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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A128281 a(n) is the least product of n distinct odd primes m=p_1*p_2*...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

Original entry on oeis.org

3, 21, 105, 1365, 884037
Offset: 1

Views

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

Comments

From Iain Fox, Aug 26 2020: (Start)
a(6) > 10^9 if it exists.
All terms are members of A076274 since the definition requires that (1+m)/2 be prime.
The number of prime factors of m congruent to 3 (mod 4) must be even except for n=1.
(End)
a(6) > 2*10^11 if it exists. - David A. Corneth, Aug 27 2020
a(n) >= A070826(n+1) by definition of the sequence. - Iain Fox, Aug 28 2020

Examples

			105=3*5*7, (3*5*7+1)/2=53, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes and 105 is the least such number which is the product of 3 primes, so a(3)=3.

Crossrefs

Subsequence of A076274.
Lower bound: A070826.
Cf. A002145, A128276, A128282, A128283, A128284, A128285, A128286, A168352.

Programs

  • PARI
    a(n)=if(n==1, return(3)); my(p=prod(k=1, n, prime(k+1))); forstep(m=p+if(p%4-1, 2), +oo, 4, if(bigomega(m)==n && omega(m)==n, fordiv(m, d, if(!isprime((d+m/d)/2), next(2))); return(m))) \\ Iain Fox, Aug 27 2020

Extensions

Definition corrected by Iain Fox, Aug 25 2020

A344646 Array read by antidiagonals T(n,k) = ((n+k+1)^2 - (n+k+1) mod 2)/4 + min(n,k) for n and k >= 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 5, 5, 4, 6, 7, 8, 7, 6, 9, 10, 11, 11, 10, 9, 12, 13, 14, 15, 14, 13, 12, 16, 17, 18, 19, 19, 18, 17, 16, 20, 21, 22, 23, 24, 23, 22, 21, 20, 25, 26, 27, 28, 29, 29, 28, 27, 26, 25, 30, 31, 32, 33, 34, 35, 34, 33, 32, 31, 30, 36, 37, 38, 39, 40, 41, 41, 40, 39, 38, 37, 36
Offset: 0

Views

Author

Michel Marcus, May 25 2021

Keywords

Examples

			Array begins:
  0  1  2  4  6 ...
  1  3  5  7 10 ...
  2  5  8 11 14 ...
  4  7 11 15 19 ...
  6 10 14 19 24 ...
  ...

Links

Crossrefs

Cf. A128282 (another pairing function).

Programs

  • Mathematica
    T[n_, k_] := ((n + k + 1)^2 - Mod[n + k + 1, 2])/4 + Min[n, k]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 25 2021 *)
  • PARI
    T(n,k) = ((n+k+1)^2 - (n+k+1)%2)/4 + min(n,k);
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