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.

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A102250 Indices of semiprime Haüy rhombic dodecahedral numbers.

Original entry on oeis.org

2, 3, 4, 6, 12, 15, 16, 22, 34, 36, 51, 66, 87, 99, 100, 106, 117, 139, 141, 159, 166, 169, 174, 177, 180, 192, 201, 205, 232, 274, 282, 307, 337, 339, 342, 367, 370, 372, 379, 381, 411, 412, 429, 430, 432, 439, 444, 454, 460, 471, 477, 507, 510, 517, 555, 577
Offset: 1

Views

Author

Jonathan Vos Post, Feb 18 2005

Keywords

Comments

Because the Haüy rhombic dodecahedral numbers are A046142(n) = (2*n-1)(8*n^2-14*n+7) no Haüy rhombic dodecahedral number can be prime.
Integers n such that both (2*n-1) and (8*n^2-14*n+7) are primes.
Integers n such that (2*n-1)*(8*n^2-14*n+7) is an element in the intersection of A046142 and A001358.

Examples

			a(3) = 4 because the 3rd Haüy rhombic dodecahedral number is A046142(3) = (2*4-1)(8*4^2-14*4+7) = 553 and because 553 = 7 * 79 is a semiprime.

References

  • R.-J. Haüy, Essai d'une théorie sur la structure des crystaux appliquée à plusieurs genres de substances crystallisées, 1784.
  • H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.

Links

Crossrefs

Cf. A001358, A046142, A085601, A102130.

Programs

  • Magma
    [n: n in [0..600] | IsPrime(2*n-1) and IsPrime(8*n^2-14*n+7)]; // Vincenzo Librandi, Sep 22 2012
  • Mathematica
    Select[ Range[1000], PrimeQ[2# - 1] && PrimeQ[8#^2 - 14# + 7] &]
Select[Range[1000],AllTrue[{2#-1,8#^2-14#+7},PrimeQ]&] (* Harvey P. Dale, Apr 13 2025 *)

A171663 Expansion of (1 + 4*x - 6*x^2 - 16*x^3 + 20*x^4)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)).

Original entry on oeis.org

1, 5, 5, 13, 25, 41, 113, 145, 481, 545, 1985, 2113, 8065, 8321, 32513, 33025, 130561, 131585, 523265, 525313, 2095105, 2099201, 8384513, 8392705, 33546241, 33562625, 134201345, 134234113, 536838145, 536903681, 2147418113, 2147549185
Offset: 0

Views

Author

Jonathan Vos Post, Dec 14 2009

Keywords

Links

Crossrefs

Cf. A000040, A000215, A019434.
Cf. A092440, A085601 (bisections). - R. J. Mathar, Jan 25 2010

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2)) )); // G. C. Greubel, Jun 01 2019
  • Mathematica
    Flatten[Table[2^(2*n+1) + 1 + 2^(n+1) {-1, 1}, {n, 0, 40}]] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)
  • PARI
    my(x='x+O('x^40)); Vec((1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2))) \\ G. C. Greubel, Jun 01 2019
  • Sage
    ((1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 01 2019

Formula

G.f.: (1 + 4*x - 6*x^2 - 16*x^3 + 20*x^4)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)). - Colin Barker, Apr 27 2013

Extensions

More terms from R. J. Mathar and J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010
New name from Joerg Arndt, Jun 03 2019

A220985 The left Aurifeuillian factor of 10^(20n+10) + 1.

Original entry on oeis.org

3541, 904806804901, 99004980069800499001, 9990004998000699800049990001, 999900004999800006999800004999900001, 99999000004999980000069999800000499999000001, 9999990000004999998000000699999800000049999990000001
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding right Aurifeuillian factor is A220986.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979-A220990.

Programs

  • Mathematica
    Table[10^(8n+4) - 10^(7n+4) + 5 * 10^(6n+3) - 2 * 10^(5n+3) + 7 * 10^(4n+2) - 2 * 10^(3n+2) + 5 * 10^(2n+1) - 10^(n+1) + 1, {n, 0, 20}]

Formula

a(n) = 10^(8n+4) - 10^(7n+4) + 5 * 10^(6n+3) - 2 * 10^(5n+3) + 7 * 10^(4n+2) - 2 * 10^(3n+2) + 5 * 10^(2n+1) - 10^(n+1) + 1.
Aurifeuillian factorization: 10^(20n+10) + 1 = (10^(4n+2) + 1) * a(n) * A220986(n).

A220986 The right Aurifeuillian factor of 10^(20n + 10) + 1.

Original entry on oeis.org

27961, 1105207205101, 101005020070200501001, 10010005002000700200050010001, 1000100005000200007000200005000100001, 100001000005000020000070000200000500001000001, 10000010000005000002000000700000200000050000010000001
Offset: 0

Views

Author

Stuart Clary, Dec 27 2012

Keywords

Comments

The corresponding left Aurifeuillian factor is A220985.

Links

Crossrefs

Cf. A092440, A085601, A220978, A198410, A220979, A220980, A220981, A220982
Cf. A220983, A220984, A220985, A220987, A220988, A220989, A220990

Programs

  • Mathematica
    a[n_] := 10^(8n + 4) + 10^(7n + 4) + 5 * 10^(6n + 3) + 2 * 10^(5n + 3) + 7 * 10^(4n + 2) + 2 * 10^(3n + 2) + 5 * 10^(2n + 1) + 10^(n + 1) + 1

Formula

a(n) = 10^(8n + 4) + 10^(7n + 4) + 5 * 10^(6n + 3) + 2 * 10^(5n + 3) + 7 * 10^(4n + 2) + 2 * 10^(3n + 2) + 5 * 10^(2n + 1) + 10^(n + 1) + 1
Aurifeuillian factorization: 10^(20n + 10) + 1 = (10^(4n + 2) + 1) * A220985(n) * a(n)

A250198 Numbers k such that the right Aurifeuillian primitive part of 2^k+1 is prime.

Original entry on oeis.org

2, 6, 10, 14, 18, 22, 30, 34, 38, 42, 54, 58, 66, 70, 90, 102, 110, 114, 126, 138, 170, 178, 242, 294, 314, 326, 350, 378, 462, 566, 646, 726, 758, 1150, 1242, 1302, 1482, 1558, 1638, 1710, 1770, 1970, 1994
Offset: 1

Views

Author

Eric Chen, Jan 18 2015

Keywords

Comments

All terms are congruent to 2 modulo 4.
Let Phi_n(x) denote the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nM(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, and this is Phi_{2n}(2).
Let M(n) = the Aurifeuillian M-part of 2^n+1, M(n) = 2^(n/2) + 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).
Let M*(n) = GCD(M(n), J*(n)), this sequence lists all n such that M*(n) is prime.

Examples

			14 is in this sequence because the right Aurifeuillian primitive part of 2^14+1 is 29, which is prime.
26 is not in this sequence because the right Aurifeuillian primitive part of 2^26+1 is 8321, which equals 53 * 157 and is not prime.

Links

Crossrefs

Cf. A250197, A153443, A019320, A072226, A161508, A085601, A229768, A061443.

Programs

  • Mathematica
    Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) + 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]
  • PARI
    isok(n) = isprime(gcd(2^(n/2) + 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015
Previous Showing 21-25 of 25 results.

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