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 1-7 of 7 results.

A290096 Filter-sequence related to cycle-structure of permutations listed in table A055089: Least number with the same prime signature as A290095.

Original entry on oeis.org

2, 4, 12, 8, 8, 12, 60, 36, 24, 16, 16, 24, 24, 16, 60, 24, 36, 16, 16, 24, 24, 60, 16, 36, 420, 180, 180, 72, 72, 180, 120, 72, 48, 32, 32, 48, 48, 32, 120, 48, 72, 32, 32, 48, 48, 120, 32, 72, 120, 72, 48, 32, 32, 48, 420, 180, 120, 48, 48, 120, 180, 72, 48, 32, 32, 72, 72, 180, 32, 48, 72, 32, 48, 32, 120, 48, 72, 32
Offset: 0

Views

Author

Antti Karttunen, Aug 17 2017

Keywords

Links

Crossrefs

Cf. A046523, A060126, A290095, A290097 (rgs-transform of this sequence).
Other filter-sequences related to factorial base and finite permutations: A278225, A278234, A278235, A278236.

Formula

a(n) = A046523(A290095(n)).
a(n) = A278225(A060126(n)).

A072411 LCM of exponents in prime factorization of n, a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 3
Offset: 1

Views

Author

Labos Elemer, Jun 17 2002

Keywords

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 14, 168, 1779, 17959, 180665, 1808044, 18084622, 180856637, 1808585068, 18085891506, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.8085... . - Amiram Eldar, Sep 10 2022

Examples

			n = 288 = 2*2*2*2*2*3*3; lcm(5,2) = 10; Product(5,2) = 10, max(5,2) = 5;
n = 180 = 2*2*3*3*5; lcm(2,2,1) = 2; Product(2,2,1) = 4; max(2,2,1) = 2; it deviates both from maximum of exponents (A051903, for the first time at n=72), and product of exponents (A005361, for the first time at n=36).
For n = 36 = 2*2*3*3 = 2^2 * 3^2 we have a(36) = lcm(2,2) = 2.
For n = 72 = 2*2*2*3*3 = 2^3 * 3^2 we have a(72) = lcm(2,3) = 6.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(2,4) = 4.
For n = 360 = 2^3 * 3^2 * 5^1 we have a(360) = lcm(1,2,3) = 6.

Links

Crossrefs

Cf. A028234, A067029, A072412-A072414, A273058, A284569, A290103.
Similar sequences: A001222 (sum of exponents), A005361 (product), A051903 (maximal exponent), A051904 (minimal exponent), A052409 (gcd of exponents), A267115 (bitwise-and), A267116 (bitwise-or), A268387 (bitwise-xor).
Cf. also A055092, A060131.
Differs from A290107 for the first time at n=144.
After the initial term, differs from A157754 for the first time at n=360.

Programs

  • Mathematica
    Table[LCM @@ Last /@ FactorInteger[n], {n, 2, 100}] (* Ray Chandler, Jan 24 2006 *)
  • PARI
    a(n) = lcm(factor(n)[,2]); \\ Michel Marcus, Mar 25 2017
  • Python
    from sympy import lcm, factorint
def a(n):
    l=[]
    f=factorint(n)
    for i in f: l+=[f[i],]
    return lcm(l)
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 25 2017

Formula

a(1) = 1; for n > 1, a(n) = lcm(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 09 2016
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A284569(A156552(n)).
a(n) = A290103(A181819(n)).
a(A289625(n)) = A002322(n).
a(A290095(n)) = A055092(n).
a(A275725(n)) = A060131(n).
a(A260443(n)) = A277326(n).
a(A283477(n)) = A284002(n). (End)

Extensions

a(1) = 1 prepended and the data section filled up to 120 terms by Antti Karttunen, Aug 09 2016

A055091 Minimum number of transpositions needed to represent each permutation given in reversed colexicographic ordering A055089.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 3, 1, 2, 2, 3, 3, 2, 2, 1, 3, 2, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 4, 2, 3, 3, 4, 4, 3, 3, 2, 4, 3, 2, 3, 3, 4, 4, 3, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 2, 4, 3, 3, 4, 3, 4, 2, 3, 3, 4, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 3, 3, 4, 2, 3, 4, 3, 3, 2, 4, 3, 3, 2, 2
Offset: 0

Views

Author

Antti Karttunen, Apr 18 2000

Keywords

Links

Crossrefs

Cf. A046660, A055089, A055090, A055093, A060130, A290095.
Cf. also A034968 (minimum number of adjacent transpositions).

Programs

  • Maple
    with(group); [seq(count_transpositions(convert(PermRevLexUnrank(j),'disjcyc')),j=0..)];
count_transpositions := proc(l) local c,t; t := 0; for c in l do t := t + (nops(c)-1); od; RETURN(t); end;
# Procedure PermRevLexUnrank given in A055089.

Formula

a(n) = A055093(n) - A055090(n).
a(n) = A046660(A290095(n)) = A060130(A060126(n)). - Antti Karttunen, Dec 30 2017

Extensions

Entry revised by Antti Karttunen, Dec 30 2017

A055093 Number of moved (non-fixed) elements in each permutation given in reversed colexicographic ordering A055089, i.e., the sum of their cycle lengths (excluding the 1-cycles, i.e., fixed elements).

Original entry on oeis.org

0, 2, 2, 3, 3, 2, 2, 4, 3, 4, 4, 3, 3, 4, 2, 3, 4, 4, 4, 3, 3, 2, 4, 4, 2, 4, 4, 5, 5, 4, 3, 5, 4, 5, 5, 4, 4, 5, 3, 4, 5, 5, 5, 4, 4, 3, 5, 5, 3, 5, 4, 5, 5, 4, 2, 4, 3, 4, 4, 3, 4, 5, 4, 5, 5, 5, 5, 4, 5, 4, 5, 5, 4, 5, 3, 4, 5, 5, 3, 4, 2, 3, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 4, 4, 5, 4, 4, 3, 5, 5, 4, 3, 3
Offset: 0

Views

Author

Antti Karttunen, Apr 04 2000

Keywords

Comments

Also number of displacements for permutations in lexicographic order. - Joerg Arndt, Jan 22 2024

Links

Crossrefs

Cf. A055089, A055090, A055091, A055092, A060129, A275812, A290095.

Programs

  • Maple
    A055093(n) = count_nonfixed(convert(PermRevLexUnrank(j), 'disjcyc')).
count_nonfixed := l -> convert(map(nops,l), `+`);
# Procedure PermRevLexUnrank given in A055089.

Formula

a(n) = A055090(n) + A055091(n).
a(n) = A275812(A290095(n)) = A060129(A060126(n)). - Antti Karttunen, Dec 30 2017

Extensions

Entry revised by Antti Karttunen, Dec 30 2017

A055092 Order of each permutation given in reversed colexicographic ordering A055089, i.e., the least common multiple of their cycle lengths.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 2, 2, 3, 4, 4, 3, 3, 4, 2, 3, 2, 4, 4, 3, 3, 2, 4, 2, 2, 2, 2, 6, 6, 2, 3, 6, 4, 5, 5, 4, 4, 5, 3, 4, 6, 5, 5, 4, 4, 3, 5, 6, 3, 6, 4, 5, 5, 4, 2, 2, 3, 4, 4, 3, 2, 6, 4, 5, 5, 6, 6, 2, 5, 4, 6, 5, 4, 5, 3, 4, 6, 5, 3, 4, 2, 3, 2, 4, 4, 5, 2, 6, 6, 5, 5, 6, 6, 5, 2, 4, 5, 4, 4, 3, 5, 6, 4, 3, 3
Offset: 0

Views

Author

Antti Karttunen, Apr 04 2000

Keywords

Links

Crossrefs

Cf. A055089, A060131, A072411, A290095.

Programs

  • Maple
    A055092(n) = count_permorder(convert(PermRevLexUnrank(j), 'disjcyc')).
count_permorder := proc(l) local c,t; t := 1; for c in l do t := ilcm(t,nops(c)); od; RETURN(t); end;
# Procedure PermRevLexUnrank given in A055089.

Formula

a(n) = A072411(A290095(n)) = A060131(A060126(n)). - Antti Karttunen, Dec 30 2017

Extensions

Entry revised by Antti Karttunen, Dec 30 2017

A055090 Number of cycles (excluding fixed points) of the n-th finite permutation in reversed colexicographic ordering (A055089).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2
Offset: 0

Views

Author

Antti Karttunen, Apr 18 2000

Keywords

Comments

Among the first n! entries k appears A136394(n,k) times. - Tilman Piesk, Apr 06 2012

Links

Crossrefs

Cf. A195663, A195664, A055089 (ordered finite permutations).
Cf. A198380 (cycle type of the n-th finite permutation).
Cf. A136394, A055089, A055090, A055093, A055091, A060128, A290095.

Programs

  • Maple
    with(group); seq(nops(convert(PermRevLexUnrank(j),'disjcyc')),j=0..)];
# Procedure PermRevLexUnrank given in A055089.

Formula

a(n) = A055093(n) - A055091(n).
a(n) = A056170(A290095(n)) = A060128(A060126(n)). - Antti Karttunen, Dec 30 2017

Extensions

Name changed by Tilman Piesk, Apr 06 2012

A290097 Restricted growth sequence transform of A290096, related to cycle-structure of permutations listed in table A055089.

Original entry on oeis.org

1, 2, 3, 4, 4, 3, 5, 6, 7, 8, 8, 7, 7, 8, 5, 7, 6, 8, 8, 7, 7, 5, 8, 6, 9, 10, 10, 11, 11, 10, 12, 11, 13, 14, 14, 13, 13, 14, 12, 13, 11, 14, 14, 13, 13, 12, 14, 11, 12, 11, 13, 14, 14, 13, 9, 10, 12, 13, 13, 12, 10, 11, 13, 14, 14, 11, 11, 10, 14, 13, 11, 14, 13, 14, 12, 13, 11, 14, 12, 13, 9, 12, 10, 13, 13, 14, 10, 11, 11, 14, 14, 11, 11, 14, 10, 13, 14
Offset: 0

Views

Author

Antti Karttunen, Aug 17 2017

Keywords

Links

Crossrefs

Cf. A290095, A290096.
Cf. tables A055089, A195663.
Showing 1-7 of 7 results.

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