A050385 -id:A050385 - cp's OEIS frontend

A295162 Decimal expansion of a constant related to the asymptotics of A050385.

5, 4, 8, 7, 4, 5, 2, 1, 8, 8, 2, 9, 7, 4, 6, 2, 1, 4, 7, 5, 6, 7, 4, 4, 5, 2, 9, 3, 2, 3, 0, 3, 0, 9, 2, 5, 5, 3, 2, 0, 0, 4, 2, 9, 1, 0, 2, 4, 3, 6, 3, 2, 7, 2, 8, 7, 1, 0, 6, 6, 6, 2, 9, 5, 2, 9, 1, 2, 1, 6, 7, 0, 0, 2, 4, 3, 2, 7, 3, 4, 4, 8, 2, 9, 3, 2, 7, 7, 8, 2, 8, 9, 4, 6, 7, 3, 2, 0, 0, 3, 4, 5, 3, 3, 3, 0, 2, 2

Offset: 1

Vaclav Kotesovec, Nov 15 2017

			5.48745218829746214756744529323030925532004291024363272871...

I. P. Goulden, L. B. Richmond, and J. Shallit, Natural exact covering systems and the reversion of the Möbius series, arXiv:1711.04109v3 [math.NT], revision of Dec 12 2017

Cf. A050385.

Equals limit n->infinity (A050385(n))^(1/n).

A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -2, -3, -2, 3, 8, 8, -2, -16, -24, -10, 24, 59, 54, -11, -117, -174, -90, 162, 431, 449, -20, -835, -1393, -848, 1062, 3352, 3748, 317, -6257, -11134, -7583, 7294, 25956, 30786, 5217, -46545, -88132, -65062, 48534, 199234, 249263, 63034, -342174, -691679, -554002

Offset: 0

Ilya Gutkovskiy, Mar 10 2018

sign

Invert transform of A008683.

Seiichi Manyama, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms

Cf. A008683, A050385, A068341, A073776, A073777, A117209, A117210, A185694, A280194.

a:= proc(n) option remember; `if`(n=0, 1, add(
      numtheory[mobius](j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 10 2018

nmax = 47; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[MoebiusMu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]

my(N=66, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, moebius(k)*x^k))) \\ Seiichi Manyama, Apr 06 2022

a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*a(n-k))); \\ Seiichi Manyama, Apr 06 2022

G.f.: 1/(1 - Sum_{k>=1} A008683(k)*x^k).

a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * a(n-k). - Seiichi Manyama, Apr 06 2022

A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -3, -6, 5, 61, 126, -308, -2772, -5669, 25630, 224730, 486551, -3068155, -29264219, -72173176, 513535711, 5625869262, 16687752839, -113740116822, -1496118902963, -5508392724427, 31534346503605, 523333047780288, 2414704077547660, -10254467367668159

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

sign

Exponential transform of A008683.

			E.g.f.: A(x) = 1 + x/1! - 3*x^3/3! - 6*x^4/4! + 5*x^5/5! + 61*x^6/6! + 126*x^7/7! - 308*x^8/8! - 2772*x^9/9! - 5669*x^10/10! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..592
N. J. A. Sloane, Transforms

Cf. A008683, A050385, A050386, A068341, A073776, A104688, A117209, A117210, A195589, A204290, A300663.

nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[MoebiusMu[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]

a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

E.g.f.: exp(Sum_{k>=1} A008683(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022

A050386 Exponential reversion of Moebius function A008683.

Original entry on oeis.org

1, 1, 4, 25, 221, 2505, 34707, 568177, 10731571, 229706718, 5495040882, 145285035974, 4206973447847, 132410823640004, 4500857134998016, 164322352411837139, 6412953180173688644, 266421162165751276297

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

N. J. A. Sloane, Transforms
Index entries for reversions of series

Cf. A008683, A050385.

length = 40; Range[length]! InverseSeries[Sum[MoebiusMu[n] x^n/n!, {n, 1, length}] + O[x]^(length+1)][[3]] (* Vladimir Reshetnikov, Nov 07 2015 *)

seq(n)= Vec(serlaplace(serreverse(sum(k=1, n, moebius(k)*x^k/k!) + O(x*x^n)))); \\ Michel Marcus, Apr 21 2020

E.g.f. A(x) satisfies: A(x) = x - Sum_{k>=2} mu(k) * A(x)^k / k!. - Ilya Gutkovskiy, Apr 22 2020

Typo in name corrected by Sean A. Irvine, Aug 15 2021

A294496 Number of distinct minimal period lengths of periodic infinite words on n symbols having the constant gap property.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 15, 18, 31, 35, 56, 62

Offset: 1

Jeffrey Shallit, Nov 01 2017

A periodic infinite word consists of a block x repeated infinitely to the right: X = x^omega = xxx.... The minimal period length of such a word X is the length of the shortest word y such that X = y^omega. Such a word has the constant-gap property if for each letter i occurring in the word, there is a constant c_i such that two consecutive occurrences of i are separated by exactly c_i symbols. For example (0102)^omega is a constant-gap word on 3 symbols with minimal period length 4.

Alternatively, this is the number of distinct lcm's of moduli that can appear in a disjoint covering system of the integers consisting of n congruences. Disjoint covering systems and constant-gap periodic sequences are in 1-1 correspondence. For example, the covering system corresponding to (0102)^omega is x == 0 (mod 2), x == 1 (mod 4), x == 3 (mod 4), and the lcm of the moduli (2,4,4) is 4.

			For n = 3 the 3 constant gap words on 3 symbols are (0102)^omega, (0121)^omega, (012)^omega, with minimal period lengths 4,4,3, respectively, so 2 distinct period lengths.

Jeffrey Shallit, list of distinct moduli for disjoint covering systems
I. P. Goulden, L. B. Richmond, and J. Shallit, Disjoint covering systems and the reversion of the Mobius series, arxiv Preprint arXiv:1711.04109 [math.NT], November 11 2017.

Cf. A050385.

A294593 Number of natural disjoint covering systems of cardinality n, with gcd of the moduli equal to 2.

Original entry on oeis.org

0, 1, 2, 6, 22, 88, 372, 1636, 7406, 34276, 161436, 771238, 3728168, 18201830, 89622696, 444533010, 2219057382, 11139859864, 56203325212, 284828848740, 1449270351504

Offset: 1

Jeffrey Shallit, Nov 03 2017

A disjoint covering system (DCS) is a system of congruences of the form x == a_i (mod m_i) such that every integer lies in exactly one of the congruences. Here the "moduli" are the m_i. The DCS is "natural" if it can be obtained by starting with the congruence x == 0 (mod 1) and "splitting": choosing a congruence and replacing it by r congruence.

			For n = 4 the 6 possible disjoint congruence systems are
(a) x == 1 (mod 2), x == 2 (mod 4), x == 0 (mod 8), x == 4 (mod 8)
(b) x == 1 (mod 2), x == 0 (mod 4), x == 2 (mod 8), x == 6 (mod 8)
(c) x == 1 (mod 2), x == 0 (mod 6), x == 2 (mod 6), x == 4 (mod 6)
(d) x == 0 (mod 2), x == 3 (mod 4), x == 1 (mod 8), x == 5 (mod 8)
(e) x == 0 (mod 2), x == 1 (mod 4), x == 3 (mod 8), x == 7 (mod 8)
(f) x == 0 (mod 2), x == 1 (mod 6), x == 3 (mod 6), x == 5 (mod 6)

S. Porubsky and J. Schönheim, Covering systems of Paul Erdös: past, present and future, in Paul Erdös and his Mathematics, Vol. I, Bolyai Society Mathematical Studies 11 (2002), 581-627.

I. P. Goulden, Andrew Granville, L. Bruce Richmond, and J. Shallit, Natural exact covering systems and the reversion of the Möbius series, Ramanujan J. (2019) Vol. 50, 211-235.
I. P. Goulden, L. B. Richmond, and J. Shallit, Natural exact covering systems and the reversion of the Möbius series, arXiv:1711.04109 [math.NT], 2017-2018.

Cf. A050385.

A296195 Number of disjoint covering systems of cardinality n.

Original entry on oeis.org

1, 1, 3, 10, 39, 160, 691, 3081, 14095, 65757, 311695, 1496833, 7267009

Offset: 1

Jeffrey Shallit, Dec 07 2017

A disjoint covering system or DCS (also called "exact covering system" or ECS) is a system of n congruences such that every integer belongs to exactly one of the congruences.

This sequence agrees with A050385 on the first 12 terms, but differs at a(13).

			For n = 3 the a(3) = 3 DCS are
(i) x == 0 (mod 3), x == 1 (mod 3), x == 2 (mod 3)
(ii) x == 0 (mod 2), x == 1 (mod 4), x == 3 (mod 4)
(iii) x == 1 (mod 2), x == 0 (mod 4), x == 2 (mod 4)

S. Porubsky and J. Schönheim, Covering systems of Paul Erdös: past, present and future, in Paul Erdös and his Mathematics, Vol. I, Bolyai Society Mathematical Studies 11 (2002), 581-627.

I. P. Goulden, L. B. Richmond, and J. Shallit, Natural exact covering systems and the reversion of the Möbius series, arXiv:1711.04109v3 [math.NT], revision of Dec 12 2017

Cf. A050385, which counts a subset of the DCS called "natural exact covering systems".

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A295162 Decimal expansion of a constant related to the asymptotics of A050385.

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A300663 Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).

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A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

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A050386 Exponential reversion of Moebius function A008683.

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A294496 Number of distinct minimal period lengths of periodic infinite words on n symbols having the constant gap property.

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A294593 Number of natural disjoint covering systems of cardinality n, with gcd of the moduli equal to 2.

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A296195 Number of disjoint covering systems of cardinality n.

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