Luis H. Gallardo - cp's OEIS frontend

A355891 Numbers k such that k = ivgenpoly(A) for some composite polynomial A in F_2[x] that satisfies the condition sigma(A) = A + 1.

1905, 424321, 438065, 443617, 7044945, 7899377, 7925761, 26397649, 32286449, 38123521, 55759233

Offset: 1

Luis H. Gallardo, Jul 28 2022

Let A be a polynomial in F_2[x]. We let lift(A) in Z[x] denote the same polynomial, but with integer coefficients 0,1.
Let ivgenpoly(A) be the positive integer equal to the lift(A) evaluated in x=2. For example, if A = x^2+x+1 in F_2[x], we have lift(A) = x^2+x+1 in Z[x], and ivgenpoly(A) = 2^2+2+1 = 7. Similarly, for every positive integer n, we let genpoly(n) denote the unique polynomial A in F_2[x] such that n = ivgenpoly(A). The coefficients of A, are the digits of the base-2 expansion of n.
Over the integers, it is easy to check that sigma(p)=p+1 implies that p is a prime number, where sigma(n) is the sum of all positive divisors of the positive integer n. However, in F_2[x] the analogous result is false.
We denote by sigma(A) the sum of all divisors of A. The sequence shows integers k = ivgenpoly(A) such that A is a composite polynomial in F_2[x] for which sigma(A)=A+1.

			a(1) = 1905, since 1905 = ivgenpoly(A), with A = x^10+x^9+x^8+x^6+x^5+x^4+1, satisfies A = (x^3+x+1)*(x^3+x^2+1)*(x^4+x+1) so that sigma(A) = (x^3+x)*(x^3+x^2)*(x^4+x) = A+1, and for any number m with 0 < m < 1905, with m = ivgenpoly(B), one has that either sigma(B) is unequal to B+1 or B is irreducible.
Moreover, a(2) = 424321, since 424321 = ivgenpoly(A), with A = x^18+x^17+x^14+x^13+x^12+x^11+x^8+x^7+1, satisfies A = (x^4+x^3+1)*(x^4+x^3+x^2+x+1)*(x^5+x^2+1)*(x^5+x^4+x^2+1) so that sigma(A) = A+1, and for any number m with 1905 < m < 424321, with m = ivgenpoly(B), one has that either sigma(B) is unequal to B+1 or B is irreducible.

Cf. A000203.

A354226 a(n) is the number of distinct prime factors of (p^p - 1)/(p - 1) where p = prime(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 1, 4, 7, 1, 7, 5, 3, 3, 5, 3, 4, 6, 4, 10, 5, 4, 6, 6, 9, 5, 4, 5, 8, 6, 4, 11

Offset: 1

Luis H. Gallardo, May 20 2022

a(34) > 3, and depends on the full factorization of the 296-digit composite number (139^139 - 1)/138. - Tyler Busby, Jan 22 2023

Sequence continues as ?, 8, ?, 5, 8, 4, 5, ?, 8, ?, 8, 7, 6, 3, 3, ..., where ? represents uncertain terms. - Tyler Busby, Jan 22 2023

			a(3)=2, since (5^5 - 1)/(5 - 1) = 11*71.

factordb, Status of (139^139 - 1)/138.

Cf. A000040, A001039, A001221, A088790, A125135, A212552, A214812.

a(n) = my(p=prime(n)); omega((p^p-1)/(p-1)); \\ Michel Marcus, May 22 2022

from sympy import factorint, prime
def a(n): p = prime(n); return len(factorint((p**p-1)//(p-1)))
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, May 23 2022

a(n) = A001221(A001039(n)).

a(18)-a(33) from Amiram Eldar, May 20 2022

A351682 Prime numbers p such that the (p-1)-st Bell number B(p-1) is a primitive root modulo p.

Original entry on oeis.org

2, 3, 11, 13, 17, 19, 29, 31, 47, 53, 71, 103, 113, 127, 131, 137, 139, 149, 173, 179, 181, 191, 211, 233, 239, 241, 251, 257, 263, 269, 293, 317, 347, 367, 379, 401, 431, 439, 449, 461, 503, 509, 523, 541, 557, 587, 607, 617, 619, 647, 653, 683, 691, 733, 743, 761, 773, 797, 821, 823, 827, 853, 859, 881, 919, 929

Offset: 1

Luis H. Gallardo, May 04 2022

nonn

Heuristically, the density of the sequence in the primes should approach Artin's constant: 0.3739558136...

			For n = 2 one has a(2) = 3 since B(2) = 2 is a primitive root modulo 3.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A000110, A005596, A350429.

filter:= proc(p) local b;
  b:= combinat:-bell(p-1);
  numtheory:-order(b,p) = p-1
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, May 04 2023

Corrected by Robert Israel, May 04 2023

A351688 Prime numbers p such that the order of the (p-1)-th Bell number B(p-1) is a power of 2 modulo p.

Original entry on oeis.org

3, 17, 23, 37, 67, 89, 193, 227, 257, 593, 641, 769, 1889, 10331, 12289, 13441, 18433, 40961, 65537, 85121, 96769, 2752513, 3655681

Offset: 1

Luis H. Gallardo, May 05 2022

An odd prime p is a counterexample of Kurepa's conjecture if and only if B(p-1) = 1 modulo p.

			a(1)=3 since B(2)=2 has order 2 modulo 3.
a(3)=37, since B(36)=6 modulo 37 has order 4 = 2^2 modulo 37.

Cf. A000110.

Do[p = Prime[k]; m = Mod[BellB[p-1], p]; If[m != 0, f = FactorInteger[MultiplicativeOrder[m, p]]; If[Length[f] == 1 && f[[1, 1]] == 2, Print[p]]], {k, 1, 500}] (* Vaclav Kotesovec, May 06 2022 *)

A350523 Prime numbers p such that K(p) = 0! + 1! + ... + (p-1)! == -2 (mod p).

Original entry on oeis.org

2, 3, 23, 67, 227, 10331

Offset: 1

Luis H. Gallardo, Jan 03 2022

The Kurepa Conjecture says that K(p) is nonzero in F_p, the finite field with p elements. Primes for which K(p) takes some fixed nonzero value in F_p might have some interest.

No further terms < 6*10^6. - Michael S. Branicky, Jan 03 2022

Vladica Andrejić, Alin Bostan and Milos Tatarevic, Improved algorithms for left factorial residues, Information Processing Letters, Vol. 167 (2021), Article ID 106078, 4 p.; arXiv preprint, arXiv:1904.09196 [math.NT], 2019-2020.

Cf. A003422, A100612.

Subsequence of A236400.

q[p_] := PrimeQ[p] && Divisible[Sum[k!, {k, 0, p - 1}] + 2, p]; Select[Range[230], q] (* Amiram Eldar, Jan 03 2022 *)

from sympy import isprime
def K(n):
    ans, f = 0, 1
    for i in range(1, n+1):
        ans += f%n
        f = (f*i)%n
    return ans%n
def ok(n): return isprime(n) and (K(n) + 2)%n == 0
print([k for k in range(11000) if ok(k)]) # Michael S. Branicky, Jan 03 2022

# faster version for initial segment of sequence
from sympy import isprime
def afind(limit):
    f = 1 # (p-1)!
    s = 2 # sum(0! + 1! + ... + (p-1)!)
    for p in range(2, limit+1):
        if isprime(p) and s%p == p-2:
            print(p, end=", ")
        s += f*p
        f *= p
afind(11000) # Michael S. Branicky, Jan 03 2022

A350429 Prime numbers p for which there exists at least one integer k < p such that p divides the k-th Bell number.

Original entry on oeis.org

5, 7, 13, 19, 23, 29, 37, 47, 53, 61, 67, 71, 73, 89, 101, 107, 131, 137, 139, 157, 163, 167, 173, 179, 181, 191, 193, 211, 223, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 313, 331, 349, 353, 367, 401, 419, 431, 433, 439, 443, 449, 467, 491, 499, 509, 541

Offset: 1

Luis H. Gallardo, Dec 30 2021

nonn

Igor Shparlinski proved in 1991 that k < (1/2)*binomial(2*p,p) (see A290059).

			a(1)=5 since modulo 5 we have B(0)=1, B(1)=1, B(2)=2, and B(3)=0.

I. E. Shparlinskiy, On the Distribution of Values of Recurring Sequences and the Bell Numbers in Finite Fields, European Journal of Combinatorics, Vol. 12, No. 1 (1991), pp. 81-87.

Cf. A000110, A290059.

q[p_] := Module[{k = 1}, While[k < p && ! Divisible[BellB[k], p], k++]; k < p]; Select[Range[500], PrimeQ[#] && q[#] &] (* Amiram Eldar, Dec 30 2021 *)

A335383 a(n) is the number of irreducible Mersenne polynomials in GF(2)[x] that have degree n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 0, 4, 2, 2, 2, 0, 2, 6, 0, 6, 2, 0, 2, 2, 2, 4, 0, 4, 0, 0, 8, 2, 2, 8, 0, 4, 2, 2, 2, 0, 0, 6, 0, 4, 0, 0, 2, 0, 2, 8, 0, 8, 0, 0, 8, 0, 0, 4, 0, 8, 2, 0, 8, 0, 2, 8, 0, 4, 0, 0, 4, 0, 0, 10, 0, 6, 2, 0, 2, 0, 0, 4, 0, 6, 0, 0, 6, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 4, 0, 8, 4, 0, 6

Offset: 2

Luis H. Gallardo, Jun 04 2020

nonn

A Mersenne polynomial is a binary (i.e., an element of GF(2)[x]) polynomial M, of degree > 1, such that M+1 has only 0 and 1 as roots in a fixed algebraic closure of GF(2).
If for some positive integers  a,b, M = x^a(x+1)^b+1 is an irreducible Mersenne polynomial, then gcd(a,b)=1. This condition is not sufficient.
There is no known formula for a(n). Of course it is bounded above by the total number of prime (irreducible) binary polynomials of degree n, but this is a too weak upper bound. A trivial, better upper bound, is simply n-1, the total number of Mersenne polynomials (prime or not) of degree n.

			For n = 5 one has a(5) = 2 since there are 2 irreducible Mersenne polynomials of degree 5. Namely, x^2*(x+1)^3+1 and x^3*(x+1)^2+1.
For n = 8, a(8) = 0 since there are no irreducible Mersenne polynomial of degree 8.

L. H. Gallardo and O. Rahavandrainy, On even (unitary) perfect polynomials over F_2, Finite Fields Appl. 18, no. 5, (2012), 920-932.
L. H. Gallardo and O. Rahavandrainy, Characterization of Sporadic perfect polynomials over F_2, Funct. Approx. Comment. Math.,(2016), 7-21.
L. H. Gallardo and O. Rahavandrainy, On Mersenne polynomials over F_2, Finite Fields Appl. 59 (2019), 284-296.
L. H. Gallardo and O. Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.

Cf. A267918, A272486, A162570.

a(n) = sum(k=1, n-1, polisirreducible(Mod(1, 2)*(x^(n-k)*(x+1)^k+1))); \\ Michel Marcus, Jun 07 2020

a(A272486(n)) = 0. - Michel Marcus, Jun 07 2020

A335379 a(n) is the number of Mersenne prime (irreducible) polynomials M = x^k(x+1)^(n-k)+1 of degree n in GF(2)[x] (k goes from 1 to n-1) such that Phi_7(M) has an odd number of prime divisors (omega(Phi_7(M)) is odd).

Original entry on oeis.org

1, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 2

Luis H. Gallardo, Jun 03 2020

nonn

Phi_7(x)=1+x+x^2+x^3+x^4+x^5+x^6, is the 7th cyclotomic polynomial; omega(P(x)) counts the 2 X 2 distinct irreducible divisors of the binary polynomial P(x) in GF(2)[x].

It is surprising that a(n) be so small (conjecturally it is always 1 or 2). The sequence appeared when working the special case p=7 of a conjecture (see Links) about prime divisors in GF(2)[x] of the composed cyclotomic polynomial Phi_p(M), where p is a prime number and M is a Mersenne irreducible polynomial.

			For n=4 a(4)= 0 (the sequence begins a(2)=1,a(3)=2,...), since there is no Mersenne polynomial M of degree 4 in GF(2)[x] such that omega(Phi_7(M)) is odd.

Antti Karttunen, Table of n, a(n) for n = 2..1700
L. H. Gallardo and O. Rahavandrainy, On Mersenne polynomials over GF(2), Finite Fields Appl. 59 (2019), 284-296.

Cf. A057460 (A074710), A071642, A267918, A272486, A162570.

a(n)={my(phi7=polcyclo(7)); sum(k=1, n-1, my(p=Mod(x^k * (x+1)^(n-k) + 1, 2)); polisirreducible(p) && #(factor(subst(phi7, x, p))~)%2)} \\ Andrew Howroyd, Jun 04 2020

Terms a(22) and beyond from Andrew Howroyd, Jun 04 2020

A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^kf(k), where f(k) = harmonic(2^k) - klog(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.

Original entry on oeis.org

2, 7, 2, 3, 4, 3, 5, 8, 7, 7, 0, 7, 5, 9, 6, 7, 6, 4, 7, 8, 4, 0, 7, 0, 6, 7, 6, 9, 2, 3, 9, 5, 5, 5, 7, 8, 7, 4, 8, 2, 2, 5, 1, 0, 8, 0, 6, 4, 3, 9, 5, 8, 7, 1, 6, 4, 5, 3, 8, 9, 6, 2, 0, 4, 1, 2, 8, 3, 7, 5, 9, 7, 0, 0, 5, 7, 2, 9, 6, 5, 1, 1, 5, 0, 1, 2, 9, 8, 4, 6, 1, 7, 7, 3, 1, 3, 1, 7, 3, 9, 8, 0, 2, 7

Offset: 0

Luis H. Gallardo, Apr 20 2019

			0.272343587707596764784070676923955578748225108064395871645389620412837597...

Cf. A001620 (Euler-Mascheroni), A001008/A002805 (harmonic), A002162 (log(2)), A094640 (alternate Euler's constant), A256921 (a similar constant).

evalf(Sum((-1)^k*(harmonic(2^k) - k*log(2) - gamma), k=0..infinity), 120); # Vaclav Kotesovec, Apr 30 2019

digits = 104; s = NSum[(-1)^k*(HarmonicNumber[2^k] - k*Log[2] - EulerGamma), {k, 0, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10]; RealDigits[s, 10, digits][[1]] (* Jean-François Alcover, Apr 28 2019 *)

default(realprecision, 120); sumalt(k=0, (-1)^k*(psi(2^k+1) - k*log(2))) \\ Vaclav Kotesovec, Apr 30 2019

A280717 Given a prime number p, let b = -p and c = p^2. Assuming that the polynomial P(x) := x^2+b*x+c takes at least one prime value for some positive integer x

Original entry on oeis.org

3, 7, 43, 1693, 2864557, 8205572225569, 67331415548799635795058613, 4533519519805137360312930667312809111343819483374997, 20552799236454203238557860425684304712780972342513397945121797314302926172950212696842909492430773376197

Offset: 1

Luis H. Gallardo, Jan 07 2017

The next term is only defined if the set S above is not empty.
Conjecture: the sequence is well defined.
a(13) has 1654 digits. If S is not empty, then its maximal element is P(x) where x is the least positive integer x <= p/2 such that P(x) is prime. - Chai Wah Wu, Jan 09 2017

			a(2) = 7, since 7 = max S_3, where S_3 = {x^2-3x+9 : x is an integer with 0<x<2, and x^2-3x+9 is a prime number}. Clearly, S_3={7}, thus a(2)=7. Now we explain why a(3)=43. We have 43 = max S_7. S_7 := {x^2-7x+49 : x is an integer, 0 <x<7, and x^2-7x+49 is a prime number}. By computations S_7 = {37,43}. Thus a(3) = max S_7 = 43. We explain also why a(4) = 1693. One has 1693 = max S_43, where S_43 = {x^2-43x+43^2 : x is an integer, 0 <x < 43, and x^2-43x+43^2 is a prime number}. By computations S_43 = {1399,1429,1459,1543,1597,1627,1693}. Thus a(3) = max S_43 = 1693.

Chai Wah Wu, Table of n, a(n) for n = 1..12

with(numtheory):
xa := proc(aa) local P,x,a,a2,mi,mm; a:= aa; a2 := a^2; mi := 0; for x from 1 to a-1 do P := x^2-a*x+a2; if isprime(P) then mi := max(P,mi); fi; od;; mi; end;
F := proc(n) option remember if n=1 then return(3); fi; if n=2 then xa(3); else xa(F(n-1)); fi; end;

P[p_, x_] := x^2 - p x + p^2;
A280717[1] = 3;
A280717[n_] := A280717[n] = P[A280717[n - 1], NestWhile[# - 1 &, A280717[n - 1] - 1, # > A280717[n - 1]/2 && ! PrimeQ@P[A280717[n - 1], #] &]];
A280717 /@ Range[5] (* Davin Park, Feb 06 2017 *)

from _future_ import division
from sympy import isprime
A280717_list, n = [3], 3
for _ in range(10):
    for i in range(1,n//2+1):
        j = i**2+n*(n-i)
        if isprime(j):
            n = j
            A280717_list.append(n)
            break # Chai Wah Wu, Jan 09 2017

a(5) corrected and a(6)-a(9) added by Chai Wah Wu, Jan 09 2017

cp's OEIS Frontend

User: Luis H. Gallardo

A355891 Numbers k such that k = ivgenpoly(A) for some composite polynomial A in F_2[x] that satisfies the condition sigma(A) = A + 1.

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A354226 a(n) is the number of distinct prime factors of (p^p - 1)/(p - 1) where p = prime(n).

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A351682 Prime numbers p such that the (p-1)-st Bell number B(p-1) is a primitive root modulo p.

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A351688 Prime numbers p such that the order of the (p-1)-th Bell number B(p-1) is a power of 2 modulo p.

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Mathematica

A350523 Prime numbers p such that K(p) = 0! + 1! + ... + (p-1)! == -2 (mod p).

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A350429 Prime numbers p for which there exists at least one integer k < p such that p divides the k-th Bell number.

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Mathematica

A335383 a(n) is the number of irreducible Mersenne polynomials in GF(2)[x] that have degree n.

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PARI

Formula

A335379 a(n) is the number of Mersenne prime (irreducible) polynomials M = x^k(x+1)^(n-k)+1 of degree n in GF(2)[x] (k goes from 1 to n-1) such that Phi_7(M) has an odd number of prime divisors (omega(Phi_7(M)) is odd).

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A307671 Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^kf(k), where f(k) = harmonic(2^k) - klog(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.