A066699 -id:A066699 - cp's OEIS frontend

A125245 Numbers k such that binomial(6k, k) + 1 is prime.

0, 1, 2, 4, 10, 15, 98, 111, 118, 236, 280, 512, 1284, 1303, 1818, 2525, 2692, 4620, 8405, 11539, 13190, 21525, 30338, 48069, 50687, 56208, 56620, 81091, 101488

Offset: 1

Alexander Adamchuk, Nov 25 2006

a(25) > 50000. - Robert Price, May 13 2019

Cf. A125244 = numbers n such that binomial(6n, n) - 1 is prime. Cf. A066699 = numbers n such that binomial(2n, n) + 1 is prime. Cf. A066726 = numbers n such that binomial(2n, n) - 1 is prime. Cf. A125220, A125221, A125240, A125241, A125242, A125243.

Do[f=Binomial[6n, n]+1; If[PrimeQ[f], Print[n]], {n, 1, 1000}]

More terms from Ryan Propper, Mar 28 2007
a(1)=0 and a(20)-a(24) from Robert Price, May 13 2019
a(25)-a(29) from Georg Grasegger, Jun 23 2025

A085793 Numbers k such that (k-1)*binomial(2k,k) + 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 13, 17, 18, 22, 23, 28, 31, 48, 49, 52, 80, 99, 167, 201, 295, 372, 381, 391, 638, 653, 720, 779, 887, 1047, 1454, 1647, 1719, 2405, 3234, 3257, 3542, 3623, 3765, 3796, 4337, 4490, 5228, 6507, 8544, 9990, 10000, 12478, 13479, 15487, 17115

Offset: 1

Ed Pegg Jr, Jul 23 2003

nonn

a(69) > 10^5 - Robert Price, Sep 15 2024

			9999 * 20000!/(10000!)^2 + 1 is prime

Robert Price, Table of n, a(n) for n = 1..68
Ed Pegg Jr, Binomial Primes.

Cf. A066699, A066726.

is(n)=ispseudoprime((n-1)*binomial(2*n,n)+1) \\ Charles R Greathouse IV, May 22 2017

a(53)-a(68) from Robert Price, Sep 15 2024

A066727 Least factor of n^phi(n) - 1.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 17, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 13, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3

Offset: 1

Robert G. Wilson v, Jan 15 2002

nonn

n^Phi(n)-1 is never prime. This sequence is an outgrowth of Euler's generalization to Fermat's little theorem.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A020639, A066916, A066699.

a = {}; Do[ a = Append[a, FactorInteger[ n^EulerPhi[n] - 1, FactorComplete -> False][[1, 1]]], {n, 1, 100}]; a

A020639(n) = if(1==n,n,forprime(p=2,,if(!(n%p),return(p))));
A066727(n) = if(1==n,0,A020639((n^eulerphi(n))-1)); \\ Antti Karttunen, Oct 24 2024

For n > 1, a(n) = A020639(A066916(n)). - Antti Karttunen, Oct 24 2024

A067316 a(n) is the number of values of j, 0 <= j <= n, such that 1 + binomial(n,j) is prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 4, 2, 5, 6, 6, 6, 6, 4, 5, 2, 6, 8, 8, 6, 6, 4, 4, 2, 11, 4, 4, 8, 8, 8, 4, 2, 6, 4, 8, 14, 8, 4, 5, 6, 12, 10, 4, 6, 9, 8, 8, 4, 6, 8, 6, 10, 6, 6, 12, 6, 8, 4, 12, 2, 6, 8, 4, 2, 8, 18, 8, 2, 6, 14, 10, 16, 10, 6, 4, 10, 13, 8, 12, 4, 8, 2, 8, 14, 2, 6, 4, 10, 10, 16, 10, 10, 9

Offset: 0

Labos Elemer, Jan 15 2002

nonn

			For n = 8, the primes are 2, 29, 71, 29, 2, so a(n) = 5.
a(n) = 6 for n = 9, 10, 11, 12. Also, a(n) = 10 for n = 149, ..., 154.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A066699, A067317.

a[n_] := Count[Table[PrimeQ[Binomial[n, w]+1], {w, 0, n}], True]

a(n) = sum(j=0, n, isprime(1 + binomial(n,j))); \\ Michel Marcus, Oct 30 2018

a(n) = 2 * sum(k=0, (n-1)\2, isprime(binomial(n, k) + 1)) + if(!(n%2), isprime(binomial(n, n/2) + 1)); \\ Amiram Eldar, Jul 18 2024

A075840 Primes of the form (2*n)!/(n!)^2+1.

Original entry on oeis.org

2, 3, 7, 71, 3433, 2704157, 35345263801, 2104098963721, 6892620648693261354601, 410795449442059149332177041, 1520803477811874490019821888415218657, 5949105755928259715106809205795376486501, 1480212998448786189993816895482588794876101

Offset: 1

Donald S. McDonald, Oct 14 2002

nonn

			7 is a term because C(4,2)+1 = 6+1 = 7 is prime.

New Zealand Science Monthly, Bulletin Board, Feb. 1999. Binomial(300,150)+185 = nextprime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..25

Cf. A092751 = n such that (2*n)!/(n!)^2+1 is prime, A112858 = primes of the form (2*n)!/(n!)^2-1.

Cf. A000984, n's are in A066699.

[a: n in [0..100] | IsPrime(a) where a is Factorial(2*n) div Factorial(n)^2+1]; // Vincenzo Librandi Mar 17 2015

a = Select[ Range[100], PrimeQ[Binomial[2#, # ] + 1] & ]; Binomial[2a, a] + 1
Select[Table[(2 n)! / (n!)^2 + 1, {n, 0, 80}], PrimeQ] (* Vincenzo Librandi, Mar 17 2015 *)

v=[]; for(n=0,100,x=bin(2*n,n)+1; if(isprime(x), v=concat(v,x),)); v

Edited by Robert G. Wilson v, Oct 15 2002
Definition corrected by Alexander Adamchuk, Nov 30 2007
Edited by N. J. A. Sloane, Nov 30 2007
a(13) from Vincenzo Librandi, Mar 17 2015

A234963 Number of ways to write n = k + m with k > 0 and m > 2 such that C(2*sigma(k) + phi(m), sigma(k) + phi(m)/2) - 1 is prime, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 0, 3, 2, 2, 3, 3, 5, 3, 4, 3, 3, 3, 2, 3, 0, 3, 3, 4, 3, 0, 1, 2, 3, 1, 2, 3, 3, 1, 3, 3, 4, 1, 2, 3, 3, 2, 6, 4, 1, 4, 2, 3, 2, 2, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 0, 2, 3, 1, 3, 1, 2, 0, 3, 1, 4, 4, 4, 1, 0, 5, 2, 1, 3, 2, 2, 1, 2, 1

Offset: 1

Zhi-Wei Sun, Jan 01 2014

nonn

Conjecture: a(n) > 0 for all n >= 180.
Clearly, this implies that there are infinitely many primes of the form C(2*n,n) - 1. We have verified the conjecture for n up to 10000.
Note that every n = 400, ..., 9123 can be written as k + m with k > 0 and m > 0 such that f(k, m) = sigma(k) + phi(m) is even and C(f(k, m) + 2, f(k, m)/2 + 1) + 1 is prime, but this fails for n = 9124.

			a(5) = 1 since 5 = 1 + 4 with C(2*sigma(1) + phi(4), sigma(1) + phi(4)/2) - 1 = C(4, 2) - 1 = 5 prime.
a(28) = 1 since 28 = 2 + 26 with C(2*sigma(2) + phi(26), sigma(2) + phi(26)/2) - 1 = C(18, 9) - 1 = 48619 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Cf. A000010, A000040, A000203, A000984, A066726, A066699, A232270, A233544, A234451, A234470, A234503, A234514, A234567, A234615.

sigma[n_] := DivisorSigma[1, n];
f[n_,k_] := Binomial[2*sigma[k] + EulerPhi[n-k], sigma[k] + EulerPhi[n-k]/2] - 1;
a[n_] := Sum[If[PrimeQ[f[n,k]], 1, 0], {k, 1, n-3}];
Table[a[n], {n, 1, 100}]

A066732 Least k such that the least factor of k^Phi(k) -1 is the n-th prime.

Original entry on oeis.org

3, 4, 6, 18, 150, 60, 30, 22440, 120360, 44880, 5610, 11730, 8160, 473280, 277440, 131070, 548760, 920040, 750720, 440130, 329970, 27030, 5689560, 522240, 1020, 3028890, 2639760, 6866130, 251430, 134130, 7481190, 2390880, 2664240, 9926130, 279480, 9730290

Offset: 1

Robert G. Wilson v, Jan 15 2002

nonn

			18^Phi(18)-1 = 18^6-1 = 34012223 = 7^3 * 17 * 19 * 307. Therefore since 7, the fourth prime, is the least prime in the factorization, a(4) = 18.

Robert Price, Table of n, a(n) for n = 1..56

Cf. A066699.

a = Table[0, {53} ]; Do[b = 1; While[ PowerMod[n, EulerPhi[n], Prime[b]] != 1, b++ ]; If[ a[[b]] == 0, a[[b]] = n], {n, 3, 10^6} ]

a(35)-a(36) from Robert Price, Nov 06 2023

cp's OEIS Frontend

A125245 Numbers k such that binomial(6k, k) + 1 is prime.

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A085793 Numbers k such that (k-1)*binomial(2k,k) + 1 is prime.

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A066727 Least factor of n^phi(n) - 1.

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A067316 a(n) is the number of values of j, 0 <= j <= n, such that 1 + binomial(n,j) is prime.

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A075840 Primes of the form (2*n)!/(n!)^2+1.

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Extensions

A234963 Number of ways to write n = k + m with k > 0 and m > 2 such that C(2*sigma(k) + phi(m), sigma(k) + phi(m)/2) - 1 is prime, where sigma(k) is the sum of all positive divisors of k and phi(.) is Euler's totient function.

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A066732 Least k such that the least factor of k^Phi(k) -1 is the n-th prime.

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Extensions