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.

A330849 First element of the first maximal isospectral chain of length n.

Original entry on oeis.org

6, 84, 10980, 488880, 5385063600, 348751729800, 1524738985849800
Offset: 1

Views

Author

Walter Kehowski, Feb 08 2020

Keywords

Comments

Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of the first isospectral chain of length n if there is no integer M < N such that M is also the first element of an isospectral chain of length n. Then a(n)=N, where N is the first element of the first isospectral chain of length n.
Two integers are isospectral if they have the same spectral basis. An isospectral chain of length n is a sequence N1,...,Nn of integers with the same spectral basis such that N1=2*N2=...=n*Nn and index(Nk)=k. A chain is maximal if it cannot be extended to an isospectral chain of length n+1.
The spectral sum of an integer N with at least two prime factors is the sum of the elements of its spectral basis, and is of the form k*N+1, where k is a positive integer. Then we say that N has index k, index(N)=k.

Examples

			a(1) = 6 since 6 has spectral basis {3,4} of index 1.
a(2) = 84 since 84 = 2*42 and both 84 and 42 have the spectral basis {21, 28, 36}, and 42 has index 2. Also, 84 is maximal since 84/3 = 28 has spectral basis {21, 8}.
a(3) = 10980 since 10980 = 2*5490 = 3*3660 and all three integers 10980, 5490, 3660 have spectral basis {2745, 2440, 2196, 3600}. Also, 10980 is maximal since 10980/4 = 2475 has spectral basis {2440, 2196, 855}.
a(4) = 488880 since 488880 = 2*244440 = 3*162960 = 4*122220 and all four integers 488880, 244440, 162960, 122220 have spectral basis {91665, 108640, 97776, 69840, 120960}. Also, 488880 is maximal since 488880/5 = 97776 has spectral basis {91665, 10864, 69840, 23184}.
a(5) = 5385063600 since 5385063600 = 2*2692531800 = 3*1795021200 = 4*1346265900 = 5*1077012720, and all five integers 5385063600, 2692531800, 1795021200, 1346265900, 1077012720 have spectral basis {1009699425, 398893600, 861610176, 769294800, 850273200, 702399600, 792892800}. Also, 5385063600 is maximal since 5385063600/6 = 897510600 has spectral basis {112188825, 398893600, 861610176, 769294800, 850273200, 702399600, 792892800}.
a(6) = 348751729800 since 348751729800 = 2*174375864900 = 3*116250576600 = 4*87187932450 = 5*69750345960 = 6*58125288300, and all six integers 348751729800, 174375864900, 116250576600, 87187932450, 69750345960, 58125288300 have spectral basis {43593966225, 38750192200, 41850207576, 55066062600, 56250279000, 56196961200, 57044061000}. Also, 348751729800 is maximal since 348751729800 is not divisible by 7.
a(7) = 1524738985849800 since 1524738985849800 = 2*762369492924900 = 3*508246328616600 = 4*381184746462450 = 5*304947797169960 = 6*254123164308300 = 7*217819855121400, and all seven integers 1524738985849800, 762369492924900, 508246328616600, 381184746462450, 304947797169960, 254123164308300, 217819855121400 have spectral basis {190592373231225, 169415442872200, 182968678301976, 186702732961200, 89690528579400, 196740514303200, 193276772854200, 208868354226000, 106483588520400}. Also, 1524738985849800 is maximal since 1524738985849800/8 = 190592373231225 has spectral basis {169415442872200, 182968678301976, 186702732961200, 89690528579400, 6148141071975, 2684399622975, 18275980994775, 106483588520400}.

Links

Crossrefs

Cf. A330836, A330838, A330841, A330842.

A330836 Numbers of the form 2^(2*p-1)*3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

Original entry on oeis.org

4704, 1476096, 396386304, 6753750274277376, 442715102395357986816, 113336363243719574421504, 31901471869127420013759771876790370304, 42404329554681223873219247037048711787234652848116929825491652260298489856
Offset: 1

Views

Author

Walter Kehowski, Jan 12 2020

Keywords

Comments

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2, M_p^2*(M_p+1)^2, (M_p^2-1)^2}. The first element of the spectral basis of a(n) is A330819(n+1), the second element is A330837(n+1), and the third element is A330820(n+1). Generally, a power-spectral basis is a spectral basis that consists of primes and powers.
The spectral sum of a(n), that is, the sum of the elements of its spectral basis, is 2*a(n)+1. In this case, we say that a(n) has index 2.
a(n) is also isospectral with A330838(n), that is, a(n) and A330838(n) have the same spectral basis, but A330838(n) has index 1. Thus, A330838(n) and a(n) form an isospectral pair.
Subsequence of Zumkeller numbers (A083207), since a(n) = 2^r * 3 * s, where s is relatively prime to 6. - Ivan N. Ianakiev, Feb 03 2020

Examples

			If p = 3, then a(1) = 2^(2*3-1)*3*7^2 = 4704, and the spectral basis of 4704 is {63^2, 56^2, 48^2}, consisting of powers. The spectral sum of a(1), that is, the sum of the elements of its spectral basis, is 2*4704+1 = 9409. In this case, we say that a(1) has index 2. The number A330838(1) = 9704 has the same spectral basis as a(1), but with index 1. We say that A330838(1) and a(1) are isospectral and form an isospectral pair.

Links

Crossrefs

Cf. A000043, A000668, A133049, A139306, A330819, A330820, A330837, A330838.

Programs

  • Maple
    a := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n+1);
  m:=2^p-1;
  return 2^(2*p-1)*3*m^2;
end:
  • Mathematica
    f[p_] := 2^(2*p - 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *)

Formula

a(n) = A139306(n+1) * 3 * A133049(n+1).

A330837 a(n) = M(n)^2*(M(n)+1)^2, where M(n) = A000668(n) is the n-th Mersenne prime.

Original entry on oeis.org

144, 3136, 984064, 264257536, 4502500182851584, 295143401596905324544, 75557575495813049614336, 21267647912751613342506514584526913536, 28269553036454149248812831358032474524823101898744619883661101506865659904
Offset: 1

Views

Author

Walter Kehowski, Jan 12 2020

Keywords

Comments

a(n+1) is the second element of the power-spectral basis of both A330836(n) and A330838(n). Also, a(n) = A139256(n)^2, where A139256(n) is the sum of the divisors of the n-th perfect number, A000396(n).
Also: squares of twice the perfect numbers. - M. F. Hasler, Feb 07 2020

Examples

			If p=3, then a(2) = (7*2^3)^2 = 56^2, and the spectral basis of A330836(1) = 4704 and A330838(1) = 9408 is {63^2, 56^2, 48^2}, consisting of powers.

Links

Crossrefs

Cf. A000043, A000396, A000668, A133049, A139306, A139256, A330819, A330820, A330836.

Programs

  • Maple
    a := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n);
  m:=2^p-1;
  return m^2*(m+1)^2;
end:
  • Mathematica
    f[p_] := 2^(2p)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 12 2020 *)
  • PARI
    forprime(p=1,999,isprime(2^p-1)&&print1((2^p-1)^2<<(2*p)",")) \\ M. F. Hasler, Feb 07 2020

Formula

a(n) = A330824(n) * A133049(n).
a(n) = (2*A000396(n))^2 = (2^p-1)^2*4^p with p = A000043(n). - M. F. Hasler, Feb 07 2020
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.