A088790 -id:A088790 - cp's OEIS frontend

A054767 Period of the sequence of Bell numbers A000110 (mod n).

1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, 25239592216021, 411771, 10153, 48, 51702516367896047761, 39, 109912203092239643840221, 9372, 1784341, 85593501183, 949112181811268728834319677753, 312, 3905, 75718776648063, 117, 1647084

Offset: 1

Eric W. Weisstein, Feb 09 2002

For p prime, a(p) divides (p^p-1)/(p-1) = A023037(p), with equality at least for p up to 19.

Wagstaff shows that N(p) = (p^p-1)/(p-1) is the period for all primes p < 102 and for primes p = 113, 163, 167 and 173. For p = 7547, N(p) is a probable prime, which means that this p may have the maximum possible period N(p) also. See A088790. - T. D. Noe, Dec 17 2008

Jianing Song, Table of n, a(n) for n = 1..102 (b-file based on the Wagstaff article)
J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.
Jianing Song, A summary of several proofs in the Lunnon article.
Samuel S. Wagstaff Jr., Aurifeuillian factorizations and the period of the Bell numbers modulo a prime, Math. Comp. 65 (1996), 383-391.
Eric Weisstein's World of Mathematics, Bell Number.

Cf. A000110, A023037, A214810. A146093-A146122 gives Bell numbers read mod 3 to mod 32.

(* Warning: this program is just a verification of the existing data
 and should not be used to extend the sequence beyond a(28) *)
BellMod[k_, m_] := Mod[Sum[Mod[StirlingS2[k, j], m], {j, 1, k}], m];
BellMod[k_, 1] := BellB[k];
period[nn_List] := Module[{lgmin=2, lgmax=5, nn1},
   lg=If[Length[nn]<=lgmax, lgmin, lgmax];
   nn1 = nn[[1;;lg]];
   km=Length[nn]-lg;
   Catch[Do[If[nn1==nn[[k;;k+lg-1]], Throw[k-1]];
   If[k==km, Throw[0]], {k, 2, km}]]];
dd[n_] := SelectFirst[Table[{d, n/d},
     {d, Divisors[n][[2;;-2]]}], GCD@@#==1&];
a[1]=1;
a[p_?PrimeQ] := a[p] = (p^p-1)/(p-1);
a[n_/;n>4 && dd[n]!={}] := With[{g = dd[n]}, LCM[a[g[[1]]], a[g[[2]]]]];
a[n_/;MemberQ[FactorInteger[n][[All, 2]], 1]] := a[n]=
   With[{pp = Select[FactorInteger[n], #1[[2]] ==1 &][[All, 1]]},
      a[n/Times@@pp]*Times@@a/@pp];
a[n_/;n>4 && GCD @@ FactorInteger[n][[All, 2]]>1] := a[n]=
   With[{g=GCD @@ FactorInteger[n][[All, 2]]}, n^(1/g)*a[n^(1-1/g)]];
a[n_] := period[Table[BellMod[k, n], {k, 1, 28}]];
Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jul 31 2012, updated May 06 2024 *)

If gcd(n,m) = 1, a(n*m) = lcm(a(n), a(m)). But the sequence is not in general multiplicative; e.g. a(2) = 3, a(9) = 39 and a(18) = 39. - Franklin T. Adams-Watters, Jun 06 2006

a(2^s) = 3*2^s for s >= 2 (Theorem 6.4 in the Lunnon article). For an odd prime p, if a(p) = (p^p-1)/(p-1) (which is conjectured to hold for all p), then a(p^s) = p^(s-1)*(p^p-1)/(p-1) (Theorem 6.2 in the Lunnon article). - Jianing Song, Jun 18 2025

More information from Phil Carmody, Dec 22 2002
Extended by T. D. Noe, Dec 18 2008
a(26) corrected by Jean-François Alcover, Jul 31 2012
a(18) corrected by Charles R Greathouse IV, Jul 31 2012
a(27)-a(28) from Charles R Greathouse IV, Sep 07 2016

A023037 a(n) = n^0 + n^1 + ... + n^(n-1), or a(n) = (n^n-1)/(n-1) with a(0)=0; a(1)=1.

Original entry on oeis.org

0, 1, 3, 13, 85, 781, 9331, 137257, 2396745, 48427561, 1111111111, 28531167061, 810554586205, 25239592216021, 854769755812155, 31278135027204241, 1229782938247303441, 51702516367896047761, 2314494592664502210319, 109912203092239643840221

Offset: 0

David W. Wilson

For prime n, a(n) is conjectured to be the period of Bell numbers (mod n). See A054767. - T. D. Noe, Oct 12 2007
For prime n, a(n) is a multiple of the period of Bell numbers mod n (and conjectured to be exactly the period, as mentioned above). - Charles R Greathouse IV, Jul 31 2012
For n >= 1, a(n) is the number whose base n representation is a string of n ones. For example, 11111 in base 5 is a(5) = 781. - Melvin Peralta, May 23 2016
For n > 0, n^(a(n)-1) == 1 (mod a(n)), so for n > 1, a(n) is a prime or a Fermat pseudoprime to base n. - Thomas Ordowski, Mar 15 2021

			a(3) = 3^0 + 3^1 + 3^2 = 1+3+9 = 13.

Alois P. Heinz, Table of n, a(n) for n = 0..387 (first 101 terms from T. D. Noe)
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.

Cf. A001039, A054767, A088790 (n such that a(n) is prime), A125118.

A023037:=n->add(n^i, i=0..n-1): seq(A023037(n), n=0..25); # Wesley Ivan Hurt, May 28 2016

Join[{0,1},Table[(n^n-1)/(n-1),{n,2,20}]] (* Harvey P. Dale, Aug 01 2014 *)

a(n) = if(n==1, 1, (n^n-1)/(n-1)); \\ Altug Alkan, Oct 04 2017

def A023037(n): return (n**n-1)//(n-1) if n>1 else n # Chai Wah Wu, Sep 28 2023

[lucas_number1(n,n+1,n) for n in range(0, 19)] # Zerinvary Lajos, May 16 2009

a(n) = A125118(n,n-1) for n>1. - Reinhard Zumkeller, Nov 21 2006

a(n) = [x^n] x/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017

Entry improved by Tobias Nipkow (nipkow(AT)in.tum.de).

A070518 Value of n-th cyclotomic polynomial at n.

Original entry on oeis.org

0, 3, 13, 17, 781, 31, 137257, 4097, 532171, 9091, 28531167061, 20593, 25239592216021, 7027567, 2392743361, 4294967297, 51702516367896047761, 34006393, 109912203092239643840221, 25536159601, 7006306553612521, 25405143539623, 949112181811268728834319677753

Offset: 1

Labos Elemer, May 02 2002

nonn

a(28341) is divisible by 283411^2. What is the next n such that a(n) is not squarefree? - Jianing Song, Nov 01 2024

			n=10: 10th cyclotomic polynomial is 1-x+x^2-x^3+x^4; at x=10 it gives a(10)=9091.

G. C. Greubel, Table of n, a(n) for n = 1..250
Mathematics Stack Exchange, Is the "cyclotomic diagonalization" always squarefree?
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A070519 (indices of prime terms), A088790 (prime indices of prime terms).

a:= n-> numtheory[cyclotomic](n$2):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 05 2024

Mathematica
```
Table[Cyclotomic[w, w], {w, 1, 35}]
```

a(n) = polcyclo(n, n) \\ Michel Marcus, Apr 02 2016

A110932 Numbers k such that 2*k^k + 1 is prime.

Original entry on oeis.org

0, 1, 12, 18, 251, 82992

Offset: 0

Ray G. Opao, Sep 25 2005

As a "list of numbers such that ...", the sequence should have offset 1, but to preserve the validity of formulas referring to this sequence, the offset was set to 0 when the initial value a(0)=0 was added. - M. F. Hasler, Sep 02 2012

Cf. A110931, A121270 (= primes in A014566), A088790, A160360, A160600.

The primes 2n^n+1, for k<4, n=a(k)<251, are listed at A216148(k) = A216147(a(k)). - M. F. Hasler, Sep 02 2012

Join[{0}, Select[Range[1000], PrimeQ[2*#^# + 1] &]] (* Robert Price, Mar 27 2019 *)

is_A110932(n)=ispseudoprime(n^n*2+1) \\ M. F. Hasler, Sep 02 2012

a(5) from Serge Batalov, Apr 08 2018

A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

Original entry on oeis.org

2, 3, 4, 6, 10, 12, 14, 19, 31, 46, 74, 75, 98, 102, 126, 180, 236, 310, 368, 1770, 1858, 3512, 4878, 5730, 7547, 7990, 8636, 9378, 11262

Offset: 1

Labos Elemer, May 02 2002

When n is prime, then the solutions are given in A088790.
No term of this sequence is congruent to 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 1 (mod 4), then Cyclotomic(k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019
All terms <= 1858 have been proven with PARI's implementation of ECPP.  All larger terms are BPSW PRPs.  There are no further terms <= 30000. - Lucas A. Brown, Dec 28 2020

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Aurifeuillean factorization

Cf. A070518, A070520, A088790 ((k^k-1)/(k-1) is prime), A088817 (cyclotomic(2k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Cf. A117544, A085398.

Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[n]], {n, 2, 256}]

for(n=2,10^9,if(ispseudoprime(polcyclo(n,n)),print1(n,", "))); \\ Joerg Arndt, Jan 22 2015

More terms from T. D. Noe, Oct 17 2003

a(29) from Charles R Greathouse IV, May 05 2011

A056826 Primes p such that (p^p + 1)/(p + 1) is a prime.

Original entry on oeis.org

3, 5, 17, 157

Offset: 1

Robert G. Wilson v, Aug 29 2000

Note that (k^k+1)/(k+1) is prime only if k is prime, in which case it equals cyclotomic(2k,k), the 2k-th cyclotomic polynomial evaluated at x=k. This sequence is a subsequence of A088817. Are there only a finite number of these primes? - T. D. Noe, Oct 20 2003
(3^2 + 5^2)/2 = 17, (5^2 + 17^2)/2 = 157. - Thomas Ordowski, Jul 28 2013
Let b(0) = 1, b(1) = 3; b(n+1) = (b(n)^2 + b(n-1)^2)/2. Conjecture: if b(n) = p is prime, then (p^p+1)/(p+1) is prime. Note that b(1) = 3, b(2) = 5, b(3) = 17, b(4) = 157 and b(9) is also prime. - Thomas Ordowski, Jul 29 2013
Next term > 3000. - Seiichi Manyama, Mar 24 2018
No more terms through 6000. - Jon E. Schoenfield, Mar 25 2018
No more terms through 20000. - Michael S. Branicky, Jul 30 2024

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 157, p. 51, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Theory of Numbers, 1994 A3.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A088790 ((n^n-1)/(n-1) is prime), A088817 (cyclotomic(2n, n) is prime).

Do[ If[ PrimeQ[ (Prime[ n ]^Prime[ n ] + 1)/(Prime[ n ] + 1) ], Print[ Prime[ n ] ] ], {n, 1, 213} ]
Do[p=Prime[n]; If[PrimeQ[(p^p+1)/(p+1)], Print[p]], {n, 100}] (* T. D. Noe, Oct 20 2003 *)

forprime(p=3, 1000, if(isprime((p^p+1)/(p+1)), print1(p", "))) \\ Seiichi Manyama, Mar 24 2018

Definition corrected by Alexander Adamchuk, Nov 12 2006

A160600 Numbers k such that 3*(2k)^(2k)+1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 143, 225

Offset: 1

M. F. Hasler, Jul 10 2009

This corresponds to the numbers such that 3m^m+1 is prime, but these must all be even, m=2k, and therefore it is more natural to record the sequence of k=m/2.

Next term > 15000. - Matevz Markovic, Oct 09 2012

			a(1) = 1, because 2^2*3+1 = 13 is the smallest prime of this form.
a(2) = 2, because 4^4*3+1 = 769 is the next smallest prime of this form. a(3) = 3, because 6^6*3+1 = 139969 is again a prime.

Cf. A160360 (3n^n+2 is prime), A121270 = primes among Sierpinski numbers A014566(n)=n^n+1; A216148 = A216147(A110932): primes 2n^n+1; A088790, A065798.

q:= k-> isprime(3*(2*k)^(2*k)+1):
select(q, [$1..225])[];  # Alois P. Heinz, Aug 04 2025

for(i=1,9999,ispseudoprime(i^i*3+1)&print1(i/2,","))

A374173 a(n) is the smallest prime whose base-n representation contains a run of at least n identical digits.

Original entry on oeis.org

3, 13, 683, 3907, 55987, 960803, 19173967, 435848051, 11111111113, 1540683021299, 19453310068921, 328114698808283, 45302797058044219, 469172025408063623, 19676527011956855059, 878942778254232811943, 120353718818554114936591, 109912203092239643840221

Offset: 2

Robert P. P. McKone, Jun 30 2024

a(2) to a(18) are all increasing, but a(19) is smaller than a(18).

a(n) = A023037(n) for n in A088790. - Robert Israel, Dec 31 2024

			a(2) = 3 = 11_2.
a(3) = 13 = 111_3.
a(11) = 1540683021299 = 544444444444_11.
a(18) = 120353718818554114936591 = 3111111111111111111_18.
a(19) = 109912203092239643840221 = 1111111111111111111_19.

Robert Israel, Table of n, a(n) for n = 2..385

Cf. A023037, A088790.

f:= proc(n) local t,Q,i,j;
  t:= (n^n-1)/(n-1);
  if isprime(t) then return t fi;
  for i from 1 to n-1 do
    Q:= select(isprime, [seq(i*t*n+j,j=1..n-1),
         seq(i*n^n+j*t,j=1..n-1)]);
    if Q <> [] then return min(Q) fi;
  od;
  FAIL
end proc:
map(f, [$2..20]); # Robert Israel, Dec 31 2024

d[n_]:=d[n]=Table[Table[m,n],{m,0,n-1}];
dpre[n_]:=Flatten[Table[{m}~Join~#&/@d[n],{m,0,n-1}],1];
dpost[n_]:=Flatten[Table[Map[#~Join~{m}&,d[n]],{m,0,n-1}],1];
dprepost[n_]:=Flatten[Table[Map[{j}~Join~#~Join~{m}&,d[n]],{m,0,n-1},{j,0,n-1}],2];
c[n_]:=c[n]=DeleteDuplicates[Sort[Select[FromDigits[#,n]&/@Join[d[n],dpre[n],dpost[n],dprepost[n]],#>n&]]];
a[n_]:=a[n]=Do[If[PrimeQ[q],Return[q];Break[];],{q,c[n]}];
Table[a[n],{n,2,19}]

A102604 Numbers k such that ((2k)^k - 1)/(2k - 1) is prime.

Original entry on oeis.org

2, 3, 7, 41, 43, 79, 421

Offset: 1

Pierre CAMI, Jan 29 2005

The next k in the sequence is > 4261, if it exists.
Note that (b^k - 1)/(b-1) is prime only if k is prime, so all the elements in this sequence must be primes. - Marco Bodrato (marco2007(AT)bodrato.it), Oct 31 2007
a(8) > 20000, if it exists. - Michael S. Branicky, Aug 12 2024

			(((2*2)^2) - 1)/(2*2 - 1) = 15/3 = 5 is prime so a(1)=2.

Cf. A088790.

 Select[Prime[Range[100]],PrimeQ[((2#)^#-1)/(2#-1)]&] (* Harvey P. Dale, Mar 09 2022 *)

lista(nn) = {forprime(n = 2, nn, if (isprime(((2*n)^n-1)/(2*n-1)), print1(n, ", ")););} \\ Michel Marcus, Feb 05 2014

A283515 Numbers k such that sigma(k^(k-1)) is a prime.

Original entry on oeis.org

2, 3, 4, 16, 19, 31, 7547

Offset: 1

Jaroslav Krizek, Mar 10 2017

sigma(k) is the sum of the divisors of k (A000203).
Numbers k such that A000203(A000169(k)) is a prime.
a(8) > 10^4.
Corresponding values of k^(k-1): 2, 9, 64, 1152921504606846976, ...
Corresponding values of sigma(k^(k-1)): 3, 13, 127, 2305843009213693951, ...
Subsequence of A280257 (numbers k such that tau(k^(k-1)) is prime).
Prime terms are in A088790.
For k < 1000, sigma(k^(k+1)) is prime only for k = 5: sigma(5^6) = sigma(15625) = 19531 (prime).

			sigma(4^3) = sigma(64) = 127 (prime).

Cf. A000169, A000203, A088790, A280257.

[n: n in [1..500] | IsPrime(SumOfDivisors(n^(n-1)))];

fQ[n_] := PrimeQ[DivisorSigma[1, n^(n - 1)]]; Select[Range@1000, fQ] (* Robert G. Wilson v, Mar 10 2017 *)

isok(n) = isprime(sigma(n^(n-1))); \\ Michel Marcus, Mar 10 2017

a(7) from Giovanni Resta, Mar 10 2017

cp's OEIS Frontend

A054767 Period of the sequence of Bell numbers A000110 (mod n).

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A023037 a(n) = n^0 + n^1 + ... + n^(n-1), or a(n) = (n^n-1)/(n-1) with a(0)=0; a(1)=1.

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A070518 Value of n-th cyclotomic polynomial at n.

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A110932 Numbers k such that 2*k^k + 1 is prime.

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A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

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A056826 Primes p such that (p^p + 1)/(p + 1) is a prime.

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Mathematica

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A160600 Numbers k such that 3*(2k)^(2k)+1 is prime.

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A102604 Numbers k such that ((2k)^k - 1)/(2k - 1) is prime.