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.

User: Matthew Niemiro

Matthew Niemiro's wiki page.

Matthew Niemiro has authored 4 sequences.

A338413 Number of 2 X 2 matrices with integer entries in [-n,n] that are diagonalizable over the complex numbers.

Original entry on oeis.org

65, 569, 2281, 6313, 14265, 28033, 49921, 82545, 128945, 192809, 277849, 388185, 528617, 704049, 919857, 1181393, 1495569, 1868249, 2306921, 2818441, 3410809, 4091937, 4870273, 5754449, 6753233, 7877641, 9136441, 10540633, 12101001, 13828465, 15734545, 17830353, 20129713, 22644553, 25387929
Offset: 1

Views

Author

Matthew Niemiro, Nov 07 2020

Keywords

Comments

A diagonalizable matrix A is one which can be expressed as XDY, where D is a diagonal matrix and X = Y^-1 are square matrices. By 'diagonalizable over C,' it is meant that the matrix D has complex entries.
The nondiagonalizable 2 x 2 matrices are the nondiagonal ones whose characteristic polynomial has discriminant 0. - Robert Israel, Nov 12 2020

Links

Crossrefs

a(1) is given by A091470(2).

Programs

  • Maple
    N:= 30: # for a(1)..a(N)
V:= Vector(N):
for t from 1 to N do
  for d in select(`<=`,numtheory:-divisors(t^2),N) do
    for n from max(d, t^2/d) to N do
      V[n]:= V[n] + (8*(n-t)+4)
od od od:
for n from 1 to N do V[n]:= (2*n+1)^4 - (V[n] + 4*n*(2*n+1)) od:
convert(V,list); # Robert Israel, Nov 12 2020
  • Mathematica
    a[n_] := Length[Select[Tuples[Tuples[Range[-n, n], 2], 2], DiagonalizableMatrixQ]];

Extensions

More terms from Robert Israel, Nov 12 2020

A331547 Numbers k such that k and k! - 1 have the same number of divisors.

Original entry on oeis.org

3, 7, 8, 10, 26, 27, 34, 85, 93, 104, 143, 152
Offset: 1

Views

Author

Matthew Niemiro, Jan 20 2020

Keywords

Comments

The sequence also includes: 143, 152, 186, 230, 379, 381, 543, 573, 602. - Daniel Suteu, Jan 21 2020
The sequence also includes 2881. Even though the complete factorization of 136!-1 is not known, 136 is not a term, since 136!-1 is known to be the product of 2 distinct primes and a composite number, so it has at least 4 prime factors and 3 distinct prime factors, thus the number of divisors >= 12, whereas 136 has 8 divisors. - Chai Wah Wu, Feb 26 2020
Similar reasoning (considering only small prime factors of k! - 1) shows that the next terms (> a(12) = 152) can only be within the set {154, 160, 162, 164, 176, 180, 182, 186, 187, 188, 192, 195, 196, 198, 204, ...}. - M. F. Hasler, Feb 26 2020

Links

Crossrefs

Supersequence of A103317.
Cf. A000005, A002982, A064145.

Programs

  • Mathematica
    Select[Range[50], DivisorSigma[0, #] - DivisorSigma[0, Factorial[#] - 1] == 0 &]
  • PARI
    isok(k) = k>1 && numdiv(k)==numdiv(k!-1); \\ Jinyuan Wang, Jan 20 2020
  • PARI
    {is(n)=my(f); n>2&& numdiv(n)>=numdiv(f=factor(n!-1,0))&& if( ispseudoprime(vecmax(f[,1])), numdiv(n)==numdiv(f), numdiv(n)<2*numdiv(f), 0, numdiv(n)==numdiv(n!-1))} \\ Avoids complete factorization if possible. - M. F. Hasler, Feb 26 2020

Formula

A331547 = { n > 1 | A000005(n) = A064145(n) }. - M. F. Hasler, Feb 26 2020

Extensions

a(8)-a(9) from Jinyuan Wang, Jan 20 2020
a(10) from Amiram Eldar, Jan 20 2020
a(11)-a(12) from Chai Wah Wu, Feb 26 2020

A330715 a(1), a(2), a(3) = 1; a(n) = (a(n-1) mod a(n-3)) + a(n-2) + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 6, 7, 10, 12, 16, 19, 24, 28, 34, 39, 46, 52, 60, 67, 76, 84, 94, 103, 114, 124, 136, 147, 160, 172, 186, 199, 214, 228, 244, 259, 276, 292, 310, 327, 346, 364, 384, 403, 424, 444, 466, 487, 510, 532, 556, 579, 604, 628, 654, 679, 706, 732
Offset: 1

Views

Author

Matthew Niemiro, Dec 27 2019

Keywords

Links

Programs

  • Mathematica
    Nest[Append[#, Mod[#[[-1]], #[[-3]] ] + #[[-2]] + 1] &, {1, 1, 1}, 57] (* Michael De Vlieger, Dec 27 2019 *)
nxt[{a_,b_,c_}]:={b,c,Mod[c,a]+b+1}; NestList[nxt,{1,1,1},60][[;;,1]] (* Harvey P. Dale, Jul 18 2025 *)
  • Python
    x = 1
y = 1
z = 1
for i in range(4, 1001):
    new = z % x + y + 1
    print(str(i) +" " + str(new))
    x = y
    y = z
    z = new

Formula

a(1), a(2), a(3) = 1; a(n) = (a(n-1) mod a(n-3)) + a(n-2) + 1.
Conjectures from Colin Barker, Dec 28 2019: (Start)
G.f.: x*(1 - x - x^2 + 2*x^3 - x^4 + x^6 - 2*x^7 + 2*x^8) / ((1 - x)^3*(1 + x)).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>9.
a(n) = (99 - 3*(-1)^n - 24*n + 2*n^2) / 8 for n>5.
(End)

Extensions

More terms from Michael De Vlieger, Dec 27 2019

A330305 Decimal expansion of simple continued fraction transform of golden ratio.

Original entry on oeis.org

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

Views

Author

Matthew Niemiro, Dec 13 2019

Keywords

Comments

Decimal expansion of 1+1/(6+1/(1+1/(8+1/(...)))).

Examples

			1.14453286514339078255387896629437609713997399753057764689245626533862191268948...

Crossrefs

Cf. A001622 (golden ratio).

Programs

  • Mathematica
    RealDigits[N[FromContinuedFraction[RealDigits[GoldenRatio, 10, 1000][[1]]], 120]][[1]] (* Vaclav Kotesovec, Dec 23 2019 *)

Extensions

More digits from Vaclav Kotesovec, Dec 23 2019

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