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.

A120292 Absolute value of numerator of determinant of n X n matrix with elements M[i,j] = prime(i)/(1+prime(i)) if i=j and 1 otherwise.

Original entry on oeis.org

2, 1, 1, 5, 1, 23, 1, 1, 1, 23, 17, 13, 5, 1, 1, 1, 1, 37, 293, 47, 61, 29, 1, 29, 271, 593, 43, 233, 29, 811, 1, 941, 101, 1, 1, 1231, 131, 29, 1, 521, 1, 109, 1, 149, 509, 89, 59, 107, 617, 1, 1, 47, 173, 3067, 47, 1, 3463, 3599, 89, 431, 4021, 521, 2161, 2239, 103, 1, 1
Offset: 1

Views

Author

Alexander Adamchuk, Jul 08 2006, Jul 04 2008

Keywords

Comments

Some a(n) are equal to 1 (n = 2, 3, 5, 7, 8, 9, 14, 15, 16, 17, 23, 31, 34, 35, 39, 41, 43, 50, 51, 56, ...).
a(58) = 3599 = 59*61 is not prime. - T. D. Noe, Nov 15 2006
Most terms are prime or 1.
Numbers n such that a(n)>1 and is not prime are listed in A141779(n) = {58, 282, 367, 743, 808, 1015, 1141, 1299, 1962, 2109, 2179, 2397, 2501, ...}.
Composite terms are listed in A141781 = {3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, ...}.
Note that all listed terms of A141781 are semiprime, for example: 3599 = 59*61, 118477 = 257*461, 210589 = 251*839, 971573 = 643*1511.
Conjecture: All composite terms are semiprime.

Links

Crossrefs

Cf. A024528, A118679, A125716, A141780, A141779, A141781.

Programs

  • Mathematica
    Abs[Numerator[Table[Det[DiagonalMatrix[Table[Prime[i]/(Prime[i]+1)-1,{i,1,n}]]+1],{n,1,60}]]]
Table[Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]],{n,1,282}]
  • PARI
    a(n)=abs(numerator(matdet(matrix(n,n,i,j,if(i==j,prime(i)/(1+prime(i)),1))))) \\ Charles R Greathouse IV, Feb 07 2013

A141779 Numbers k such that A120292(k) is composite.

Original entry on oeis.org

58, 282, 367, 743, 808, 1015, 1141, 1299, 1962, 2109, 2179, 2397, 2501, 3704, 3825, 3912, 3932, 3935, 4016, 4049, 4247, 4327, 4598, 4915, 4977, 5210, 5266, 5396, 5420, 5512, 5562, 5773, 5981, 6031, 6249, 6616, 6984, 7117, 7121, 7304, 7338, 7424, 7653
Offset: 1

Views

Author

Alexander Adamchuk, Jul 04 2008

Keywords

Comments

Composite terms of A120292 are listed in A141781 = {3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, ...}.
Note that all listed terms correspond to semiprimes, for example: 3599 = 59*61, 118477 = 257*461, 210589 = 251*839, 971573 = 643*1511.
Conjecture: All composite terms of A120292 are semiprime.

Crossrefs

Cf. A120292 = Absolute value of numerator of determinant of n X n matrix with elements M[i, j] = prime(i)/(1+prime(I)) if i=j and 1 otherwise.
Cf. A125716 (k such that A120292(k) = 1).
Cf. A141780 (k such that A120292(k) is prime).
Cf. A141781 (terms of A120292 that are greater than 1 and are not prime; or A120292(A141779)).

Programs

  • Mathematica
    Do[f=Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, n}])/Product[Prime[k] + 1, {k, 1, n}] ]];If[ !PrimeQ[f]&&!(f==1),Print[{n,f,FactorInteger[f]}]],{n,1,8212}]
  • PARI
    for(n=1,100,t=abs(numerator(matdet(matrix(n,n,i,j,if(i==j, prime(i)/(1+prime(i)),1))))); if(t>3 && !isprime(t), print1(n", "))) \\ Charles R Greathouse IV, Feb 07 2013

Formula

A141781(n) = A120292( a(n) ).

A141780 Numbers n such that A120292(n) is prime.

Original entry on oeis.org

1, 4, 6, 10, 11, 12, 13, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 57, 59, 60, 61, 62, 63, 64, 65, 68, 69, 72, 73, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 99, 100, 102
Offset: 1

Views

Author

Alexander Adamchuk, Jul 04 2008

Keywords

Crossrefs

Cf. A120292 = absolute value of the numerator of the determinant of n X n matrix M with M[i, j] = prime[i]/(1 + prime[i]) if i = j, and 1 otherwise.
Cf. A125716 = numbers n such that A120292(n) = 1.
Cf. A141779 = numbers n such that A120292(n) > 1 and is not prime.
Cf. A141781 = terms of A120292 that are greater than 1 and are not prime; or A120292(A141779).

Programs

  • Mathematica
    Select[Range[200],PrimeQ[Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, #}])/Product[Prime[k] + 1, {k, 1, #}] ]]]&]
  • PARI
    isok(n) = isprime(abs(numerator(matdet(matrix(n, n, i, j, if(i==j, prime(i)/(1+prime(i)), 1)))))); \\ Michel Marcus, May 10 2020
Showing 1-3 of 3 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.