cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A321488 Nonsemiprimes in A306097 = A121707 \ A267999.

Original entry on oeis.org

2057, 2873, 3689, 4015, 4991, 6137, 6851, 9401, 10795, 11033, 11501, 11837, 11849, 12341, 12593, 13481, 13795, 14297, 15113, 15695, 17155, 17633, 18011, 18377, 18469, 18941, 19097, 20009, 21463, 21641, 22661, 22919, 23273, 24089, 24521, 25721, 25993, 26381
Offset: 1

Views

Author

M. F. Hasler, Nov 11 2018

Keywords

Comments

Equivalently, terms of A321487 not in A267999, or intersection of A321487 and A306097.

Links

Crossrefs

Cf. A306097 = A121707 \ A267999, A321487 \ A267999, A321487 n A306097.

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

Original entry on oeis.org

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

Views

Author

Alexander Adamchuk, Aug 16 2006

Keywords

Comments

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

Links

Crossrefs

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_{k
Cf. A326478 for a conjectured connection with the Bernoulli numbers.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    is(n)=my(n3=n^3);sum(k=1,n-1,Mod(k,n3)^n)==0 \\ Charles R Greathouse IV, May 09 2013
  • PARI
    for(n=2, 1000, if(sum(k=1, n-1, k^n) % n^3 == 0, print1(n", "))) \\ Altug Alkan, Oct 15 2015
  • Sage
    # 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

Extensions

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

A267999 Numbers n > 1 such that gcd(n, 2^n - 2) = 1.

Original entry on oeis.org

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, 623
Offset: 1

Views

Author

Thomas Ordowski and Robert Israel, Jan 24 2016

Keywords

Comments

Odd numbers n > 1 such that gcd(n, 2^(n-1)-1) = 1.
Conjecture: this is a subsequence of A121707. Tested for all terms <= 10^5.
For n > 1, if gcd(n, 2^n-2) = 1, then n is an "anti-Carmichael number" defined: p-1 does not divide n-1 for every prime p dividing n. Generally: for k > 1, gcd(k, b^k-b) = 1 for some integer b if and only if k is an "anti-Carmichael number". - Thomas Ordowski, Aug 14 2018

Links

Crossrefs

Cf. A121707.
Cf. A306097 for terms of A121707 not in this sequence.

Programs

  • Magma
    [n: n in [2..800] | Gcd(n, 2^n-2) eq 1]; // Vincenzo Librandi, Jan 24 2016
  • Maple
    select(n -> igcd(n, 2&^n-2 mod n)=1, [seq(i,i=3..10000, 2)]);
  • Mathematica
    Select[Range[2, 768], GCD[#, 2^# - 2] == 1 &] (* or *)
Select[Range[2, 768], OddQ@ # && GCD[#, 2^(# - 1) - 1] == 1 &] (* Michael De Vlieger, Jan 24 2016 *)
  • PARI
    lista(nn) = for(n=2, nn, if(gcd(n, 2^n - 2) == 1, print1(n, ", "))); \\ Altug Alkan, Jan 24 2016

Formula

a(n) = A121707(n) for n < 62. A121707(62) = 697 = A306097(1) is the first term of A121707 not in this sequence. - 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

Views

Author

M. F. Hasler, Nov 11 2018

Keywords

Comments

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.

Links

Crossrefs

Cf. A121706, A267999, A319386, A306097.
Showing 1-4 of 4 results.

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