A157658 -id:A157658 - cp's OEIS frontend

A073776 a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.

1, 1, 2, 3, 6, 9, 17, 28, 50, 83, 147, 249, 435, 742, 1288, 2207, 3819, 6561, 11333, 19497, 33640, 57915, 99874, 172020, 296550, 510886, 880580, 1517226, 2614889, 4505745, 7765094, 13380640, 23059193, 39735969, 68476885, 118001888

Offset: 0

Paul D. Hanna, Aug 10 2002

Recurrence relation involves the Moebius function.
Radius of convergence of A(x) is r=0.5802946238073267...
Related limits are
  lim_{n->infinity} a(n) r^n = 0.6303632342... and
  lim_{n->infinity} a(n+1)/a(n) = 1.723262561763844...
From Gary W. Adamson, Aug 11 2016: (Start)
The definition in the heading follows from the INVERTi transform of (1, 2, 3, 6, 9, 17, ...) equals -mu(n) for n >= 2 (cf. A157658).
Then for example, a(6) = 17 = (1, 1, 0, 1, -1, 1) dot (9, 6, 3, 2, 1, 1) = (9 + 6 + 0 + 2 - 1 + 1); in agreement with the first example. (End)

			a(6) = -mu(2)a(5) - mu(3)a(4) - mu(4)a(3) - mu(5)a(2) - mu(6)a(1) - mu(7)a(0) = 9 + 6 + 0 + 2 - 1 + 1 = 17.
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 17*x^6 + 28*x^7 + 50*x^8 + 83*x^9 + 147*x^10 + 249*x^11 + 435*x^12 + ...
where
1/A(x) = 1 - x - x^2 - x^4 + x^5 - x^6 + x^9 - x^10 - x^12 + x^13 + x^14 - x^16 - x^18 + x^20 + x^21 - x^22 + x^25 - x^28 - x^29 - x^30 + ... + mu(n)*x^n +...
Also, g.f. A(x) satisfies:
x*A(x) = x*A(x)/A(x*A(x)) + x^2*A(x)^2/A(x^2*A(x)^2) + x^3*A(x)^3/A(x^3*A(x)^3) + x^4*A(x)^4/A(x^4*A(x)^4) + x^5*A(x)^5/A(x^5*A(x)^5) + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.

Cf. A073777, A068341, A070965, A008683, A185694, A157658.

a073776 n = a073776_list !! (n-1)
a073776_list = 1 : f [1] where
   f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms
   ms = map negate $ tail a008683_list
-- Reinhard Zumkeller, Nov 03 2015

a[0] = 1; a[n_] := a[n] = Sum[-MoebiusMu[k + 1]*a[n - k], {k, 1, n}]; Array[a,35,0] (* Jean-François Alcover, Apr 11 2011 *)

{a(n) = my(A=[1,1],F); for(i=1,n, A=concat(A,0); F=Ser(A); A = Vec(sum(m=1,#A, subst(x/F, x, x^m*F^m))) ); A[n+1]}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 19 2016

G.f.: A(x) = x / (Sum_{n>=1} mu(n)*x^n), A(0)=1, where mu(n) = Moebius function of n.

G.f. A(x) satisfies: x*A(x) = Sum_{n>=1} x^n*A(x)^n / A( x^n*A(x)^n ). - Paul D. Hanna, Apr 19 2016

A157928 a(n) = 0 if n < 2, = 1 otherwise.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Jaroslav Krizek, Mar 09 2009

A characteristic function which indicates whether n has a prime factorization n = product p_i^e_i where p_i are primes (A000040) and e_i nonnegative exponents, at least one e_i nonzero.
a(n), n>=1, is also generated by the following Dirichlet convolutions:
  a(n) = A157658(n) * A000012(n),
  a(n) = A008683(n) * A032741(n).
a(n) appears as a factor in the following Dirichlet convolutions:
  a(n) * A000010(n) = A051953(n),
  a(n) * A000027(n) = A001065(n),
  a(n) * A000012(n) = A032741(n).
a(n) is also both the number of disconnected 0-regular graphs on n vertices and the number of disconnected 1-regular graphs on 2n vertices. - Jason Kimberley, Sep 27 2011
Partial sums of A185012. - Jason Kimberley, Oct 15 2011
Decimal expansion of 1/900. - Elmo R. Oliveira, May 05 2024

J. S. Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000010, A000012, A000027, A000040, A001065, A008683, A032741, A051953, A057427, A157658, A185012.

PadRight[{0,0},120,{1}] (* Harvey P. Dale, Jun 03 2019 *)

a(n) = A057427(n-1) for n >= 2.
From Elmo R. Oliveira, Jul 20 2024: (Start)
G.f.: x^2/(1-x).
E.g.f.: exp(x) - x - 1. (End)

Definition simplified by R. J. Mathar, May 17 2010

A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 5, 10, 1, 30, 1, 12, 11, 32, 1, 36, 1, 38, 13, 16, 1, 92, 7, 18, 19, 46, 1, 74, 1, 80, 17, 22, 15, 140, 1, 24, 19, 116, 1, 90, 1, 62, 51, 28, 1, 256, 9, 62, 23, 70, 1, 136, 19, 140, 25, 34, 1, 286, 1, 36, 61, 192, 21, 122, 1, 86, 29, 114

Offset: 1

Paolo P. Lava, Feb 19 2015

nonn

a(n) = 1 if n is prime.

			The aliquot parts of 8 are 1, 2, 4 and their sum is 7.
Now, let us calculate the aliquot parts of 1, 2 and 4:
1 => 0;  2 => 1;  4 => 1, 2.  Their sum is 0 + 1 + 1 + 2 = 4.
Let us calculate the aliquot parts of 1, 1, 2:
1 => 0;  1 = > 0; 2 => 1. Their sum is 1.
We have left 1: 1 => 0.
Finally, 7 + 4 + 1 = 12. Therefore a(8) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)
Jon Maiga, Computer-generated formulas for A255242, Sequence Machine.

Cf. A001047, A001065, A001787, A006516, A016127, A016130, A016135, A255243, A330575.

Sequences that appear in the convolution formulas: A000203, A001065, A051953, A062790, A067824, A074206, A157658, A174725, A174726, A191161, A211779, A245211, A323910, A253249.

with(numtheory): P:=proc(q) local a,b,c,k,n,t,v;
for n from 1 to q do b:=0; a:=sort([op(divisors(n))]); t:=nops(a)-1;
while add(a[k],k=1..t)>0 do b:=b+add(a[k],k=1..t); v:=[];
for k from 2 to t do c:=sort([op(divisors(a[k]))]); v:=[op(v),op(c[1..nops(c)-1])]; od;
a:=v; t:=nops(a); od; print(b); od; end: P(10^3);

f[s_] := Flatten[Most[Divisors[#]] & /@ s]; a[n_] := Total@Flatten[FixedPointList[ f, {n}]] - n; Array[a, 100] (* Amiram Eldar, Apr 06 2019 *)

ali(n) = setminus(divisors(n), Set(n));
a(n) = my(list = List(), v = [n]); while (#v, my(w = []); for (i=1, #v, my(s=ali(v[i])); for (j=1, #s, w = concat(w, s[j]); listput(list, s[j]));); v = w;); vecsum(Vec(list)); \\ Michel Marcus, Jul 15 2023

a(1) = 0.
a(2^k) = k*2^(k-1) = A001787(k), for k>=1.
a(n^k) = (n^k-2^k)/(n-2), for n odd prime and k>=1.
In particular:
a(3^k)  = A001047(k-1);
a(5^k)  = A016127(k-1);
a(7^k)  = A016130(k-1);
a(11^k) = A016135(k-1).
From Antti Karttunen, Nov 22 2024: (Start)
a(n) = A330575(n) - n.
Also, following formulas were conjectured by Sequence Machine:
a(n) = (A191161(n)-n)/2.
a(n) = Sum_{d|n} A001065(d)*A074206(n/d). [Compare to David A. Corneth's Apr 13 2020 formula for A330575]
a(n) = Sum_{d|n} A051953(d)*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A174726(n/d).
a(n) = Sum_{d|n} A062790(d)*A253249(n/d).
a(n) = Sum_{d|n} A157658(d)*A191161(n/d).
a(n) = Sum_{d|n} A174725(d)*A211779(n/d).
a(n) = Sum_{d|n} A245211(d)*A323910(n/d).
(End)

A144734 Triangle read by rows, A054533 * transpose(A101688) (matrix product) provided A101688 is read as a square array by antidiagonals upwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 2, 0, 1, 2, 3, 4, 0, -1, 0, 2, 3, 2, 0, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 3, 3, 3, 6, 6, 6, 0, -1, 0, -1, 0, 4, 5, 4, 5, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, -2, -2, 0, 0, 4, 4, 6, 6, 4, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, -1, 0

Offset: 1

Gary W. Adamson, Sep 20 2008

Right border = A000010, phi(n).

Row sums = A023896: (1, 1, 3, 4, 10, 6, 21, ...).

			First few rows of the triangle are as follows:
  1;
  0,  1;
  0,  1,  2;
  0,  0,  2,  2;
  0,  1,  2,  3,  4;
  0, -1,  0,  2,  3,  2;
  0,  1,  2,  3,  4,  5,  6;
  0,  0,  0,  0,  4,  4,  4,  4;
  0,  0,  0,  3,  3,  3,  6,  6,  6;
  0, -1,  0, -1,  0,  4,  5,  4,  5,  4;
  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10;
  ...
row 4 = (0, 0, 2, 2) = partial sums from the right of row 4 of triangle A054533: (0, -2, 0, 2).

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A000010, A023896, A054533, A101688, A157658 (column 2).

Triangle read by rows, A054533 * transpose(A101688) (matrix product); i.e., partial sums from of the right of triangle A054533 (because A101688 can be viewed as an upper triangular matrix of 1's).
From Petros Hadjicostas, Jul 28 2019: (Start)
T(n,k) = Sum_{m = k..n} A054533(n,m) = Sum_{d|n} d * mu(n/d) * ((n/d) - ceiling(k/d) + 1) for n >= 1 and 1 <= k <= n.
T(n,k) = phi(n) - Sum_{d|n} d * mu(n/d) * ceiling(k/d) for n >= 2 and 1 <= k <= n.
(End)

Name edited by and more terms from Petros Hadjicostas, Jul 28 2019

A157657 a(1) = 1, a(n) = -mu(n) for n >= 2.

Original entry on oeis.org

1, 1, 1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, -1, 0, 1, 0, 1, 0, -1, -1, 1, 0, 0, -1, 0, 0, 1, 1, 1, 0, -1, -1, -1, 0, 1, -1, -1, 0, 1, 1, 1, 0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, -1, 1, 0, 1, -1, 0, 0, -1, 1, 1, 0, -1, 1, 1, 0, 1, -1, 0, 0, -1, 1, 1, 0, 0, -1, 1, 0, -1, -1, -1, 0, 1, 0, -1, 0, -1, -1, -1, 0, 1, 0, 0, 0

Offset: 1

Jaroslav Krizek, Mar 03 2009

sign

Apparently the Dirichlet inverse of A114006. [R. J. Mathar, Jul 15 2010]

Not multiplicative; for example a(2)*a(3) != a(6). - R. J. Mathar, Mar 31 2012

Antti Karttunen, Table of n, a(n) for n = 1..5000
Index entries for sequences computed from exponents in factorization of n

Cf. A008683, A157658 (same except for a(1)).

a[1]:=1; a[n_]:=-MoebiusMu[n]; Array[a,100] (* Stefano Spezia, May 03 2025 *)

a(n)=if(n==1,1, -moebius(n)); \\ Joerg Arndt, Aug 25 2014

(define (A157657 n) (if (= 1 n) 1 (- (A008683 n)))) ;; Antti Karttunen, Jul 26 2017

Added more terms, Joerg Arndt, Aug 25 2014

cp's OEIS Frontend

A073776 a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.

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A157928 a(n) = 0 if n < 2, = 1 otherwise.

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A255242 Calculate the aliquot parts of a number n and take their sum. Then repeat the process calculating the aliquot parts of all the previous aliquot parts and add their sum to the previous one. Repeat the process until the sum to be added is zero. Sequence lists these sums.

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A144734 Triangle read by rows, A054533 * transpose(A101688) (matrix product) provided A101688 is read as a square array by antidiagonals upwards.

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A157657 a(1) = 1, a(n) = -mu(n) for n >= 2.

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