A267999 -id:A267999 - cp's OEIS frontend

A121707 Numbers n > 1 such that n^3 divides Sum_{k=1..n-1} k^n = A121706(n).

35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 275, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 455, 473, 475, 493, 497, 515, 517, 527, 533, 535, 539, 551, 559, 575, 581, 583, 589, 611

Offset: 1

Alexander Adamchuk, Aug 16 2006

nonn

All terms belong to A038509 (Composite numbers with smallest prime factor >= 5). Many but not all terms belong to A060976 (Odd nonprimes, c, which divide Bernoulli(2*c)).
Many terms are semiprimes:
- the non-semiprimes are {275, 455, 475, 539, 575, 715, 775, 875, 935, ...}: see A321487;
- semiprime terms that are multiples of 5 have indices {7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, ...} = A002145 (Primes of form 4*k + 3, except 3, or k > 0; or Primes which are also Gaussian primes);
- semiprime terms that are multiples of 7 have indices {5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, ...} = A003627 (Primes of form 3*k - 1, except 2, or k > 1);
- semiprime terms that are multiples of 11 have indices {5, 7, 13, 17, 19, 23, 37, 43, 47, 53, 59, 67, 73, 79, 83, ...} = Primes of the form 4*k + 1 and 4*k - 1. [Edited by Michel Marcus, Jul 21 2018, M. F. Hasler, Nov 09 2018]
Conjecture: odd numbers n > 1 such that n divides Sum_{k=1..n-1} k^(n-1). - Thomas Ordowski and Robert Israel, Oct 09 2015. Professor Andrzej Schinzel (in a letter to me, dated Dec 29 2015) proved this conjecture. - Thomas Ordowski, Jul 20 2018
Note that n^2 divides Sum_{k=1..n-1} k^n for every odd number n > 1. - Thomas Ordowski, Oct 30 2015
Conjecture: these are "anti-Carmichael numbers" defined; n > 1 such that p - 1 does not divide n - 1 for every prime p dividing n. Equivalently, odd numbers n > 1 such that n is coprime to A027642(n-1). A number n > 1 is an "anti-Carmichael" if and only if gcd(n, b^n - b) = 1 for some integer b. - Thomas Ordowski, Jul 20 2018
It seems that these numbers are all composite terms of A317358. - Thomas Ordowski, Jul 30 2018
a(62) = 697 is the first term not in A267999: see A306097 for all these terms. - M. F. Hasler, Nov 09 2018
If the conjecture from Thomas Ordowski is true, then no term is a multiple of 2 or 3. - Jianing Song, Jan 27 2019
Conjecture: an odd number n > 1 is a term iff gcd(n, A027642(n-1)) = 1. - Thomas Ordowski, Jul 19 2019
Conjecture: Sequence consists of numbers n > 1 such that r = b^n + n - b will produce a prime for one or more integers b > 1. Only when n is in this sequence do one or more prime factors of n fail to divide r for all b.  Also, n and b must be coprime for r to be prime. The above also applies to r = b^n - n - b, ignoring n=3, b=2. - Richard R. Forberg, Jul 18 2020
Odd numbers n > 1 such that Sum_{k(even)=2..n-1}2*k^(n-1) == 0 (mod n). - Davide Rotondo, Oct 28 2020
What is the asymptotic density of these numbers? The numbers A267999 have a slightly lower density. The difference between the densities is equal to the density of the numbers A306097. - Thomas Ordowski, Feb 15 2021
The asymptotic density of this sequence is in the interval (0.253, 0.265) (Ordowski, 2021). - Amiram Eldar, Feb 26 2021

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1371 terms from Robert Israel)
T. Ordowski, Density of anti-Carmichael numbers, SeqFan Mailing List, Feb 17 2021.
Don Reble, Comments on A121707

Cf. A000312, A002145, A002997, A027642, A031971, A038509, A060976, A121706, A267999 (probably a subsequence).
Cf. A306097 for terms of this sequence A121707 not in sequence A267999, A321487 for terms which are not semiprimes.
Cf. A191677 (n divides Sum_{kCf. A326478 for a conjectured connection with the Bernoulli numbers.

filter:= n -> add(k &^ n mod n^3, k=1..n-1) mod n^3 = 0:
select(filter, [$2..1000]); # Robert Israel, Oct 08 2015

fQ[n_] := Mod[Sum[PowerMod[k, n, n^3], {k, n - 1}], n^3] == 0; Select[
Range[2, 611], fQ] (* Robert G. Wilson v, Apr 04 2011 and slightly modified Aug 02 2018 *)

is(n)=my(n3=n^3);sum(k=1,n-1,Mod(k,n3)^n)==0 \\ Charles R Greathouse IV, May 09 2013

for(n=2, 1000, if(sum(k=1, n-1, k^n) % n^3 == 0, print1(n", "))) \\ Altug Alkan, Oct 15 2015

# after Andrzej Schinzel
def isA121707(n):
    if n == 1 or is_even(n): return False
    return n.divides(sum(k^(n-1) for k in (1..n-1)))
[n for n in (1..611) if isA121707(n)] # Peter Luschny, Jul 18 2019

Sequence corrected by Robert G. Wilson v, Apr 04 2011

A108574 Range of A000790 (primary pretenders).

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 38, 39, 46, 49, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 106, 111, 118, 121, 122, 123, 129, 133, 134, 141, 142, 145, 146, 158, 159, 166, 169, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 217, 218, 219, 226, 237, 249, 254, 259, 262, 265, 267, 274, 278, 289, 291, 298, 301, 302, 303, 305, 309, 314, 321, 326, 327, 334, 339, 341, 346, 358, 361, 362, 365, 381, 382, 386, 393, 394, 398, 411, 417, 422, 427, 445, 446, 447, 451, 453, 454, 458, 466, 469, 471, 478, 481, 482, 485, 489, 501, 502, 505, 511, 514, 519, 526, 529, 537, 538, 542, 543, 545, 553, 554, 561

Offset: 1

David W. Wilson, Jun 10 2005

All terms except for the last term, 561, are semiprimes (A001358). Semiprimes up to 559 that are not here: 35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 473, 493, 497, 515, 517, 527, 533, 535, 551, 559. - Zak Seidov, Jan 08 2015
The LCM of all terms is 23# * 277# (where # denotes the primorial function A034386), the period of A000790, and therefore also of the related sequence b(n) = gcd(A000790(n), n). - M. F. Hasler, Feb 16 2018
Range of A295997. - Thomas Ordowski, Feb 27 2018
These numbers k < 561 are semiprimes k = pq such that p-1 | q-1, where primes p <= q. Equivalent condition is p-1 | k-1. - Thomas Ordowski, Aug 18 2018
This shows that all even semiprimes < 561 are in this sequence. The odd semiprimes not in this sequence are the semiprimes (equivalently: all terms but 275, 455, 475, 539) less than 561 in A267999 (which equals A121707 up to 695). - M. F. Hasler, Nov 09 2018

J. H. Conway, R. K. Guy, W. A. Schneeberger and N. J. A. Sloane, The Primary Pretenders, Acta Arith. 78 (1997), 307-313.
J. H. Conway, R. K. Guy, W. A. Schneeberger and N. J. A. Sloane, The Primary Pretenders, arXiv:math/0207180 [math.NT], 2002.

Cf. A000790, A001358, A295997.

pp[n_] := For[c = 4, True, c = If[PrimeQ[c+1], c+2, c+1], If[PowerMod[n, c, c] == Mod[n, c], Return[c]]];seq[n_] := seq[n] = Table[pp[k], {k, 0, 2^n}] // Union; seq[10]; seq[n = 11]; While[ Print["n = ", n, " more terms: ", Complement[seq[n], seq[n-1]]]; seq[n] != seq[n-1], n++]; A108574 = seq[n] (* Jean-François Alcover, Oct 18 2013 *)

my(A=List(561)); forprime(q=2,561\2, forprime(p=2,min(q,561\q), (q-1)%(p-1)|| listput(A, p*q))); A108574=Set(A) \\ M. F. Hasler, Nov 09 2018

A321487 Numbers in A121707 (n^3 > 1 divides Sum_{k=1..n-1} k^n) which are not semiprimes.

Original entry on oeis.org

275, 455, 475, 539, 575, 715, 775, 875, 935, 1075, 1127, 1175, 1235, 1295, 1375, 1421, 1463, 1475, 1495, 1547, 1595, 1615, 1675, 1715, 1775, 1859, 1955, 1975, 2009, 2015, 2035, 2057, 2075, 2093, 2135, 2255, 2261, 2299, 2303, 2375, 2387, 2555, 2575, 2597, 2635, 2639, 2675, 2717, 2783

Offset: 1

M. F. Hasler, Nov 11 2018

nonn

Most terms of A121707 and its (conjectured) subsequence A267999 are semiprimes. This sequence lists the exceptions.
At first, it looked as if most terms were multiples of 5. The first exceptions are a({4, 11, 16}) = {539, 1127, 1421}. However, after the first 30 terms, almost every other term is not divisible by 5.
The first terms not in A267999 are {2057, 2873, 3689, 4015, 4991, 6137, ...}, cf. A321488.

M. F. Hasler, Table of n, a(n) for n = 1..2500

Cf. A121706, A267999, A319386, A306097.

A214606 a(n) = gcd(n, 2^n - 2).

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 14, 29, 2, 31, 2, 3, 2, 1, 2, 37, 2, 3, 2, 41, 2, 43, 2, 15, 2, 47, 2, 7, 2, 3, 2, 53, 2, 1, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 14, 71, 2, 73, 2, 3, 2, 1, 2, 79

Offset: 1

Alex Ratushnyak, Jul 22 2012

Greatest common divisor of n and 2^n - 2.
a(n)=n iff n=1 or n is prime or n is Fermat pseudoprime to base 2 or even pseudoprime to base 2. - Corrected by Thomas Ordowski, Jan 25 2016
Indices of 1's: A121707 preceded by 1. - False, see A267999.
Numbers n such that a(n) does not equal A020639(n) (the least prime factor of n): A146077.

			a(3) = 3 because 2^3 - 2 = 6 and gcd(3, 6) = 3.
a(4) = 2 because 2^4 - 2 = 14 and gcd(4, 14) = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000918, A064535, A121707, A020639, A146077.

import java.math.BigInteger;
public class A214606 {
  public static void main (String[] args) {
    BigInteger c1 = BigInteger.valueOf(1);
    BigInteger c2 = BigInteger.valueOf(2);
    for (int n=0; n<222; n++) {
      BigInteger bn=BigInteger.valueOf(n),pm2=c1.shiftLeft(n).subtract(c2);
      System.out.printf("%s, ", bn.gcd(pm2).toString());
    }
  }
}

[GCD(n, 2^n-2): n in [1..80]]; // Vincenzo Librandi, Jan 26 2016

seq(igcd(n, (2&^n - 2) mod n), n=1 .. 1000); # Robert Israel, Jan 26 2016

Table[GCD[n, 2^n - 2], {n, 1, 59}] (* Alonso del Arte, Jul 22 2012 *)

a(n)=gcd(n,lift(Mod(2,n)^n-2)) \\ Charles R Greathouse IV, May 29 2014

A316111 a(n) is the smallest k > 1 such that gcd(k, n^k - n) = 1, for n > 1.

Original entry on oeis.org

35, 35, 77, 77, 143, 55, 55, 77, 119, 119, 35, 55, 187, 143, 77, 35, 35, 77, 143, 247, 95, 35, 77, 77, 77, 55, 55, 143, 77, 77, 35, 35, 247, 143, 143, 35, 35, 77, 77, 143, 55, 95, 119, 119, 77, 35, 55, 143, 143, 77, 35, 35, 119, 299, 221, 55, 35, 77, 77, 77, 55, 55, 187, 119

Offset: 2

Thomas Ordowski, Aug 13 2018

nonn

Conjecture: all the terms are in A121707. If k is a term, then k is an "anti-Carmichael number": p-1 does not divide k-1 for every prime p dividing k.
The sequence is unbounded, since a(m!) > m.
Prediction: a(n) < n for all sufficiently large n.
GCD(n, a(n)) = 1. a(n) is odd. Is a(n) squarefree? - David A. Corneth, Aug 13 2018

Cf. A121707, A267999.

a(n) = {my(k=2); while (gcd(k, n^k - n) != 1, k++); k;} \\ Michel Marcus, Aug 13 2018

a(n) = {my(k=3); while (gcd(k, n^k - n) != 1, k+=2; while(gcd(n, k) > 1, k+=2)); k} \\ David A. Corneth, Aug 13 2018

More terms from Michel Marcus, Aug 13 2018

A316348 a(n) is the smallest k > 1 such that gcd(k, m^k - m) = 1 for all m = 2,...,n.

Original entry on oeis.org

35, 35, 77, 77, 143, 143, 143, 143, 299, 299, 323, 323, 323, 323, 437, 437, 667, 667, 667, 667, 899, 899, 899, 899, 899, 899, 1457, 1457, 1739, 1739, 1739, 1739, 1739, 1739, 1763, 1763, 1763, 1763, 2021, 2021, 2491, 2491, 2491, 2491, 3127, 3127, 3127, 3127, 3127

Offset: 2

Thomas Ordowski, Aug 13 2018

nonn

Conjecture: all the terms are in A121707.
From David A. Corneth, Aug 13 2018: (Start)
GCD(n, a(n)) = 1. a(n) is odd.
Is a(n) squarefree?
a(n+1) >= a(n) by definition. (End)
It seems that a(prime(n+1)-1) > a(prime(n)-1) for n > 1. - Thomas Ordowski, Aug 13 2018

Michel Marcus, Table of n, a(n) for n = 2..306

Cf. A121707, A267999, A316111.

isok(k, n)= {for (m=2, n, if (gcd(k, m^k - m) != 1, return (0));); return(1);}
a(n) = {my(k=2); while (! isok(k, n), k++); k;} \\ Michel Marcus, Aug 13 2018

Conjecture: a(n) ~ n^2.

More terms from Michel Marcus, Aug 13 2018

A318055 Numbers k such that gcd(k, 2^k - 2) = 1 and gcd(k, 3^k - 3) > 1.

Original entry on oeis.org

247, 403, 559, 715, 871, 1027, 1339, 1495, 1651, 1807, 1963, 2009, 2035, 2119, 2587, 2743, 2899, 2993, 3055, 3211, 3523, 3649, 3679, 3835, 3977, 3991, 4147, 4303, 4331, 4453, 4615, 4633, 4699, 4771, 4927, 5239, 5395, 5617, 5707, 5863, 5995, 6019, 6031, 6161, 6331, 6487, 6799, 6929, 6955, 7081, 7111

Offset: 1

Thomas Ordowski, Aug 14 2018

nonn

Odd numbers k such that gcd(k,2^(k-1)-1) = 1 and gcd(k,3^(k-1)-1) > 1.
It seems that a(n) == 91 (mod 156) for infinitely many n.
Fermat pseudoprimes to base 3 (A005935) in this sequence are 16531, 49051, 72041, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A267999 and probably of A121707.
Cf. A139613(2n+1): it gives many terms of the sequence.
Cf. A005935.

Filtered([1..10000],k->Gcd(k,2^k-2) = 1 and Gcd(k,3^k-3) > 1);  # Muniru A Asiru, Oct 07 2018

select(k->gcd(k,2^k-2) = 1 and gcd(k,3^k-3) > 1,[$1..10000]); # Muniru A Asiru, Oct 07 2018

Select[Range[8000], GCD[#, 2^# - 2] == 1 && GCD[#, 3^# - 3] > 1 &] (* Amiram Eldar, Mar 31 2024 *)

isok(k) = (gcd(k,2^k-2) == 1) && (gcd(k,3^k-3) != 1); \\ Michel Marcus, Aug 14 2018

More terms from Michel Marcus, Aug 14 2018

A260867 Least k > 1 that divides A260868(n) + 2^k - 2.

Original entry on oeis.org

35, 161, 55, 35, 115, 35, 115, 35, 77, 209, 473, 253, 55, 77, 35, 235, 247, 55, 35, 35, 899, 119, 1003, 415, 143, 35, 335, 95, 299, 497, 203, 575, 35, 247, 323, 95, 77, 437, 901, 35, 55, 473, 35, 1457, 77, 55, 517, 35, 235, 493, 161, 535, 209, 115, 95, 1067, 689, 323, 35, 1199, 1313, 355, 77, 815, 635, 869, 235, 119, 551, 55, 115

Offset: 1

M. F. Hasler, Aug 11 2015

nonn

For all numbers N not listed in A260868, the least k > 1 that divides N + 2^k - 2 is equal to the least prime factor of N.

It appears that the range of this sequence is A267999. For example, 155 occurs first somewhat late for N = 2729. - Corrected by Thomas Ordowski, Oct 27 2018

my(aa(n)=for(k=2,9e9,Mod(2,k)^k+n-2||return(k)));for(n=2,1e5,aa(n)==factor(n)[1,1]||print1(aa(n)","))

Showing 1-10 of 10 results.

cp's OEIS Frontend

A306097 Terms of A121707 not in A267999.

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A321488 Nonsemiprimes in A306097 = A121707 \ A267999.

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A121707 Numbers n > 1 such that n^3 divides Sum_{k=1..n-1} k^n = A121706(n).

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A108574 Range of A000790 (primary pretenders).

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A321487 Numbers in A121707 (n^3 > 1 divides Sum_{k=1..n-1} k^n) which are not semiprimes.

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A214606 a(n) = gcd(n, 2^n - 2).

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A316111 a(n) is the smallest k > 1 such that gcd(k, n^k - n) = 1, for n > 1.

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A316348 a(n) is the smallest k > 1 such that gcd(k, m^k - m) = 1 for all m = 2,...,n.

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