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: Jens Voß

Jens Voß's wiki page.

Jens Voß has authored 76 sequences. Here are the ten most recent ones:

A289274 Numbers k such that the deficiency of k^2 is itself a square > 1.

Original entry on oeis.org

46, 284, 1633, 149728, 242656, 260495, 298057, 1056752, 9587584, 17706256, 914429696, 985501822, 1074266048, 1484820224, 4241800921, 12147056128, 109548719577, 287291764736, 360499817799
Offset: 1

Views

Author

Jens Voß, Jun 30 2017

Keywords

Comments

The sequence of square roots of the deficiencies of this sequence is A288144.
The disjoint union of the current sequence with the powers of 2 (A000079) is A289275, the sequence of numbers k for which the deficiency of k^2 is a square (including 1).

Examples

			The deficiency of 46^2 is 2*46^2 - sigma(46^2) = 19^2, so 46 is a term of the sequence.

Crossrefs

Cf. A000079, A033879, A288144, A289275.

Programs

  • Maple
    issq := n -> evalb(n>1 and issqr(n)):
A033879 := n -> 2*n - numtheory[sigma](n):
isa := n -> issq(A033879(n^2)):
select(isa, [$1..2000]); # Peter Luschny, Jul 25 2017
  • PARI
    isok(n) = issquare(d = 2*n^2 - sigma(n^2)) && (d!=1); \\ Michel Marcus, Jul 25 2017

Extensions

a(10) from Chai Wah Wu, Jul 26 2017
a(11)-a(19) from Giovanni Resta, Jul 27 2017

A288145 Square roots of the deficiencies of the squares of A289275.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 53, 1, 1, 1583, 1, 1, 1, 1, 1, 1, 1, 3509, 6479, 223309, 1, 291205, 1, 1, 65791, 1, 1, 1, 56179, 1, 50039, 1, 1, 1, 1, 1, 3505843, 456456275, 1, 16781311, 5734169, 1, 4144461731, 1, 1, 23461111, 1, 1, 1, 56585278013
Offset: 1

Views

Author

Jens Voß, Jul 01 2017

Keywords

Comments

The terms different from 1 in this sequence form sequence A288144.

Links

Crossrefs

Cf. A289275, A288144, A289274.

Extensions

a(35)-a(54) from Giovanni Resta, Jul 27 2017

A288144 Square roots of the deficiencies of the squares of A289274.

Original entry on oeis.org

19, 53, 1583, 3509, 6479, 223309, 291205, 65791, 56179, 50039, 3505843, 456456275, 16781311, 5734169, 4144461731, 23461111, 56585278013, 50656373, 164136392937
Offset: 1

Views

Author

Jens Voß, Jul 01 2017

Keywords

Comments

This sequence consists of the terms of A288145 different from 1.

Examples

			The deficiency of 46^2 is 2*46^2 - sigma(46^2) = 19^2, so this sequence starts with 19 (since 46 is the first term of the A289274).

Crossrefs

Cf. A289274, A289275, A288145.

Extensions

a(10) from Chai Wah Wu, Jul 26 2017
a(11)-a(19) from Giovanni Resta, Jul 27 2017

A289275 Numbers k such that the deficiency of k^2 is itself a square.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 46, 64, 128, 256, 284, 512, 1024, 1633, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 149728, 242656, 260495, 262144, 298057, 524288, 1048576, 1056752, 2097152, 4194304, 8388608, 9587584, 16777216, 17706256, 33554432, 67108864, 134217728
Offset: 1

Views

Author

Jens Voß, Jun 30 2017

Keywords

Comments

The sequence of square roots of the deficiencies of this sequence is A288145.
The current sequence is the disjoint union of the powers of 2 (A000079) and the sequence A289274.

Links

Crossrefs

Disjoint union of the sequences A000079 and A289274.
Square roots of deficiencies of squares is A288145.

Programs

  • Mathematica
    Select[Range[10^5], IntegerQ@ Sqrt[2 #^2 - DivisorSigma[1, #^2]] &] (* Michael De Vlieger, Jul 04 2017 *)
  • PARI
    isok(n) = issquare(2*n^2 - sigma(n^2)); \\ Michel Marcus, Jul 01 2017

Extensions

a(35)-a(38) from Giovanni Resta, Jul 27 2017

A284210 Number of subgroups of order n of the symmetric group Sym(n) on n symbols.

Original entry on oeis.org

1, 1, 1, 7, 6, 280, 120, 25335, 11200, 276696, 362880, 374838255, 39916800, 2414617920, 11721790080
Offset: 1

Views

Author

Jens Voß, Mar 23 2017

Keywords

Comments

The diagonal of A243748. - R. J. Mathar, Mar 30 2017 [edited by Peter Munn, Mar 06 2025]

Examples

			The group Sym(4) contains 3 cyclic groups of order 4, 3 non-normal elementary abelian groups of order 4 and one normal group of order 4, so A284210(4) = 3 + 3 + 1 = 7.

Programs

  • GAP
    List([1..14], n -> Sum(List(Filtered(ConjugacyClassesSubgroups(SymmetricGroup(n)), c -> Size(Representative(c)) = n)), c -> Size(c)));

Formula

If n is prime, A284210(n) = (n-2)!.

A272135 Numbers of ways of placing the numbers 1, ..., n on a circle (not counting rotations and reflections) such that for each s in {1, ..., n(n+1)/2}, there exists a connected subset S of the circle such that the numbers covered by S add up to s.

Original entry on oeis.org

1, 1, 1, 1, 2, 10, 41, 126, 537, 3956, 19776, 76340, 388047, 2775155, 15013424, 54188455, 272147013
Offset: 0

Views

Author

Jens Voß, Apr 21 2016

Keywords

Examples

			Out of the 3 essentially different arrangements (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4) of four points on a circle, only (1, 2, 3, 4) and (1, 3, 2, 4) yield all sums in {1, ..., 10}, so a(4) = 2.

Links

Programs

  • Mathematica
    ok[w_] := Block[{v = Join[w,w], n = Length@w}, n(n+1)/2 == Length@ Union@ Flatten@ Table[ Total@ Take[v, {i, i+k}], {i,n}, {k, 0, n-1}]]; a[n_] := If[n<3, 1, Sum[ Length@ Select[ Permutations@ Complement[ Range@n, e], ok@ Join[e, #] &], {e, Flatten[ Table[{a,1,b}, {a,2,n}, {b,a+1,n}], 1]}]]; a /@ Range[0, 9] (* Giovanni Resta, Apr 21 2016 *)

Extensions

a(15) from Giovanni Resta, Apr 21 2016
a(16) from Giovanni Resta, Apr 22 2016

A251739 Smallest k such that n * sum(i=0..k, binomial(k,i) mod (n-1) ) <= 2^n.

Original entry on oeis.org

1, 4, 3, 6, 5, 8, 7, 8, 9, 10, 10, 9, 10, 10, 10, 11, 11, 11, 12, 11, 12, 11, 11, 12, 13, 12, 11, 13, 12, 13, 13, 13, 12, 13, 13, 14, 13, 14, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 14, 14, 15, 14, 15, 15, 15, 15, 16, 15
Offset: 2

Views

Author

Jens Voß, Dec 07 2014

Keywords

Comments

Aside from the third value, the sequence is the same as A251738.

Examples

			For n = 3,
3 * sum(i=0..1, binomial(1,i) mod 2) = 3 * (1 + 1) = 6 > 2^1,
3 * sum(i=0..2, binomial(2,i) mod 2) = 3 * (1 + 0 + 1) = 6 > 2^2,
3 * sum(i=0..3, binomial(3,i) mod 2) = 3 * (1 + 1 + 1 + 1) = 12 > 2^3,
3 * sum(i=0..4, binomial(4,i) mod 2) = 3 * (1 + 0 + 0 + 0 + 1) = 6 <= 2^4,
so A251739(3) = 4.

Crossrefs

Cf. A251738.

A251738 Smallest k such that n * sum(i=0..k, binomial(k,i) mod (n-1) ) < 2^n.

Original entry on oeis.org

1, 4, 6, 6, 5, 8, 7, 8, 9, 10, 10, 9, 10, 10, 10, 11, 11, 11, 12, 11, 12, 11, 11, 12, 13, 12, 11, 13, 12, 13, 13, 13, 12, 13, 13, 14, 13, 14, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 14, 14, 15, 14, 15, 15, 15, 15, 16, 15
Offset: 2

Views

Author

Jens Voß, Dec 07 2014

Keywords

Comments

Aside from the third value, the sequence is the same as A251739.

Examples

			For n = 3,
3 * sum(i=0..1, binomial(1,i) mod 2) = 3 * (1 + 1) = 6 >= 2^1,
3 * sum(i=0..2, binomial(2,i) mod 2) = 3 * (1 + 0 + 1) = 6 >= 2^2,
3 * sum(i=0..3, binomial(3,i) mod 2) = 3 * (1 + 1 + 1 + 1) = 12 >= 2^3,
3 * sum(i=0..4, binomial(4,i) mod 2) = 3 * (1 + 0 + 0 + 0 + 1) = 6 < 2^4,so A251738(3) = 4.

Crossrefs

Cf. A251739.

A234974 Expected lengths of random walks along the edges of a Platonic solid (in the order cube, octahedron, dodecahedron, icosahedron) from one vertex to an opposing one.

Original entry on oeis.org

10, 6, 35, 15
Offset: 1

Views

Author

Jens Voß, Jan 02 2014

Keywords

Comments

For all Platonic solids (excluding the tetrahedron), the expected number of steps of a random walk from one vertex to its opposite vertex is indeed an integer.

Crossrefs

Cf. comment to A063723

A229430 Number of ways to label the cells of a 2 X n grid such that no (orthogonally) adjacent cells have adjacent labels.

Original entry on oeis.org

1, 0, 0, 24, 1660, 160524, 21914632, 4065598248, 987830372684, 304870528356476, 116578000930637000, 54116343193686469960, 29984241542575292762940, 19548555813018460134901516, 14815308073366437897483622056, 12915964646307201385492841052040
Offset: 0

Views

Author

Jens Voß, Sep 23 2013

Keywords

Comments

a(n) is the number of Hamiltonian paths in the complement of the n-ladder graph. - Andrew Howroyd, Feb 14 2020

Examples

			The A(3) = 24 valid labelings of a 2 X 3 grid are
   153   163   135   513   415   416
   426   425   462   246   263   253
together with their 18 reflections and rotations.

Links

Crossrefs

Row n=2 of A229429.
Cf. A002464.

Programs

  • PARI
    seq(n)={my(gf=(1 - x)*(1 + (3*y - 2)*x + (y + 1)*x^2)/(1 + (-y^2 + 5*y - 3)*x + (y^3 - 3*y^2 + 3)*x^2 + (-2*y^3 + 5*y^2 - 3*y - 1)*x^3 + (y^3 - y^2 + 2*y)*x^4)); [subst(serlaplace(p*y^0),y,1) | p <- Vec(gf + O(x*x^n))]} \\ Andrew Howroyd, Feb 16 2020

Extensions

Terms a(9) and beyond from Andrew Howroyd, Feb 14 2020

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