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.

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A126105 Prime(n)^2*prime(n+1)...*prime(a(n)) is the least product of consecutive primes which is abundant. Note that only the first term is squared.

Original entry on oeis.org

2, 5, 10, 20, 34, 50, 72, 97, 129, 165, 203, 248, 295, 346, 405, 469, 537, 607, 685, 766, 853, 949, 1049, 1155, 1264, 1376, 1494, 1620, 1754, 1897, 2048, 2193, 2346, 2503, 2669, 2836, 3012, 3193, 3378, 3572, 3770, 3973, 4186, 4400, 4624, 4855, 5098, 5339, 5578
Offset: 1

Views

Author

Walter Kehowski, Mar 04 2007

Keywords

Examples

			a(3)=10 since x=5^2*7*11*13*17*19*23*29=5391411025 is abundant with sigma(x)=10799308800 and sigma(x)-2*x=16486750.

Links

Crossrefs

Cf. A005101, A007684 (a very similar sequence), A007708, A007741.

Programs

  • Mathematica
    a[n_] := Module[{p = Prime[n]}, c = 1; pr = 1 + 1/p + 1/p^2; While[pr < 2, p = NextPrime[p]; pr *= (1 + 1/p); c++]; c + n - 1]; Array[a, 50] (* Amiram Eldar, Aug 14 2019 *)

Extensions

More terms from Stefan Steinerberger, May 11 2007
a(21) corrected and more terms added by Amiram Eldar, Aug 14 2019

A286042 Largest prime factor of A285993(n), the largest odd abundant number (A005231) equal to the product of n consecutive primes.

Original entry on oeis.org

13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353
Offset: 5

Views

Author

M. F. Hasler, May 01 2017

Keywords

Comments

The smallest term is a(5), there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.
The corresponding abundant numbers are A285993(n) = prime(k-n+1)*...*prime(k), with prime(k) = a(n).

Examples

			For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*a(n) with a(n) = prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*a(n) with a(n) = prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*a(n) with a(n) = prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*a(n) with a(n) = prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*a(n) with a(n) = prime(n+5).

Links

Crossrefs

Cf. A285993, A005231, A006038, A007707, A007708, A007741.

Programs

  • PARI
    a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));o[#o]} \\ Intentionally throws an error when n < 5.

Formula

a(n) = A006530(A285993(n)) >= A151800(a(n-1)) = nextprime(a(n-1)), with strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = nextprime(nextprime(a(n-1))). This is the case if A285993(n) is in A007741.

Extensions

a(66) corrected by Amiram Eldar, Sep 24 2019
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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