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 1-4 of 4 results.

A125716 Numbers n such that A120292(n) = 1.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 14, 15, 16, 17, 23, 31, 34, 35, 39, 41, 43, 50, 51, 56, 66, 67, 70, 71, 75, 77, 92, 96, 97, 98, 101, 107, 112, 115, 117, 123, 131, 132, 153, 155, 156, 160, 163, 165, 166, 170, 172, 182, 185, 196, 198, 200, 203, 204, 207, 212, 218, 231, 243, 246, 249
Offset: 1

Views

Author

Alexander Adamchuk, Feb 02 2007

Keywords

Comments

A120292(n) = {2, 1, 1, 5, 1, 23, 1, 1, 1, 23, 17, 13, 5, 1, 1, 1, 1, 37, ...} 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.

Crossrefs

Cf. A120292.

Programs

  • Mathematica
    Do[f=Abs[Numerator[Det[DiagonalMatrix[Table[Prime[i]/(Prime[i]+1)-1,{i,1,n}]]+1]]];If[ f == 1, Print[n]],{n,1,256}]

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

A141781 Composite terms of A120292: a(n) = A120292(A141779(n)).

Original entry on oeis.org

3599, 118477, 210589, 971573, 1164103, 1901959, 2446681, 3230069, 2603767, 9114493, 9772927, 1497767, 6558967, 4323827, 32405449, 33992009, 11453957, 34417541, 35938783, 36569077, 40473001, 42110911, 47901839, 55183769
Offset: 1

Views

Author

Alexander Adamchuk, Jul 04 2008

Keywords

Comments

Corresponding indices are listed in A141779(n) = {58, 282, 367, 743, 808, 1015, 1141, 1299, 1962, 2109, 2179, 2397, 2501, ...}.
Note that all listed terms are semiprime, for example: a(1) = 3599 = 59*61, a(2) = 118477 = 257*461, a(3) = 210589 = 251*839, a(4) = 971573 = 643*1511.
Conjecture: All terms 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 = Numbers n such that A120292(n) = 1. Cf. A141780 = Numbers n such that A120292(n) is prime. Cf. A141779 = Numbers n such that A120292(n)>1 and is not prime.

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(t", "))) \\ Charles R Greathouse IV, Feb 07 2013

Formula

a(n) = A120292(A141779(n)).
Showing 1-4 of 4 results.

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