A335082 -id:A335082 - cp's OEIS frontend

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

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.

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.

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.

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.

A335085 First elements of maximal isospectral chains of length 6.

Original entry on oeis.org

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.

cp's OEIS Frontend

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

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A335081 First elements of maximal isospectral chains of length 2.

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A335083 First elements of maximal isospectral chains of length 4.

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A335084 First elements of maximal isospectral chains of length 5.

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A335085 First elements of maximal isospectral chains of length 6.

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