A187767 -id:A187767 - cp's OEIS frontend

A127804 a(2n) = 2^(2n), a(2n+1) = 2^(2n+1) + a(n).

1, 3, 4, 11, 16, 36, 64, 139, 256, 528, 1024, 2084, 4096, 8256, 16384, 32907, 65536, 131328, 262144, 524816, 1048576, 2098176, 4194304, 8390692, 16777216, 33558528, 67108864, 134225984, 268435456, 536887296, 1073741824, 2147516555, 4294967296, 8590000128

Offset: 0

Paul Barry, Jan 29 2007

From Tilman Piesk, Jun 30 2025: (Start)
The binary expansion of a(n), with bits least to most significant, is row n of A115361.
Number of 1's in binary expansion of a(n) is A001511(n+1).
Row sums of triangle A128807.
Row sums of triangle A127803.(End)

Tilman Piesk, Table of n, a(n) for n = 0..1023 (terms 0..61 from R. J. Mathar)
Tilman Piesk, Illustrated triangles with 32 rows:
related to necklaces (similar to A385665),
binary (A115361),
signed (A127803)
Tilman Piesk, Serration of Boolean functions (Wikiversity)

Cf. A045654, A128807, A127803, A115361, A187767, A001511.

A127804 := proc(n)
    add( A127803(n,k),k=0..n) ;
end proc: # R. J. Mathar, Feb 12 2024
# second Maple program:
a:= proc(n) option remember;
      2^n+`if`(n::odd, a((n-1)/2), 0)
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Jul 01 2025

rows = 30;
A[n_, k_] := If[k <= n, If[n <= 2 k, 1/(2*2^n - 1), 0], 0];
T = Table[A[n, k], {n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
a[n_] := T[[n+1]] // Total;
Table[a[n], {n, 0, rows-1}] (* Jean-François Alcover, Jul 03 2018 *)

Conjecture: a(n) = 1 + A187767(n+1). - Andrew Howroyd, Jun 02 2017
From Tilman Piesk, Jun 30 2025: (Start)
a(n) = Sum_{i=0..A001511(n+1)-1} 2^((n+1) / 2^i - 1)
     = Sum_{i=0..A001511(n+1)-1} 2^floor(n / 2^i).
a(n) = A045654(n+1) / 2. (End)

Name changed by Tilman Piesk, Jun 30 2025

A373279 Expansion of Sum_{k>=0} x^(3^k) / (1 - 3*x^(3^k)).

Original entry on oeis.org

1, 3, 10, 27, 81, 246, 729, 2187, 6571, 19683, 59049, 177174, 531441, 1594323, 4783050, 14348907, 43046721, 129140409, 387420489, 1162261467, 3486785130, 10460353203, 31381059609, 94143181014, 282429536481, 847288609443, 2541865834900, 7625597484987

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373280, A373281, A373282, A373283.

Cf. A000244, A046211, A051064, A364222.

b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
a(n, k=3) = sumdiv(n, d, d*b(d, k));

G.f. A(x) satisfies A(x) = x/(1 - 3*x) + A(x^3).
If n == 0 (mod 3), a(n) = 3^n + a(n/3) otherwise a(n) = 3^n.
a(n) = Sum_{d|n} d * A046211(d).

A373280 Expansion of Sum_{k>=0} x^(4^k) / (1 - 4*x^(4^k)).

Original entry on oeis.org

1, 4, 16, 65, 256, 1024, 4096, 16388, 65536, 262144, 1048576, 4194320, 16777216, 67108864, 268435456, 1073741889, 4294967296, 17179869184, 68719476736, 274877907200, 1099511627776, 4398046511104, 17592186044416, 70368744178688, 281474976710656, 1125899906842624

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373281, A373282, A373283.

Cf. A000302, A115362.

G.f. A(x) satisfies A(x) = x/(1 - 4*x) + A(x^4).

If n == 0 (mod 4), a(n) = 4^n + a(n/4) otherwise a(n) = 4^n.

A373281 Expansion of Sum_{k>=0} x^(5^k) / (1 - 5*x^(5^k)).

Original entry on oeis.org

1, 5, 25, 125, 626, 3125, 15625, 78125, 390625, 1953130, 9765625, 48828125, 244140625, 1220703125, 6103515650, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328250, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373280, A373282, A373283.

Cf. A000351, A054662, A055457.

b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
a(n, k=5) = sumdiv(n, d, d*b(d, k));

G.f. A(x) satisfies A(x) = x/(1 - 5*x) + A(x^5).
If n == 0 (mod 5), a(n) = 5^n + a(n/5) otherwise a(n) = 5^n.
a(n) = Sum_{d|n} d * A054662(d).

A373282 Expansion of Sum_{k>=0} x^(6^k) / (1 - 6*x^(6^k)).

Original entry on oeis.org

1, 6, 36, 216, 1296, 7777, 46656, 279936, 1679616, 10077696, 60466176, 362797062, 2176782336, 13060694016, 78364164096, 470184984576, 2821109907456, 16926659444772, 101559956668416, 609359740010496, 3656158440062976, 21936950640377856

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373280, A373281, A373283.

Cf. A000400, A373216.

G.f. A(x) satisfies A(x) = x/(1 - 6*x) + A(x^6).

If n == 0 (mod 6), a(n) = 6^n + a(n/6) otherwise a(n) = 6^n.

A373283 Expansion of Sum_{k>=0} x^(7^k) / (1 - 7*x^(7^k)).

Original entry on oeis.org

1, 7, 49, 343, 2401, 16807, 117650, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010414, 678223072849, 4747561509943, 33232930569601, 232630513987207, 1628413597910449, 11398895185373143, 79792266297612050, 558545864083284007

Offset: 1

Seiichi Manyama, May 30 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A187767, A373279, A373280, A373281, A373282.

Cf. A000420, A373217, A373277.

b(n, k) = sumdiv(n, d, (gcd(d, k)==1)*(moebius(d)*k^(n/d)))/(k*n);
a(n, k=7) = sumdiv(n, d, d*b(d, k));

G.f. A(x) satisfies A(x) = x/(1 - 7*x) + A(x^7).
If n == 0 (mod 7), a(n) = 7^n + a(n/7) otherwise a(n) = 7^n.
a(n) = Sum_{d|n} d * A373277(d).

cp's OEIS Frontend

A127804 a(2n) = 2^(2n), a(2n+1) = 2^(2n+1) + a(n).

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A373279 Expansion of Sum_{k>=0} x^(3^k) / (1 - 3*x^(3^k)).

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A373280 Expansion of Sum_{k>=0} x^(4^k) / (1 - 4*x^(4^k)).

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A373281 Expansion of Sum_{k>=0} x^(5^k) / (1 - 5*x^(5^k)).

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PARI

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A373282 Expansion of Sum_{k>=0} x^(6^k) / (1 - 6*x^(6^k)).

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A373283 Expansion of Sum_{k>=0} x^(7^k) / (1 - 7*x^(7^k)).

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Author

Keywords

Links

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Programs

PARI

Formula