A028815 -id:A028815 - cp's OEIS frontend

A299763 a(n) = 1 + A182986(n).

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

Omar E. Pol, Mar 14 2018

Are these the indices of the rows of A299762 where there is a record?

G. C. Greubel, Table of n, a(n) for n = 1..10000

First differences are in A054541.
Cf. A008864, A028815, A182986, A299762.
Essentially the same as A008864, A028815, A055670, A135731, A175216.

A299763:= func< n | n eq 1 select 1 else NthPrime(n-1) + 1 >;
[A299763(n): n in [1..70]]; // G. C. Greubel, Aug 05 2024

A299763[n_]:= If[n==1, 1, Prime[n-1] +1];
Table[A299763[n], {n,70}] (* G. C. Greubel, Aug 05 2024 *)

def A299763(n): return 1 if n==1 else nth_prime(n-1) + 1
[A299763(n) for n in range(1,71)] # G. C. Greubel, Aug 05 2024

a(n) = A028815(n-1) - [n=1].

a(n) = A008864(n-1) for n >= 2, with a(1) = 1.

A309633 G.f.: x * Sum_{k>=1} x^k / (1 - a(k)*x^k).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 6, 4, 1, 7, 1, 13, 11, 6, 1, 7, 17, 7, 11, 31, 1, 45, 1, 10, 18, 6, 146, 34, 1, 9, 27, 141, 1, 261, 1, 78, 364, 8, 1, 44, 730, 537, 18, 145, 1, 255, 1281, 2203, 51, 33, 1, 2213, 1, 47, 7461, 221, 4722, 1159, 1, 85, 38, 27948, 1, 2342, 1, 36, 17060, 347, 63146, 3427, 1

Offset: 1

Ilya Gutkovskiy, Aug 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..14665

Cf. A028815 (positions of 1's), A087909, A127525, A309634.

a[n_] := a[n] = SeriesCoefficient[x Sum[x^k/(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 80}]
a[n_] := a[n] = Sum[a[d]^((n - 1)/d - 1) , {d, Divisors[n - 1]}]; a[1] = 0; a[2] = 1; Table[a[n], {n, 1, 80}]

seq(n)={my(v=vector(n)); for(n=1, #v-1, v[n+1]=sumdiv(n, d, v[d]^(n/d-1))); v} \\ Andrew Howroyd, Aug 10 2019

a(1) = 0; a(n+1) = Sum_{d|n} a(d)^(n/d-1).

A309634 G.f.: x * Sum_{k>=1} x^k / (1 - x^k)^a(k).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 5, 4, 1, 7, 1, 9, 8, 6, 1, 7, 6, 7, 8, 14, 1, 18, 1, 9, 12, 6, 23, 17, 1, 9, 17, 17, 1, 35, 1, 31, 41, 8, 1, 23, 29, 24, 12, 44, 1, 33, 47, 49, 30, 16, 1, 61, 1, 20, 120, 40, 84, 105, 1, 35, 23, 85, 1, 68, 1, 19, 115, 88, 151, 160, 1

Offset: 1

Ilya Gutkovskiy, Aug 10 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1001 terms from Antti Karttunen)

Cf. A028815 (positions of 1's), A157019, A309633.

a:= proc(n) option remember; uses numtheory;
      add(binomial((n-1)/d+a(d)-2, a(d)-1), d=divisors(n-1))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 27 2025

a[n_] := a[n] = SeriesCoefficient[x Sum[x^k/(1 - x^k)^a[k], {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 80}]
a[n_] := a[n] = Sum[Binomial[(n - 1)/d + a[d] - 2, a[d] - 1], {d, Divisors[n - 1]}]; a[1] = 0; Table[a[n], {n, 1, 80}]

seq(n)={my(v=vector(n)); v[2]=1; for(n=2, #v-1, v[n+1] = sumdiv(n, d, binomial(n/d + v[d] - 2, v[d] - 1))); v} \\ Andrew Howroyd, Aug 10 2019

a(1) = 0; a(n+1) = Sum_{d|n} binomial(n/d+a(d)-2,a(d)-1).

cp's OEIS Frontend

A299763 a(n) = 1 + A182986(n).

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A309633 G.f.: x * Sum_{k>=1} x^k / (1 - a(k)*x^k).

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A309634 G.f.: x * Sum_{k>=1} x^k / (1 - x^k)^a(k).

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