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 1-8 of 8 results.

A127600 Integer part of cube root of product of first n primes.

Original entry on oeis.org

1, 1, 3, 5, 13, 31, 79, 213, 606, 1863, 5853, 19505, 67257, 235631, 850352, 3194167, 12434883, 48949883, 198812307, 823245530, 3440622312, 14763161313, 64397952985, 287520444756, 1321070444052, 6152237618431, 28838910052201
Offset: 1

Views

Author

Artur Jasinski, Jan 19 2007

Keywords

Links

Crossrefs

Cf. A002110, A060797.

Programs

  • Maple
    Res:= NULL:
p:= 1: r:= 1:
for n from 1 to 50 do
  p:=nextprime(p);
  r:= r*p;
  Res:= Res, floor(r^(1/3));
od:
Res; # Robert Israel, Nov 10 2017
  • Mathematica
    a = {}; Do[b = 1; Do[b = b Prime[x], {x, 1, n}]; AppendTo[a, Floor[b^(1/3)]], {n, 1, 100}]; a (* Artur Jasinski *)
Floor[Surd[#,3]]&/@Rest[FoldList[Times,1,Prime[Range[30]]]] (* Harvey P. Dale, Jun 23 2014 *)
  • PARI
    a(n) = sqrtnint(prod(k=1, n, prime(k)), 3); \\ Michel Marcus, Nov 10 2017

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

A215658 Primes p such that the smallest positive integer k for which p# + k is square satisfies p# + k = k^2, where p# = 2*3*5*7*11*...*p is a primorial.

Original entry on oeis.org

2, 3, 5, 7, 17
Offset: 1

Views

Author

Jonathan Sondow, Sep 02 2012

Keywords

Comments

The corresponding values of k are 2, 3, 6, 15, 715 = A215659.
The equation p# + k = k^2 has an integer solution k if and only if 1 + 4*p# is a square.
Conjecture: Not the same sequence as A192579, which is finite.
When p is in this sequence, p# = k(k-1) is in A161620, the intersection of A002110 and A002378. - Jeppe Stig Nielsen, Mar 27 2018

Examples

			The smallest square > 17# = 510510 is 715^2 = 17# + 715, so 17 is a member.

Links

Crossrefs

Cf. A002110, A060797, A145781, A216144, A215659, A161620.

Programs

  • Mathematica
    t = {}; pm = 1; Do[pm = pm*p; s = Floor[Sqrt[pm]]; If[pm == s*(s+1), AppendTo[t, p]], {p, Prime[Range[100]]}]; t (* T. D. Noe, Sep 05 2012 *)
  • PARI
    for (n=1, 10, if (ceil(sqrt(prod(i=1, n, prime(i))))^2 - prod(i=1, n, prime(i)) - ceil(sqrt(prod(i=1, n, prime(i)))) == 0, print(prime(n)));); \\ Michel Marcus, Sep 05 2012
  • Python
    from sympy import primorial, integer_nthroot, prime
A215658_list = [prime(i) for i in range(1,10**2) if integer_nthroot(4*primorial(i)+1,2)[1]] # Chai Wah Wu, Apr 01 2021

Formula

A145781(n) = A216144(n) if and only if prime(n) is a member.
a(n)# = A215659(n)*(A215659(n)-1).

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

A216144 Square root of smallest square greater than the product of first n primes.

Original entry on oeis.org

2, 3, 6, 15, 49, 174, 715, 3115, 14937, 80435, 447840, 2724104, 17442772, 114379900, 784149082, 5708691486, 43849291331, 342473913400, 2803269796342, 23620771158595, 201815957246322, 1793779464521956, 16342108667160302, 154171144824008980, 1518409682511777987
Offset: 1

Views

Author

Michel Marcus, Sep 02 2012

Keywords

Comments

Known values such that a(n)=A145781(n) are a(n)=2,3,6,15 and 715, i.e. for primes p=2,3,5,7 and 17.
(The relation a(n)=A145781(n) means that a(n)(a(n)-1) is a primorial number.) - M. F. Hasler, Sep 02 2012, - corrected by Jonathan Sondow, Sep 02 2012

Examples

			a(2) = sqrt(2*3 + A145781(2))= sqrt(2*3 + 3) = sqrt(9) = 3.

Links

Crossrefs

Cf. A002110, A060797, A145781, A215658, A215659.

Programs

  • PARI
    j=[];for (n=1, 30, p = prod(i=1, n, prime(i)); j=concat(j, floor(sqrt((ceil(sqrt(p))^2))));); j
  • PARI
    A216144(n)=sqrtint(prod(k=1,n,prime(k)))+1 \\ - M. F. Hasler, Sep 02 2012

Formula

a(n)=sqrt(A002110(n) + A145781(n)).
a(n)=A060797(n)+1. - M. F. Hasler, Sep 02 2012

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 1-8 of 8 results.

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