A064136 -id:A064136 - cp's OEIS frontend

A064135 Number of divisors of 10^n + 1 that are relatively prime to 10^m + 1 for all 0 < m < n.

2, 2, 2, 4, 4, 2, 2, 2, 4, 4, 4, 8, 2, 4, 8, 8, 32, 8, 2, 2, 4, 8, 8, 16, 2, 8, 4, 4, 4, 4, 8, 2, 16, 4, 8, 4, 8, 8, 4, 16, 4, 4, 4, 8, 4, 8, 8, 8, 16, 4, 16, 4, 4, 2, 8, 16, 8, 4, 16, 8, 2, 4, 4, 4, 8, 4, 8, 2, 4, 8, 32, 4, 4, 8, 64, 2, 16, 64, 8, 8, 16, 16, 4

Offset: 0

Robert G. Wilson v, Sep 10 2001

nonn

			1001 = 7 * 11 * 13 and has 8 divisors, but only {1, 7, 13, 91} are relatively prime to 11 and 101, so a(3) = 4. - _Bernard Schott_, May 27 2019

Sean A. Irvine, Table of n, a(n) for n = 0..312
Sam Wagstaff, Cunningham Project, Factorizations of 10^n+1, n<=330

Cf. A064131, A064132, A064133, A064134, A064136, A064137.

a = {1}; Do[ d = Divisors[ 10^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 10^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 46} ]

a(n) = if (n==0, 2, sumdiv(10^n+1, d, vecsum(vector(n-1, k, gcd(d, 10^k+1) == 1)) == n-1)); \\ Michel Marcus, Jun 24 2018

More terms from Michel Marcus, Jul 02 2018
a(73)-a(82) from Robert Price, May 26 2019
a(73) corrected by Sean A. Irvine, May 26 2019

A064131 Number of divisors of 3^n + 1 that are relatively prime to 3^m + 1 for all 0 < m < n.

Original entry on oeis.org

2, 3, 2, 2, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 4, 4, 2, 8, 2, 4, 2, 4, 4, 2, 8, 4, 4, 4, 4, 8, 2, 4, 2, 4, 4, 2, 2, 8, 4, 16, 4, 4, 4, 2, 8, 4, 4, 8, 4, 4, 8, 8, 4, 4, 2, 4, 8, 8, 4, 4, 8, 4, 8, 16, 2, 2, 2, 4, 8, 32, 8, 32, 4, 4, 8, 16, 16, 2, 4, 8, 8, 32, 8, 16, 32, 8, 32, 32, 8, 8, 4

Offset: 0

Robert G. Wilson v, Sep 10 2001

nonn

Sam Wagstaff, Cunningham Project, Factorizations of 3^n-1, n odd, n<540

Cf. A064132, A064133, A064134, A064135, A064136, A064137.

a[0]=2; a[n_] := Length@ Select[Divisors[3^n+1], GCD[Times @@ (3^Range[1, n-1] + 1), #] == 1 &]; Array[a, 91, 0] (* Giovanni Resta, Jul 02 2018 *)

a(n) = if (n==0, 2, sumdiv(3^n+1, d, vecsum(vector(n-1, k, gcd(d, 3^k+1) == 1)) == n-1)); \\ Michel Marcus, Jun 24 2018

a(1) corrected and extended by Michel Marcus, Jul 02 2018

A064132 Number of divisors of 5^n + 1 that are relatively prime to 5^m + 1 for all 0 < m < n.

Original entry on oeis.org

2, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 4, 2, 4, 4, 8, 2, 4, 4, 8, 8, 4, 16, 4, 8, 8, 4, 4, 4, 16, 4, 16, 2, 2, 2, 8, 4, 8, 8, 16, 8, 8, 2, 2, 16, 4, 2, 16, 2, 16, 4, 16, 8, 8, 4, 2, 32, 8, 4, 8, 4, 8, 8, 16, 8, 4, 16, 16, 8, 8, 16, 8, 8, 16, 8, 8, 16, 8, 8, 4, 4, 8, 16, 8, 8, 32, 16, 2, 16

Offset: 0

Robert G. Wilson v, Sep 10 2001

nonn

From Robert Israel, Jun 26 2018: (Start)
a(n) = Product_{j: A211241(j)=2*n} (1 + e_j) where e_j is the Prime(j)-adic valuation of 5^n+1.  In most cases, each e_j = 1 and a(n) is a power of 2, but a(20243) is divisible by 3 since the multiplicative order of 5 mod 40487 is 40486 and 5^20243+1 is divisible by 40487^2.
(End)

Sam Wagstaff, Cunningham Project, Factorizations of 5^n-1, n odd, n<=375

Cf. A064131, A064133, A064134, A064135, A064136, A064137.

Cf. A211241.

f:= n -> nops(select(t -> andmap(m -> igcd(t,5^m+1)=1,[$1..n-1]), numtheory:-divisors(5^n+1))):
map(f, [$0..100]); # Robert Israel, Jun 25 2018

a[n_] := Count[Divisors[5^n+1], d_ /; AllTrue[5^Range[n-1]+1, CoprimeQ[d, #]&]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 100}] (* Jean-François Alcover, Jun 27 2018 *)

a(n) = if (n==0, 2, sumdiv(5^n+1, d, vecsum(vector(n-1, k, gcd(d, 5^k+1) == 1)) == n-1)); \\ Michel Marcus, Jun 24 2018

More terms from Robert Israel, Jun 25 2018

Incorrect Mma program deleted by Editors, Jul 02 2018

A064133 Number of divisors of 6^n + 1 that are relatively prime to 6^m + 1 for all 0 < m < n.

Original entry on oeis.org

2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 4, 2, 2, 8, 4, 2, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 8, 4, 16, 8, 16, 2, 8, 8, 4, 32, 8, 8, 4, 16, 8, 8, 4, 2, 4, 2, 16, 2, 16, 4, 8, 16, 8, 16, 16, 8, 16, 8, 4, 2, 4, 4, 2, 8, 8, 4, 32, 16, 16, 4, 4, 8, 2, 32, 8, 16, 16, 2, 16, 32, 8

Offset: 0

Robert G. Wilson v, Sep 10 2001

nonn

Sam Wagstaff, Cunningham Project, Factorizations of 6^n-1, n odd, n<330

Cf. A064131, A064132, A064134, A064135, A064136, A064137.

a = {1}; Do[ d = Divisors[ 6^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 6^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 66} ]

a(n) = if (n==0, 2, sumdiv(6^n+1, d, vecsum(vector(n-1, k, gcd(d, 6^k+1) == 1)) == n-1)); \\ Michel Marcus, Jun 24 2018

a(67)-a(80) from Giovanni Resta, Jun 26 2018

A064134 Number of divisors of 7^n + 1 that are relatively prime to 7^m + 1 for all 0 < m < n.

Original entry on oeis.org

2, 4, 3, 2, 2, 4, 4, 4, 4, 2, 4, 4, 8, 4, 2, 2, 4, 2, 2, 4, 4, 4, 8, 2, 2, 2, 4, 2, 16, 2, 4, 8, 8, 4, 8, 16, 8, 4, 2, 8, 8, 16, 2, 128, 8, 16, 16, 2, 8, 128, 16, 8, 8, 16, 8, 32, 32, 8, 16, 16, 4, 2, 8, 32, 8, 4, 16, 8, 8, 8, 8, 4, 32, 8, 2, 8, 32, 32, 4, 16, 8, 16

Offset: 0

Robert G. Wilson v, Sep 10 2001

nonn

Sam Wagstaff, Cunningham Project, Factorizations of 7^n+1, n<=301

Cf. A064131, A064132, A064133, A064135, A064136, A064137.

a = {1}; Do[ d = Divisors[ 7^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 7^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 53} ]

a(n) = if (n==0, 2, sumdiv(7^n+1, d, vecsum(vector(n-1, k, gcd(d, 7^k+1) == 1)) == n-1)); \\ Michel Marcus, Jun 24 2018

a(1) corrected and more terms from Michel Marcus, Jul 02 2018

A064137 Number of divisors of 12^n + 1 that are relatively prime to 12^m + 1 for all 0 < m < n.

Original entry on oeis.org

2, 2, 4, 4, 4, 2, 2, 4, 8, 4, 2, 2, 4, 8, 4, 8, 4, 4, 4, 8, 16, 8, 16, 8, 8, 16, 8, 32, 2, 4, 2, 4, 4, 32, 4, 2, 8, 8, 8, 16, 2, 16, 2, 16, 4, 8, 2, 16, 8, 4, 32, 16, 8, 16, 32, 16, 64, 16, 32, 32, 4, 16, 16, 16, 32, 8, 16, 8, 8, 64, 16, 4, 16, 16, 64, 64, 8, 4, 8

Offset: 0

Robert G. Wilson v, Sep 10 2001

nonn

Sam Wagstaff, Cunningham Project, Factorizations of 12^n+1, n<=240

Cf. A064131, A064132, A064133, A064134, A064135, A064136.

a = {1}; Do[ d = Divisors[ 12^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 12^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 48} ]

a(n) = if (n==0, 2, sumdiv(12^n+1, d, vecsum(vector(n-1, k, gcd(d, 12^k+1) == 1)) == n-1)); \\ Michel Marcus, Jun 24 2018

More terms from Michel Marcus, Jul 02 2018

cp's OEIS Frontend

A064135 Number of divisors of 10^n + 1 that are relatively prime to 10^m + 1 for all 0 < m < n.

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A064131 Number of divisors of 3^n + 1 that are relatively prime to 3^m + 1 for all 0 < m < n.

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A064132 Number of divisors of 5^n + 1 that are relatively prime to 5^m + 1 for all 0 < m < n.

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A064133 Number of divisors of 6^n + 1 that are relatively prime to 6^m + 1 for all 0 < m < n.

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A064134 Number of divisors of 7^n + 1 that are relatively prime to 7^m + 1 for all 0 < m < n.

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A064137 Number of divisors of 12^n + 1 that are relatively prime to 12^m + 1 for all 0 < m < n.

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Mathematica

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