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.

A383074 a(n) = floor(A066206(n)/A066205(n)).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16
Offset: 1

Views

Author

Clark Kimberling, Apr 19 2025

Keywords

Examples

			a(1) = floor(3/2) = 1; a(2) = floor(3*7/(2*5)) = 2; a(3) = floor(3*7*13/(2*5*11)) = 2.

Crossrefs

Cf. A000040, A066205, A066206.

Programs

  • Mathematica
    Table[Floor[Product[Prime[2 k], {k, 1, n}]/Product[Prime[2 k - 1], {k, 1, n}]], {n, 1, 200}]

A066205 a(n) = Product_{k=1..n} prime(2k-1), where prime(k) is k-th prime.

Original entry on oeis.org

2, 10, 110, 1870, 43010, 1333310, 54665710, 2569288370, 151588013830, 10156396926610, 741416975642530, 61537608978329990, 5969148070898009030, 614822251302494930090, 67015625391971947379810, 8510984424780437317235870, 1166004866194919912461314190
Offset: 1

Views

Author

Leroy Quet, Dec 16 2001

Keywords

Comments

Equivalently, a(n) is the product of the first n odd-indexed primes. - Jon E. Schoenfield, Jan 12 2022

Examples

			a(3) = prime(1) * prime(3) * prime(5) = 2 * 5 * 11 = 110.

Links

Crossrefs

Cf. A000040, A002110, A066206.

Programs

  • Maple
    a:= proc(n) option remember;
     `if`(n=0, 1, a(n-1)*ithprime(2*n-1))
    end:
seq(a(n), n=1..17);  # Alois P. Heinz, Jan 12 2022
  • Mathematica
    FoldList[Times, Array[Prime[2 # - 1] &, 17]] (* Michael De Vlieger, Jan 12 2022 *)
  • PARI
    { for (n=1, 100, p=1; for (k=1, n, p*=prime(2*k - 1)); write("b066205.txt", n, " ", p) ) } \\ Harry J. Smith, Feb 05 2010
  • PARI
    a(n) = prod(k=1, n, prime(2*k-1)); \\ Michel Marcus, Jan 13 2022

A095134 Sum of the product of the first ceiling(n/2) odd-indexed primes and the product of the first floor(n/2) even-indexed primes; a(1) = 2.

Original entry on oeis.org

2, 5, 13, 31, 131, 383, 2143, 7057, 48197, 193433, 1483733, 6898961, 60231361, 293988703, 2808611363, 15253406999, 164272132459, 925319250199, 10930128162979, 65091314708809, 796351893424729, 5081275480436251
Offset: 1

Views

Author

Robert G. Wilson v, May 27 2004

Keywords

Examples

			a(5) = 2*5*11 + 3*7 = 131, a(6) = 2*5*11 + 3*7*13 = 383;
a(7) = 2*5*11*17 + 3*7*13 = 2143, a(8) = 2*5*11*17 + 3*7*13*19 = 7057.

Links

Crossrefs

Cf. A095137, A000040, A066206, A066205.

Programs

  • Maple
    p:= 2: R:= 2: a:= 2: b:= 1:
for m from 1 to 10 do
  p:= nextprime(p); b:= b*p; R:= R,a+b;
  p:= nextprime(p); a:= a*p; R:= R,a+b;
od:
R; # Robert Israel, Dec 01 2024
  • Mathematica
    f[n_] := Product[Prime[i], {i, 2, n, 2}] + Product[Prime[i], {i, 1, n, 2}]; f[1] = 2; Table[ f[n], {n, 22}]
  • PARI
    a(n) = if (n==1, 2, vecprod(vector(floor(n/2), k, prime(2*k)))+vecprod(vector(ceil(n/2), k, prime(2*k-1)))); \\ Michel Marcus, Dec 03 2024

Formula

Sum_{i=1..n} of the product_{j=2..n, 2} p_j (A066206) and the product_{k=1..n, 2} p_j (A066205).

A095137 Absolute difference between the product of the first floor(n/2) even-indexed primes and the product of the first floor(n/2) odd-indexed primes.

Original entry on oeis.org

2, 1, 7, 11, 89, 163, 1597, 3317, 37823, 107413, 1182887, 4232341, 49100059, 184657283, 2329965377, 10114830259, 138903895201, 622143222539, 9382665690241, 44778520855589, 686482057860331, 3598441529151191
Offset: 1

Views

Author

Robert G. Wilson v, May 28 2004

Keywords

Examples

			a(5) = 2*5*11 - 3*7 = 89, a(6) = 3*7*13 - 2*5*11 = 163;
a(7) = 2*5*11*17 - 3*7*13 = 1597, a(8) = 3*7*13*19 - 2*5*11*17 = 3317.

Crossrefs

Cf. A095134, A000040, A066206, A066205.

Programs

  • Mathematica
    PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1} ] & /@ FactorInteger[n]]; f[n_] := Abs[ Product[ Prime[i], {i, 2, n, 2}] + Product[ Prime[i], {i, 1, n, 2}]]; f[1] = 2; Table[ f[n], {n, 24}]
Join[{2},Table[Abs[Times@@Prime[Range[1,Floor[n/2],2]]-Times@@Prime[Range[ 2,Floor[ n/2 ],2]]],{n,4,45,2}]] (* Harvey P. Dale, Jan 11 2023 *)

Formula

The absolute difference of Product_{j=1..floor(n/2)} p_(2j) (A066206) and Product_{k=1..floor(n/2)} p_(2j-1) (A066205).

A356021 Positive numbers k such that, for any consecutive prime numbers p, q <= A006530(n), the p-adic and q-adic valuations of n are different.

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 12, 16, 18, 20, 21, 24, 27, 32, 40, 45, 48, 50, 54, 63, 64, 72, 75, 80, 81, 84, 90, 96, 100, 108, 110, 126, 128, 135, 144, 147, 160, 162, 168, 189, 192, 200, 220, 243, 250, 256, 270, 273, 288, 300, 320, 324, 336, 350, 360, 375, 378, 384
Offset: 1

Views

Author

Rémy Sigrist, Jul 23 2022

Keywords

Comments

Equivalently, these are fixed points of A356014.
This sequence is infinite as it contains A066205 and A066206.
If m is a term, then m^k is a term (for any k >= 0).

Crossrefs

Cf. A006530, A066205, A066206, A356014.

Programs

  • PARI
    is(n) = { my (v=-1); forprime (p=2, oo, if (n==1, return (1), v==v=valuation(n,p), return (0), n\=p^v)) }
Showing 1-5 of 5 results.

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