Walter Kehowski - cp's OEIS frontend

A335085 First elements of maximal isospectral chains of length 6.

1400839158600, 2902429341000, 3949885485000, 9000942048000, 10563097053600, 13554828003600, 18867199233600, 26976351213000, 37127826792000, 42966550125000, 50742170640000, 54497942553600, 56675647917000, 191546420284800, 259917211125000, 294509464704000

Offset: 1

Walter Kehowski, May 24 2020

nonn

Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of a maximal isospectral chain of length n if it is not part of an isospectral chain of length greater than 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.

			a(1) = 1400839158600 since all six numbers, 1400839158600/k, k=1..6, have spectral basis {175104894825, 184472646400, 224134265376, 200119879800, 227163106800, 179924295600, 209920069800}, while index(1400839158600/k)=k, k=1..6.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A330849, A335080, A335081, A335082, A335083, A335084.

A335084 First elements of maximal isospectral chains of length 5.

Original entry on oeis.org

5385063600, 5978343600, 6789558600, 12965853600, 31967238600, 32035143600, 37418554800, 37884558600, 44580472200, 50221710000, 69733758600, 75900423600, 77102532000, 84093966000, 85348494000, 88147278000, 89292423600, 92472078600, 98119278000, 103449198600

Offset: 1

Walter Kehowski, May 24 2020

nonn

Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of a maximal isospectral chain of length n if it is not part of an isospectral chain of length greater than 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.

			a(1) = 5385063600 since all five numbers, 5385063600/k, k=1..5, have spectral basis {1009699425, 398893600, 861610176, 769294800, 850273200, 702399600, 792892800}, while index(5385063600/k)=k, k=1..5.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A330849, A335080, A335081, A335082, A335083, A335085.

A335083 First elements of maximal isospectral chains of length 4.

Original entry on oeis.org

488880, 1525680, 2870280, 4930272, 5890248, 6374664, 8862984, 9658080, 9739080, 10338480, 10544544, 12719880, 13985712, 14777280, 15543216, 16109280, 16293600, 16682400, 16747848, 17722080, 19376136, 20822472, 22178736, 22842288, 25517232, 26056368, 26927280

Offset: 1

Walter Kehowski, May 24 2020

nonn

Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of a maximal isospectral chain of length n if it is not part of an isospectral chain of length greater than 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.

			a(1) = 488880 since all four numbers, 488880/k, k=1..4, have spectral basis {91665, 108640, 97776, 69840, 120960}, while index(488880/k)=k, k=1..4.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A330849, A335080, A335081, A335082, A335084, A335085.

A335082 First elements of maximal isospectral chains of length 3.

Original entry on oeis.org

10980, 35280, 36180, 43380, 46980, 47268, 52164, 59508, 71604, 73476, 75780, 87444, 92880, 94500, 100980, 101700, 108180, 122580, 132480, 139284, 150948, 151956, 172980, 176580, 179172, 198576, 201168, 202464, 215424, 235188, 237384, 237780, 241380, 245556

Offset: 1

Walter Kehowski, May 24 2020

nonn

Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of a maximal isospectral chain of length n if it is not part of an isospectral chain of length greater than 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.

			a(1) = 10980 since the three numbers 10980, 10980/2 = 5490, and 10980/3 = 3660 all have spectral basis {2745, 2440, 2196, 3600}, while index(10980) = 1, index(5490) = 2, and index(3660) = 3.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A330849, A335080, A335081, A335083, A335084, A335085.

A335081 First elements of maximal isospectral chains of length 2.

Original entry on oeis.org

84, 228, 280, 340, 372, 408, 468, 480, 516, 624, 740, 792, 804, 840, 868, 880, 948, 984, 1012, 1188, 1200, 1204, 1236, 1240, 1364, 1380, 1440, 1456, 1488, 1496, 1524, 1624, 1652, 1668, 1672, 1700

Offset: 1

Walter Kehowski, May 24 2020

nonn

Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of a maximal isospectral chain of length n if it is not part of an isospectral chain of length greater than 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.

			a(1) = 84 since both 84 an 84/2 = 42 have spectral basis {21,28,36}, while index(84) = 1 and index(42) = 2.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A330849, A335080, A335082, A335083, A335084, A335085.

A335080 First elements of maximal isospectral chains of length 1, or, equivalently, numbers with spectral basis of index 1.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106

Offset: 1

Walter Kehowski, May 24 2020

nonn

Isospectral Chain Conjecture: There exist isospectral chains of any positive length.
A number N is the first element of a maximal isospectral chain of length n if it is not part of an isospectral chain of length greater than 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.

			a(1) = 6 since 6 has spectral basis {3,4} and, since 3+4=1*6+1, index(6) = 1.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A330849, A335081, A335082, A335083, A335084, A335085.

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

Walter Kehowski, Feb 08 2020

nonn

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.

			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}.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A330836, A330838, A330841, A330842.

A330841 Numbers of the form 2^(2p-3)9*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

3528, 1107072, 297289728, 5065312705708032, 332036326796518490112, 85002272432789680816128, 23926103901845565010319828907592777728, 31803247166010917904914435277786533840425989636087697369118739195223867392

Offset: 1

Walter Kehowski, Jan 25 2020

nonn

a(1) = 3528 has power-spectral basis {21^2, 28^2, 48^2}, of index 1. If n > 1, then a(n) has power-spectral basis {M^2*(M+2)^2, (1/4)*M^2*(M+1)^2, (M^2-1)^2}, with index 2, where M=A000668(n+1) is the (n+1)-st Mersenne prime. The first element of the spectral basis of a(n), n > 1, is A330819(n+1), the second element is A133051(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 a(1) + 1 whenever n = 1, and 2*a(n)+1 whenever n > 1. In this case, we say that a(n) has index 1 and index 2, respectively.
a(n), n > 1, is also isospectral with 9*A133051(n), that is, a(n) and 9*A133051(n) have the same spectral basis, but 9*A133051(n) has index 1. Thus 9*A133051(n) and a(n) form an isospectral pair.

			a(2) = 2^(2*5-3)*9*31^2 = 2^7*9*31^2 = 1107072 has spectral basis {1023^2, 496^2, 960^2}, consisting of powers. The spectral sum of a(2), that is, the sum of the elements of its spectral basis, is 2*a(2)+1 = 2214145. In this case we say that a(2) has index 2. The number 9 * A330817(2) = 2^(2*5-2)*9*31^2 = 2^8*9*31^2 = 2214144 has the same spectral basis as a(2), but with index 1. We say that 9 * A330817(2) and a(2) are isospectral and form an isospectral pair.

G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A133051, A152921, A152922, A330818, A330819, A330820, 9*A133051, 9*A330817, A330837.

a := proc(n::posint)
local p, m;
p:=NumberTheory[IthMersenne](n+1);
m:=2^p-1;
return 2^(2*p-3)*9*m^2;
end;

f[p_] := 9*2^(2*p - 3)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A152922(n+1) * 9 * A133049(n+1).

A330840 a(n) = 4M(n)^2(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

Original entry on oeis.org

576, 12544, 3936256, 1057030144, 18010000731406336, 1180573606387621298176, 302230301983252198457344, 85070591651006453370026058338107654144, 113078212145816596995251325432129898099292407594978479534644406027462639616

Offset: 1

Walter Kehowski, Jan 23 2020

nonn

Also a(n+1) is the second element of the power-spectral basis of A330839(n), where by power-spectral we mean that the spectral basis consists of primes and powers.

			a(2) = 4*7^2*2^(2*3) = 2^8*7^2 = 112^2, and the spectral basis of A330839(1) = 18816 is {63^2, 112^2, 48^2}, consisting only of powers.

Garret Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly, Vol. 108, No. 4 (April 2001), pp. 336-346.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A330818, A330819, A330820, A330839.

A330840 := proc(n::posint)
  local p, m;
  p:=NumberTheory[IthMersenne](n);
  m:=2^p-1;
  return 4*m^2*(m+1)^2;
end:

f[p_] := 2^(2*p + 2)*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[9] (* Amiram Eldar, Jan 24 2020 *)

a(n) = 4 * A133049(n) * A330824(n).

A330839 Numbers of the form 2^(2p+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

18816, 5904384, 1585545216, 27015001097109504, 1770860409581431947264, 453345452974878297686016, 127605887476509680055039087507161481216, 169617318218724895492876988148194847148938611392467719301966609041193959424

Offset: 1

Walter Kehowski, Jan 21 2020

nonn

Also numbers with power-spectral basis {M_p^2*(M_p+2)^2, 4*M_p^2*(M_p+1)^2, (M_p^2-1)^2}, where by power-spectral basis we mean a spectral basis that consists of primes and powers. The first element of the power-spectral basis is A330819(n+1), the second element is A330840(n+1), and the third element is A330820(n+1).

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

			a(1) = 2^(2*3+1) * 3 * 7^2 = 18816, and 18816 has spectral basis {63^2, 112^2, 48^2}, consisting of powers.

G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.
Wikipedia, Idempotent (ring theory)
Wikipedia, Peirce decomposition

Cf. A000043, A000668, A133049, A330818, A330819, A330820, A330840.

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:

f[p_] := 2^(2p + 1)*3*(2^p - 1)^2; f /@ MersennePrimeExponent /@ Range[2, 9] (* Amiram Eldar, Jan 22 2020 *)

a(n) = A330818(n+1) * 3 * A133049(n+1).

cp's OEIS Frontend

User: Walter Kehowski

A335085 First elements of maximal isospectral chains of length 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A335084 First elements of maximal isospectral chains of length 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A335083 First elements of maximal isospectral chains of length 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A335082 First elements of maximal isospectral chains of length 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A335081 First elements of maximal isospectral chains of length 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A335080 First elements of maximal isospectral chains of length 1, or, equivalently, numbers with spectral basis of index 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A330841 Numbers of the form 2^(2*p-3)*9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330840 a(n) = 4*M(n)^2*(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A330841 Numbers of the form 2^(2p-3)9*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.

A330840 a(n) = 4M(n)^2(M(n)+1)^2, where M(n) is the n-th Mersenne prime, A000668.

A330839 Numbers of the form 2^(2p+1)3*M_p^2, where p > 2 is a Mersenne exponent, A000043, and M_p is the corresponding Mersenne prime, A000668.