A141779 -id:A141779 - cp's OEIS frontend

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

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

Alexander Adamchuk, Jul 04 2008

nonn

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.

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.

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}]

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

a(n) = A120292(A141779(n)).

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

Alexander Adamchuk, Jul 08 2006, Jul 04 2008

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.

Alexander Adamchuk, Jul 04 2008, Table of n, a(n) for n = 1..282

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

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}]

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

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

Alexander Adamchuk, Jul 04 2008

nonn

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).

Select[Range[200],PrimeQ[Numerator[Abs[(1 - Sum[Prime[k] + 1,{k, 1, #}])/Product[Prime[k] + 1, {k, 1, #}] ]]]&]

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

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

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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.

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

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