A053636 -id:A053636 - cp's OEIS frontend

A063776 Number of subsets of {1,2,...,n} which sum to 0 modulo n.

2, 2, 4, 4, 8, 12, 20, 32, 60, 104, 188, 344, 632, 1172, 2192, 4096, 7712, 14572, 27596, 52432, 99880, 190652, 364724, 699072, 1342184, 2581112, 4971068, 9586984, 18512792, 35791472, 69273668, 134217728, 260301176, 505290272, 981706832

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 16 2001

From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} that are empty or contain n and have integer mean. If the subsets are not required to contain n, we get A327475. For example, the a(1) = 2 through a(6) = 12 subsets are:
  {}   {}   {}       {}       {}           {}
  {1}  {2}  {3}      {4}      {5}          {6}
            {1,3}    {2,4}    {1,5}        {2,6}
            {1,2,3}  {2,3,4}  {3,5}        {4,6}
                              {1,3,5}      {1,2,6}
                              {3,4,5}      {1,5,6}
                              {1,2,4,5}    {2,4,6}
                              {1,2,3,4,5}  {4,5,6}
                                           {1,2,3,6}
                                           {1,4,5,6}
                                           {2,3,5,6}
                                           {2,3,4,5,6}
(End)

			G.f. = 2*x + 2*x^2 + 4*x^3 + 4*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + 32*x^8 + 60*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..3333 (first 200 terms from T. D. Noe)
T. Amdeberhan, M. Can and V. Moll, Broken bracelets, Molien series, paraffin wax and the elliptic curve 48a4, SIAM Journal of Discrete Math., v.25, 2011, p. 1843. See equation (1.2).
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
N. Kitchloo and L. Pachter, An interesting result about subset sums, preprint (pdf file), 1993.
N. Kitchloo and L. Pachter, An interesting result about subset sums, preprint (pdf file), 1993.

The superdiagonal of A068009.
Cf. A000010, A000013, A051293, A053633, A053634, A053636, A054539, A082550.
Cf. A000016, A065795, A327475, A327481.

a063776 n = a053636 n `div` n  -- Reinhard Zumkeller, Sep 13 2013

Table[a = Select[ Divisors[n], OddQ[ # ] &]; Apply[Plus, 2^(n/a)*EulerPhi[a]]/n, {n, 1, 35}]
a[ n_] := If[ n < 1, 0, 1/n Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]]; (* Michael Somos, May 09 2013 *)
Table[Length[Select[Subsets[Range[n]],#=={}||MemberQ[#,n]&&IntegerQ[Mean[#]]&]],{n,0,10}] (* Gus Wiseman, Sep 14 2019 *)

{a(n) = if( n<1, 0, 1 / n * sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))}; /* Michael Somos, May 09 2013 */

a(n) = sumdiv(n, d, (d%2)* 2^(n/d)*eulerphi(d))/n; \\ Michel Marcus, Feb 10 2016

from sympy import totient, divisors
def A063776(n): return (sum(totient(d)<>(~n&n-1).bit_length(),generator=True))<<1)//n # Chai Wah Wu, Feb 21 2023

a(n) = (1/n) * Sum_{d divides n and d is odd} 2^(n/d) * phi(d).
a(n) = (1/n) * A053636(n). - Michael Somos, May 09 2013
a(n) = 2 * A000016(n).
For odd n, a(n) = A000031(n).
G.f.: -Sum_{m >= 0} (phi(2*m + 1)/(2*m + 1)) * log(1 - 2*x^(2*m + 1)). - Petros Hadjicostas, Jul 13 2019
a(n) = A082550(n) + 1. - Gus Wiseman, Sep 14 2019

More terms from Vladeta Jovovic, Aug 20 2001

A053635 a(n) = Sum_{d|n} phi(d)*2^(n/d).

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 84, 140, 288, 540, 1080, 2068, 4224, 8216, 16548, 32880, 65856, 131104, 262836, 524324, 1049760, 2097480, 4196412, 8388652, 16782048, 33554600, 67117128, 134218836, 268452240, 536870968, 1073777040, 2147483708, 4295033472, 8589938808

Offset: 0

N. J. A. Sloane, Mar 23 2000

nonn

Dirichlet convolution of phi(n) and 2^n. - Richard L. Ollerton, May 06 2021

Seiichi Manyama, Table of n, a(n) for n = 0..3321
James East and Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
James East and Ron Niles, Integer polygons of given perimeter, Bull. Aust. Math. Soc. 100(1) (2019), 131-147.
T. Pisanski, D. Schattschneider, and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180. See v(n).

Cf. A000031, A053634, A053636. Twice A034738.

Column k=2 of A185651.

[0] cat  [&+[EulerPhi(d)*2^(n div d): d in Divisors(n)]: n in [1..40]]; // Vincenzo Librandi, Jul 20 2019

with(numtheory); A053685:=n->add( phi(n/d)*2^d, d in divisors(n)); # N. J. A. Sloane, Nov 21 2009

a[0] = 0; a[n_] := Sum[EulerPhi[d] 2^(n/d), {d, Divisors[n]}];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Aug 30 2018 *)

a(n) = if (n, sumdiv(n, d, eulerphi(d)*2^(n/d)), 0); \\ Michel Marcus, Sep 20 2017

a(n) = n * A000031(n).
a(n) = Sum_{k=1..n} 2^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021
a(n) = Sum_{k=1..n} 2^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021

A309122 Sum of the sizes of all subsets of [n] whose sum is divisible by n.

Original entry on oeis.org

1, 1, 6, 6, 20, 34, 70, 124, 270, 516, 1034, 2060, 4108, 8198, 16440, 32760, 65552, 131142, 262162, 524312, 1048740, 2097162, 4194326, 8388856, 16777300, 33554444, 67109418, 134217764, 268435484, 536872072, 1073741854, 2147483632, 4294969404, 8589934608

Offset: 1

Alois P. Heinz, Jul 13 2019

nonn

The bivariate g.f. of array T(n,k) = A267632(n,k) is Sum_{n, k >= 1} T(n,k) * x^n * y^k = -x/(1 - x) - Sum_{s >= 1} (phi(s)/s) * log(1 - x^s + (-x*y)^s). Differentiating w.r.t. y and setting y = 1, we get the g.f. of a(n) = k * Sum_{1 <= k <= n} T(n,k) (see below). - Petros Hadjicostas, Jul 13 2019

			a(5) = 20 = 0 + 1 + 2 + 2 + 3 + 3 + 4 + 5 = |{}| + |{5}| + |{1,4}| + |{2,3}| + |{1,4,5}| + |{2,3,5}| + |{1,2,3,4}| + |{1,2,3,4,5}|.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000010, A053636, A267632, A309128.

b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],
      b(n-1, m, s) +(g-> g+[0, g[1]])(b(n-1, m, irem(s+n, m))))
    end:
a:= proc(n) option remember; forget(b); b(n$2, 0)[2] end:
seq(a(n), n=1..40);

b[n_, m_, s_] := b[n, m, s] = If[n == 0, {If[s == 0, 1, 0], 0},
     b[n-1, m, s] + Function[g, g + {0, g[[1]]}][b[n-1, m, Mod[s+n, m]]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A267632(n,k).
From Petros Hadjicostas, Jul 13 2019: (Start)
G.f.: Sum_{s >= 1} phi(s) * (-x)^(s-1)/(1 - x^s + (-x)^s) = -Sum_{m >= 1} phi(2*m) * x^(2*m-1) + Sum_{m >= 0} phi(2*m+1) * x^(2*m)/(1 - 2*x^(2*m+1)).
a(2*m + 1) = A053636(2*m + 1)/2 = (1/2) * Sum_{d|2*m+1} phi(d) * 2^((2*m+1)/d) for m >= 0.
a(2*m) = -phi(2*m) +  A053636(2*m)/2 for m >= 1.
(End)

A309128 (1/n) times the sum of the elements of all subsets of [n] whose sum is divisible by n.

Original entry on oeis.org

1, 1, 4, 4, 12, 20, 40, 70, 150, 284, 564, 1116, 2212, 4392, 8768, 17404, 34704, 69214, 137980, 275264, 549340, 1096244, 2188344, 4369196, 8724196, 17422500, 34797476, 69505628, 138845940, 277383904, 554189344, 1107296248, 2212559996, 4421289872, 8835361488

Offset: 1

Alois P. Heinz, Jul 13 2019

nonn

			The subsets of [5] whose sum is divisible by 5 are: {}, {5}, {1,4}, {2,3}, {1,4,5}, {2,3,5}, {1,2,3,4}, {1,2,3,4,5}.  The sum of their elements is 0 + 5 + 5 + 5 + 10 + 10 + 10 + 15 = 60.  So a(5) = 60/5 = 12.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000010, A001792 (the same for all subsets), A053636, A063776, A309122, A309280.

b:= proc(n, m, s) option remember; `if`(n=0, [`if`(s=0, 1, 0), 0],
      b(n-1, m, s) +(g-> g+[0, g[1]*n])(b(n-1, m, irem(s+n, m))))
    end:
a:= proc(n) option remember; forget(b); b(n$2, 0)[2]/n end:
seq(a(n), n=1..40);

b[n_, m_, s_] := b[n, m, s] = If[n == 0, {If[s == 0, 1, 0], 0},
     b[n-1, m, s] + Function[g, g+{0, g[[1]] n}][b[n-1, m, Mod[s+n, m]]]];
a[n_] := b[n, n, 0][[2]]/n;
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

Conjecture: a(n) = (n + 1) * A063776(n)/4 - (phi(n)/2) * (1 + (-1)^n)/2 = ((n + 1)/(4*n)) * A053636(n) - (phi(n)/2) * (1 + (-1)^n)/2. - Petros Hadjicostas, Jul 20 2019

a(n) = A309280(n,n). - Alois P. Heinz, Jul 21 2019

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A063776 Number of subsets of {1,2,...,n} which sum to 0 modulo n.

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A053635 a(n) = Sum_{d|n} phi(d)*2^(n/d).

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A309122 Sum of the sizes of all subsets of [n] whose sum is divisible by n.

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A309128 (1/n) times the sum of the elements of all subsets of [n] whose sum is divisible by n.

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