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 1-6 of 6 results.

A060797 Integer part of square root of n-th primorial, A002110(n).

Original entry on oeis.org

1, 1, 2, 5, 14, 48, 173, 714, 3114, 14936, 80434, 447839, 2724103, 17442771, 114379899, 784149081, 5708691485, 43849291330, 342473913399, 2803269796341, 23620771158594, 201815957246321, 1793779464521955, 16342108667160301, 154171144824008979
Offset: 0

Views

Author

Labos Elemer, Apr 27 2001

Keywords

Comments

Integer part of square root of product of n first primes.

Examples

			n=8, q(8) = 2*3*5*7*11*13*17*19 = 9699690, a(8)=3114. This is between the 128th and 129th divisors of the 8th primorial: 3094 < A000196(9699690)=3114 < 3135.
(In general, x=A002110(n) always has 2^n divisors, and A000196(x) always lies between the k-th and (k+1)-th divisors of x, where k=ceiling(tau(x)/2) and tau(x) is the number of divisors of x.) - _M. F. Hasler_, Sep 02 2012

Links

Crossrefs

Cf. A060755, A000196, A033677.
Cf. A002110, A060797, A127600.

Programs

  • Mathematica
    a = {}; Do[b = 1; Do[b = b Prime[x], {x, 1, n}]; AppendTo[a, Floor[b^(1/2)]], {n, 1, 100}]; a (* Artur Jasinski *)
Join[{1},Floor[Sqrt[#]]&/@FoldList[Times,Prime[Range[30]]]] (* Harvey P. Dale, Nov 22 2023 *)
  • PARI
    A060797(n)=sqrtint(prod(k=1, n, prime(k))) \\ M. F. Hasler, Sep 02 2012

Formula

a(n) = A000196(A002110(n)) = floor(sqrt(A002110(n))).

Extensions

a(23) correction by Hans Havermann, Dec 02 2010
Extended to a(0)=1=sqrt(A002110(0)) by M. F. Hasler, Sep 02 2012

A127601 Integer part of 4th root of product of first n primes.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 13, 26, 55, 122, 283, 669, 1650, 4176, 10694, 28002, 75555, 209402, 585212, 1674296, 4860120, 14206194, 42353033, 127836257, 392646335, 1232237672, 3906383039, 12444691408, 40024883480, 129326254765
Offset: 0

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Comments

For n>=6, a(n) >= prime(n+1).
a(n) gives the number of pairs within A002110(n) of the form {x, x^4} where x is nonzero positive integer. - Soumyadeep Dhar, May 16 2021

Links

Crossrefs

Cf. A002110, A060797, A127600, A127602, A127603, A127604.

Programs

  • Mathematica
    a = {}; Do[b = 1; Do[b = b Prime[x], {x, 1, n}]; AppendTo[a, Floor[b^(1/4)]], {n, 1, 50}]; a
  • PARI
    a(n)={sqrtnint(vecprod(primes(n)), 4)} \\ Andrew Howroyd, Apr 27 2021

Extensions

a(0)=1 prepended by Soumyadeep Dhar, Apr 28 2021

A127602 Integer part of 5th root of product of first n primes.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 13, 24, 46, 91, 182, 375, 788, 1672, 3612, 7991, 18062, 41100, 95294, 223520, 527208, 1263303, 3057195, 7502417, 18730768, 47143287, 119120718, 303294169, 775085050, 1995101748, 5256852524, 13937345067, 37284143091
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Crossrefs

Cf. A002110, A060797, A127600, A127601, A127603, A127604.

Programs

  • Mathematica
    a = {}; Do[b = 1; Do[b = b Prime[x], {x, 1, n}]; AppendTo[a, Floor[b^(1/5)]], {n, 1, 50}]; a
Floor[Surd[#,5]]&/@FoldList[Times,Prime[Range[40]]] (* Harvey P. Dale, Nov 16 2017 *)

A127603 Integer part of 6th root of product of first n primes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 14, 24, 43, 76, 139, 259, 485, 922, 1787, 3526, 6996, 14100, 28692, 58656, 121503, 253767, 536209, 1149378, 2480370, 5370187, 11700921, 25573556, 56230361, 126067989, 284107943, 645064989, 1468157354, 3380417306
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Crossrefs

Cf. A002110, A060797, A127600, A127601, A127602, A127604.

Programs

  • Mathematica
    a = {}; Do[b = 1; Do[b = b Prime[x], {x, 1, n}]; AppendTo[a, Floor[b^(1/6)]], {n, 1, 50}]; a

A127604 Integer part of 7th root of product of first n primes.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 15, 25, 41, 68, 117, 200, 347, 613, 1097, 1975, 3601, 6621, 12221, 22814, 42891, 81443, 156560, 302701, 586897, 1144127, 2236326, 4393717, 8777595, 17613387, 35570395, 71983616, 147125801, 301280666, 620399178, 1284393250
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Crossrefs

Cf. A002110, A060797, A127600, A127601, A127602, A127603.

Programs

  • Mathematica
    a = {}; Do[b = 1; Do[b = b Prime[x], {x, 1, n}]; AppendTo[a, Floor[b^(1/7)]], {n, 1, 50}]; a

A056127 Minimum m where product_{k=1 to m}[p_k] > (p_{m+1})^n, where p_k is k-th prime.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88
Offset: 0

Views

Author

Leroy Quet, Aug 30 2000

Keywords

Examples

			a(2) = 4, since 2*3*5 < 7^2, but 2*3*5*7 > 11^2. (The product of the first 4 primes is greater than the 5th prime squared.)

Links

Crossrefs

Cf. A002110, A060797, A127600, A127601, A127602, A127603, A127604.

Programs

  • Mathematica
    a = {}; Do[x = 1; While[Prime[x + 1] >= (Product[Prime[x], {x, 1, x}])^(1/n), x++ ]; AppendTo[a, x], {n, 1, 100}]; a (* Artur Jasinski, May 11 2007 *)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.