A214288 -id:A214288 - cp's OEIS frontend

A221740 a(n) = -4((n-1)(n+1)^(n+1)+1)/(((-1)^n-3)*n^3).

1, 7, 19, 293, 1493, 38127, 293479, 10593529, 109739369, 5135610071, 66987982331, 3856048810781, 60693710471869, 4149140360751583, 76519827268721103, 6058888636862818097, 128138108936443028945, 11533996620790579909159

Offset: 1

Alexander R. Povolotsky, Jan 23 2013

nonn

Per exhaustive program, written for bases from 2 to 10, the number of permutations pairs, which have the same ratio, equal to A221740(n)/A221741(n) = (n^2*(n+1)^n - (n+1)^n + 1) / (-n^2 + n*(n+1)^n + (n+1)^n - n - 1), is: {2,2,3,3,5,3,7,5,7,...} for n >= 1 where n = r-1 and r is the base radix. Judging by above sequence it appears that the number of such permutations pairs is related to phi, which is the Euler totient function - according to A039649, A039650, A214288 (see bullet 1 of the analysis in the answer section of the Mathematics StackExchange link). - Alexander R. Povolotsky, Jan 26 2013

G. C. Greubel, Table of n, a(n) for n = 1..385
Mathematics StackExchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.
NMBRTHRY, Number of specific permutations pairs relates to Euler Phi totient function?

Cf. A221741, A051846.

Table[-4*((n - 1)*(n + 1)^(n + 1) + 1)/(((-1)^n - 3)*n^3), {n,1,50}] (* G. C. Greubel, Feb 19 2017 *)

makelist(-4*((n-1)*(n+1)^(n+1)+1)/(((-1)^n-3)*n^3),n,1,20); /* Martin Ettl, Jan 25 2013 */

for(n=1,25, print1(-4*((n - 1)*(n + 1)^(n + 1) + 1)/(((-1)^n - 3)*n^3), ", ")) \\ G. C. Greubel, Feb 19 2017

a(n) = -4*A051846(n)/((-3 + (-1)^n)*n).
From Alexander R. Povolotsky, Oct 12 2022: (Start)
floor(a(n+1)/A221741(n+1)) = n.
Limit_{n->oo} (a(n)/A221741(n) - floor(a(n)/A221741(n))) = 0. (End)

A221741 a(n) = -4(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)n).

Original entry on oeis.org

1, 5, 9, 97, 373, 7625, 48913, 1513361, 13717421, 570623341, 6698798233, 350549891889, 5057809205989, 319164643134737, 5465701947765793, 403925909124187873, 8008631808527689309, 678470389458269406421, 15287592943577781017641

Offset: 1

Alexander R. Povolotsky, Jan 23 2013

Per exhaustive program, written for bases from 2 to 10, the number of permutations pairs, which have the same ratio, equal to A221740(n)/a(n) = (n^2 (n+1)^n-(n+1)^n+1) / (-n^2+n (n+1)^n+(n+1)^n-n-1), is: {2,2,3,3,5,3,7,5,7,...} for n>=1 where n=r-1 and r is the base radix. Judging by above sequence it appears that the number of such permutations pairs is related to phi, which is the Euler totient function - according to A039649, A039650, A214288 (see bullet 1 of the analysis in the answer section of the StackExchange link). Alexander R. Povolotsky, Jan 26 2013

G. C. Greubel, Table of n, a(n) for n = 1..385
NMBRTHRY, Number of specific permutations pairs relates to Euler Phi totient function ?
StackExchange, Permutations (with no duplicates) of decimal base digits 1,2,...,8,9,0.

Cf. A221740, A023811.

Table[-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), {n,1,50}] (* G. C. Greubel, Feb 19 2017 *)

makelist(-4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n), n, 1, 20); /* Martin Ettl, Jan 25 2013 */

for(n=1,25, print1(-4*(((n + 1)^(n + 1) - (n + 1))/((n + 1) - 1)^2 - 1)/((-3 + (-1)^n)*n), ", ")) \\ G. C. Greubel, Feb 19 2017

a(n) = -4*A023811(n+1)/((-3 + (-1)^n)*n).

A214287 Primes of the form phi(n)-1 sorted by increasing n, where phi is the Euler totient function.

Original entry on oeis.org

3, 5, 3, 5, 3, 3, 11, 5, 7, 7, 5, 17, 7, 11, 7, 19, 11, 17, 11, 7, 29, 19, 23, 11, 17, 23, 11, 41, 19, 23, 41, 19, 31, 23, 17, 23, 59, 29, 31, 47, 19, 31, 43, 23, 23, 71, 59, 23, 31, 53, 23, 41, 23, 71, 43, 59, 71, 31, 41, 59, 31, 101, 47, 47, 107, 71, 47, 71, 31, 109, 59, 79, 59, 83

Offset: 1

Vincenzo Librandi, Jul 13 2012

Primes in A109606.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000010, A078892 (associated n), A214288.

Select[Table[EulerPhi[n]-1,{n,1,1000}],PrimeQ]

cp's OEIS Frontend

A221740 a(n) = -4((n-1)(n+1)^(n+1)+1)/(((-1)^n-3)*n^3).

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A221741 a(n) = -4(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)n).

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Maxima

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A214287 Primes of the form phi(n)-1 sorted by increasing n, where phi is the Euler totient function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A221740 a(n) = -4*((n-1)*(n+1)^(n+1)+1)/(((-1)^n-3)*n^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

A221741 a(n) = -4*(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)*n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

A214287 Primes of the form phi(n)-1 sorted by increasing n, where phi is the Euler totient function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A221740 a(n) = -4((n-1)(n+1)^(n+1)+1)/(((-1)^n-3)*n^3).

A221741 a(n) = -4(((n+1)^(n+1)-(n+1))/((n+1)-1)^2-1)/((-3+(-1)^n)n).