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 1-3 of 3 results.

A035192 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 10.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 0, 2, 2, 0, 2, 1, 0, 3, 0, 1, 0, 0, 0, 2, 1, 2, 4, 0, 0, 2, 2, 1, 0, 0, 0, 3, 2, 0, 4, 1, 2, 0, 2, 0, 3, 0, 0, 2, 1, 1, 0, 2, 2, 4, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 2, 0, 2, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 4, 2, 1, 5
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 40. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Links

Crossrefs

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A038880, A097955.

Programs

  • Mathematica
    a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[10, #] &]]; Table[ a[n], {n, 1, 100}] (* G. C. Greubel, Apr 27 2018 *)
  • PARI
    my(m=10); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    a(n) = sumdiv(n, d, kronecker(10, d)); \\ Amiram Eldar, Nov 18 2023

Formula

From Amiram Eldar, Nov 18 2023: (Start)
a(n) = Sum_{d|n} Kronecker(10, d).
Multiplicative with a(p^e) = 1 if Kronecker(10, p) = 0 (p = 2 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(10, p) = -1 (p is in A038880), and a(p^e) = e+1 if Kronecker(10, p) = 1 (p is in A097955).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(sqrt(10)+3)/sqrt(10) = 1.1500865228... . (End)

A097443 Half-period primes, i.e., primes p for which the decimal expansion of 1/p has period (p-1)/2.

Original entry on oeis.org

3, 13, 31, 43, 67, 71, 83, 89, 107, 151, 157, 163, 191, 197, 199, 227, 283, 293, 307, 311, 347, 359, 373, 401, 409, 431, 439, 443, 467, 479, 523, 557, 563, 569, 587, 599, 601, 631, 653, 677, 683, 719, 761, 787, 827, 839, 877, 881, 883, 911, 919, 929, 947, 991
Offset: 1

Views

Author

Julien Peter Benney (jpbenney(AT)ftml.net), Aug 23 2004

Keywords

Comments

Primes p for which 10 has multiplicative order (p-1)/2. - Robert Israel, Jul 15 2016

Examples

			13 is a half-period prime because 1/13 = 0.076923076923076923076923..., which has period 6, or (13-1)/2.

Links

Crossrefs

Almost the same as A001914.
Cf. A001913, A055628, A056157, A056210-A056217, A098680
Cf. also A000040, A007732, A006883, A097443, A097955.

Programs

  • Maple
    select(t -> isprime(t) and numtheory:-order(10, t) = (t-1)/2,
[seq(t,t = 3..1000,2)]); # Robert Israel, Jul 15 2016
  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 200]], f[ # ] == 2 &] (* Robert G. Wilson v, Sep 14 2004 *)
  • PARI
    is(n)= gcd(10,n)==1 && isprime(n) && znorder(Mod(10,n))==(n-1)/2 \\ Dana Jacobsen, Jul 19 2016
  • Perl
    use ntheory ":all"; forprimes { say if znorder(10,$) == ($-1)/2; } 1,1000; # Dana Jacobsen, Jul 19 2016

Extensions

Edited (including prepending 3), at the suggestion of Georg Fischer, by N. J. A. Sloane, Oct 19 2018

A262362 Real positive integers with more than one factorization in Z[sqrt(10)].

Original entry on oeis.org

6, 9, 10, 12, 15, 18, 20, 24, 26, 27, 30, 36, 40, 42, 45, 48, 50, 52, 54, 60, 63, 66, 70, 72, 74, 75, 78, 80, 81, 84, 86, 90, 96, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 117, 120, 126, 130, 132, 134, 135, 138, 140, 144, 148, 150, 153, 156, 159, 160, 162, 165, 166, 168, 170
Offset: 1

Views

Author

Alonso del Arte, Dec 23 2015

Keywords

Comments

To count as distinct from another factorization, a factorization must not be derived from the other by multiplication by units. For example, (4 - sqrt(10))(4 + sqrt(10)) is not distinct from (-1)(2 - sqrt(10))(2 + sqrt(10)) as a factorization of 6 because -3 - sqrt(10) is a unit and (2 - sqrt(10))(-3 - sqrt(10)) = 4 + sqrt(10).
Given a number p that is prime in Z, if x^2 == 10 mod p has solutions in Z, then some multiples of p are in this sequence. If x is the smallest solution, then x^2 - 10 gives the smallest multiple of p in this sequence not divisible by any prior term. For example, 6^2 == 10 mod 13, and 26 = 2 * 13 = (6 - sqrt(10))(6 + sqrt(10)).
If a number is in this sequence, then so are all its real positive integer multiples. The negative multiples also have more than one factorization, but of course one has to remember to put in the -1 as needed. Since Z[sqrt(10)] has units of norm -1, it is then possible to "shop" the units to include or exclude -1 from the factorization.
Z[sqrt(10)] has class number 2. This means that while a number may have more than one factorization, all factorizations have the same number of nonunit irreducible factors. If one factorization seems to have fewer factors, then it is an incomplete factorization.

Examples

			9 = 3^2 = (-1)(1 - sqrt(10))(1 + sqrt(10)), so 9 is in the sequence.
10 = 2 * 5 = (sqrt(10))^2, so 10 is in the sequence.

Crossrefs

Cf. A097955, A262828.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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