A373281 -id:A373281 - cp's OEIS frontend

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

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.

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

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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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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PARI

Formula