A373275
a(n) = Sum_{d|n} (-1)^(d-1) * 2^(n/d-1).
Original entry on oeis.org
1, 1, 5, 5, 17, 29, 65, 117, 261, 497, 1025, 2017, 4097, 8129, 16405, 32629, 65537, 130845, 262145, 523765, 1048645, 2096129, 4194305, 8386641, 16777233, 33550337, 67109125, 134209477, 268435457, 536855053, 1073741825, 2147450741, 4294968325, 8589869057
Offset: 1
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a(n) = sumdiv(n, d, (-1)^(d-1)*2^(n/d-1));
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my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x)^k/(1+x^k))/2)
A245392
Sum_{k, k|n} 2^(k-1) + Sum_{1<=k<=n, gcd(k,n)=1} 2^(k-1).
Original entry on oeis.org
2, 4, 8, 16, 32, 56, 128, 224, 480, 856, 2048, 3200, 8192, 13656, 29920, 54752, 131072, 202104, 524288, 857952, 1939168, 3495256, 8388608, 12918016, 33013248, 55924056, 124631008, 222655840, 536870912, 809850488, 2147483648, 3579172320, 7974270688, 14316557656
Offset: 1
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f:= proc(k,n) local g; g:= igcd(k,n); g = 1 or g = k end proc:
A:= n -> 1 + add(2^(k-1),k=select(f,[$1..n],n));
seq(A(n),n=1..100); # Robert Israel, Jul 21 2014
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sum(k=1, n, if (gcd(k,n)==1, 2^(k-1), 0)) + sumdiv(n, k, k*2^(k-1));
A302546
a(n) = Sum_{k = 1...n} 2^binomial(n, k).
Original entry on oeis.org
0, 2, 6, 18, 98, 2114, 1114242, 68723671298, 1180735735906024030722, 170141183460507917357914971986913657858, 7237005577335553223087828975127304179197147198604070555943173844710572689410
Offset: 0
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Table[Sum[2^Binomial[n,d],{d,n}],{n,10}]
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a(n) = sum(k=1, n, 2^binomial(n, k)); \\ Michel Marcus, Jun 21 2018
A336129
Number of strict compositions of divisors of n.
Original entry on oeis.org
1, 2, 4, 5, 6, 16, 14, 24, 31, 64, 66, 120, 134, 208, 360, 459, 618, 894, 1178, 1622, 2768, 3364, 4758, 6432, 8767, 11440, 15634, 24526, 30462, 42296, 55742, 75334, 98112, 131428, 168444, 258403, 315974, 432244, 558464, 753132, 958266, 1280840, 1621274
Offset: 1
The a(1) = 1 through a(7) = 14 compositions:
(1) (1) (1) (1) (1) (1) (1)
(2) (3) (2) (5) (2) (7)
(1,2) (4) (1,4) (3) (1,6)
(2,1) (1,3) (2,3) (6) (2,5)
(3,1) (3,2) (1,2) (3,4)
(4,1) (1,5) (4,3)
(2,1) (5,2)
(2,4) (6,1)
(4,2) (1,2,4)
(5,1) (1,4,2)
(1,2,3) (2,1,4)
(1,3,2) (2,4,1)
(2,1,3) (4,1,2)
(2,3,1) (4,2,1)
(3,1,2)
(3,2,1)
Compositions of divisors are
A034729.
Strict partitions of divisors are
A047966.
Partitions of divisors are
A047968.
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Table[Sum[Length[Join@@Permutations/@Select[IntegerPartitions[d],UnsameQ@@#&]],{d,Divisors[n]}],{n,12}]
A338695
a(n) = Sum_{d|n} 2^(d-1) * binomial(d, n/d).
Original entry on oeis.org
1, 4, 12, 34, 80, 204, 448, 1072, 2308, 5280, 11264, 25088, 53248, 116032, 245920, 527880, 1114112, 2369152, 4980736, 10508880, 22022336, 46193664, 96468992, 201469408, 419430416, 872734720, 1811960832, 3758844096, 7784628224, 16107909312, 33285996544, 68723417856, 141734089728
Offset: 1
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a[n_] := DivisorSum[n, 2^(# - 1) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)
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a(n) = sumdiv(n, d, 2^(d-1)*binomial(d, n/d));
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N=40; x='x+O('x^N); Vec(sum(k=1, N, (2+2*x^k)^k-2^k)/2)
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