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 1-3 of 3 results.

A007708 Prime(n)*...*a(n) is the least product of consecutive primes which is abundant.

Original entry on oeis.org

5, 13, 31, 73, 149, 233, 367, 521, 733, 991, 1249, 1579, 1949, 2341, 2791, 3343, 3881, 4481, 5147, 5849, 6619, 7499, 8387, 9341, 10321, 11411, 12517, 13709, 15013, 16363, 17881, 19381, 20873, 22369, 24007, 25763, 27611, 29399, 31357
Offset: 1

Views

Author

Walter Nissen

Keywords

Comments

Differs from A007686 only for n=1. - Michel Marcus, Mar 10 2013

Links

Crossrefs

Cf. A005101, A007686, A007707, A007741.

Programs

  • Mathematica
    a[n_] := Module[{p = Prime[n]}, r = 1 + 1/p; While[r <= 2,  p = NextPrime[p]; r *= 1 + 1/p]; p]; Array[a, 39] (* Amiram Eldar, Jun 29 2019 *)
  • PARI
    a(n) = {p = prime(n); sig = p+1; prd = p; while (sig <= 2*prd, p = nextprime(p+1); sig *= p+1; prd *= p;); return (p);} \\ Michel Marcus, Mar 10 2013

Extensions

More terms from Don Reble, Nov 10 2005

A007684 Prime(n)*...*prime(a(n)) is the least product of consecutive primes that is non-deficient.

Original entry on oeis.org

2, 6, 11, 21, 35, 51, 73, 98, 130, 167, 204, 249, 296, 347, 406, 471, 538, 608, 686, 768, 855, 950, 1050, 1156, 1266, 1377, 1495, 1621, 1755, 1898, 2049, 2194, 2347, 2504, 2670, 2837, 3013, 3194, 3380, 3573, 3771, 3974, 4187, 4401, 4625, 4856
Offset: 1

Views

Author

Walter Nissen

Keywords

Comments

Subscript of the smallest primorial number that when divided by the (n-1)-th primorial number gives an abundant number.
Products of consecutive primes started with prime(a) up to prime(b) result in abundant squarefree numbers if b is large enough and provides perhaps the least squarefree solutions to Rivera Puzzle 329 and its generalization.
Adding a new prime p to the product increases the relative abundancy sigma(N)/N by a factor 1+1/p. This leads to a simple and fast algorithm, see the PARI code. - M. F. Hasler, Jul 30 2016

Examples

			n=1: a(1)=2 means that primorial(2)=6 divided by primorial(1-1)=1 gives the quotient 6/1=6 which is just non-deficient (being a perfect number);
n=3: a(n)=11 because prime(3)=5, primorial(11) = 2*3*5*...*29*31, primorial(3-1) = 2*3 = 6.
p#(11)/p#(2) = 3*5*7*11*13*17*19*23*29*31 = 33426748355 = q and sigma(q)/q = 2.00097 > 2 so q is an abundant number. Also p#(10)/p#(3-1) is not yet abundant.

Links

Crossrefs

Cf. A005100, A007686, A007702, A007707 (an essentially identical sequence).
Cf. A002110, A064001, A112640, A005101, A005231.

Programs

  • Mathematica
    spr[x_, y_] :=Apply[Times, Table[(Prime[w]+1)/(Prime[w]), {w, x, y}]];
Table[Min[Flatten[Position[Table[Floor[spr[n, w]], {w, 1, 1000}], 2]]], {n, 1, 20}] (* Labos Elemer, Sep 19 2005 *)
  • PARI
    a=1;i=0;for(n=1,99,while(2>a*=1+1/prime(i++),);print1(i",");a/=1+1/prime(n)) \\ M. F. Hasler, Jul 30 2016

Formula

a(n) is the minimal x such that floor(sigma(p#(x)/p#(n-1)) / (p#(x)/p#(n-1))) = 2, where p#(w) is the w-th primorial number, the product of first w prime numbers. For a>b, the p#(a)/p#(b)=A002110(a)/A002110(b) quotients are prime(b+1)*prime(b+2)*...*prime(a).

Extensions

Additional comments from Labos Elemer, Sep 19 2005
More terms from Don Reble, Nov 10 2005
Edited by N. J. A. Sloane, Dec 22 2006

A007702 a(n) = prime(n)*...*prime(m), the least product of consecutive primes which is non-deficient.

Original entry on oeis.org

6, 15015, 33426748355, 1357656019974967471687377449, 7105630242567996762185122555313528897845637444413640621
Offset: 1

Views

Author

Walter Nissen

Keywords

Links

Crossrefs

Cf. A005100, A007741, A007684, A007686.
Cf. A107705, A108227, A285993.

Programs

  • Mathematica
    a[n_] := Module[{p = Prime[n]}, r = 1; prod = 1; While[r < 2, r *= 1 + 1/p; prod *= p; p = NextPrime[p]]; prod]; Array[a, 5] (* Amiram Eldar, Jun 29 2019 *)
  • PARI
    A007702(n, p=prime(n), s=1+1/p, P=p)={until(2<=s*=1+1/p,P*=p=nextprime(p+1));P} \\ M. F. Hasler, Jun 15 2017

Formula

a(n) = Product_{k = n..A007684(n)} prime(k) = Product_{0 <= i < A107705(n)} prime(n+i). - M. F. Hasler, Jun 15 2017

Extensions

More terms from Don Reble, Nov 10 2005
Showing 1-3 of 3 results.

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

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