula

a(n) ~ (5/3)^n * n! * 2*Pi / (3^(3/10) * Gamma(1/5) * Gamma(3/5) * n^(1/5)).
Recurrence: 27*(15*n - 32)*a(n) = 675*(n-2)*a(n-1) + 15*(75*n^2 - 255*n + 194)*a(n-2) + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a(n-3).
Showing 1-5 of 5 results.

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