A004128 -id:A004128 - cp's OEIS frontend

A381974 Primes of the form Sum_{k >= 0} floor(m/3^k) for some number m.

2, 5, 13, 17, 19, 23, 31, 41, 53, 59, 61, 67, 71, 89, 97, 101, 103, 107, 127, 131, 139, 149, 151, 157, 163, 167, 179, 191, 193, 197, 211, 223, 227, 229, 233, 251, 257, 263, 269, 277, 283, 313, 317, 331, 337, 349, 353, 373, 379, 383, 409, 419, 421, 431, 439

Offset: 1

Clark Kimberling, Apr 01 2025

nonn

			[9/1] + [9/3] + [9/9] = 13, where [ ] = floor, so 13 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A381973. Includes A076481.

f:= proc(n) local k; add(floor(n/3^k),k=0..ilog[3](n)) end proc:
select(isprime, map(f, [$2..100])); # Robert Israel, Apr 21 2025

f[n_] := Sum[Floor[n/3^k], {k, 0, Floor[Log[3, n]]}]  (* A004128 *)
u = Select[Range[400], PrimeQ[f[#]] &]  (* A381973 *)
Map[f, u]   (* A381974 *)

A306534 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Sum_{j=0..n} floor(n/k^j).

Original entry on oeis.org

0, 0, 2, 0, 1, 6, 0, 1, 3, 12, 0, 1, 2, 4, 20, 0, 1, 2, 4, 7, 30, 0, 1, 2, 3, 5, 8, 42, 0, 1, 2, 3, 5, 6, 10, 56, 0, 1, 2, 3, 4, 6, 8, 11, 72, 0, 1, 2, 3, 4, 6, 7, 9, 15, 90, 0, 1, 2, 3, 4, 5, 7, 8, 10, 16, 110, 0, 1, 2, 3, 4, 5, 7, 8, 10, 13, 18, 132, 0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 19, 156

Offset: 0

Ilya Gutkovskiy, Feb 22 2019

			Square array begins:
   0,  0,  0,  0,  0,  0,  ...
   2,  1,  1,  1,  1,  1,  ...
   6,  3,  2,  2,  2,  2,  ...
  12,  4,  4,  3,  3,  3,  ...
  20,  7,  5,  5,  4,  4,  ...
  30,  8,  6,  6,  6,  5,  ...

Columns k=1..4 give A002378, A005187, A004128, A087069.

Cf. A306533.

Table[Function[k, Sum[Floor[n/k^j], {j, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

G.f. of column k (for k > 1): (1/(1 - x)) * Sum_{j>=0} x^(k^j)/(1 - x^(k^j)).

A346502 a(n) = 3n - (sum of digits of 3n in base 3).

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 18, 20, 26, 28, 30, 34, 36, 38, 42, 44, 46, 52, 54, 56, 60, 62, 64, 68, 70, 72, 80, 82, 84, 88, 90, 92, 96, 98, 100, 106, 108, 110, 114, 116, 118, 122, 124, 126, 132, 134, 136, 140, 142, 144, 148, 150, 152, 160, 162, 164, 168, 170, 172

Offset: 0

Bernard Schott, Jul 21 2021

Terms of A344853 without repetition.
All terms are even.
A new largest gap between 2 consecutive terms is obtained between a(3^m-1) and a(3^m), m >= 0 (see formula).
In base 2, A005187(n) = 2n - (sum of digits of 2n in base 2) is also the exponent of the largest power of 2 dividing (2n)!, but here the exponent of the largest power of 3 dividing (3n)! is not a(n) but A004128(n).

			a(8) = 24 - (sum of digits of 24 in base 3); 24_10 = 220_3 and 2+2+0 = 4, so a(8) = 24-4 = 20.

Cf. A005187 (similar, with base 2).

Cf. A004128, A053735, A344853.

a[n_] := 3*n - Plus @@ IntegerDigits[3*n, 3]; Array[a, 100, 0] (* Amiram Eldar, Jul 22 2021 *)

a(n) = 3*n - sumdigits(n,3); \\ Kevin Ryde, Jul 21 2021

from sympy.ntheory.digits import digits
def a(n): return 3*n - sum(digits(3*n, 3)[1:])
print([a(n) for n in range(60)]) # Michael S. Branicky, Jul 28 2021

a(n) = 3*n - A053735(3*n).
a(n) = 2*A004128(n).
a(n) = A344853(3n).
a(3^n) - a(3^n-1) = 2*(n+1).

A354157 Numerator of generalized Catalan number c_3(n) (see Comments).

Original entry on oeis.org

1, 1, 5, 104, 836, 7315, 202895, 1949900, 19284511, 1754890501, 18058389349, 188502545504, 5973492827120, 63732573470888, 685813307216632, 22303841469480032, 243350841747362492, 2670252449037801100, 265034693078133749180, 2936064912067020698720

Offset: 0

N. J. A. Sloane, May 29 2022, based on Section 18.6 of Cosgrave (2022)

c_3(n) = (1/3)*(1/(n+1/3))*(Product_{i=0..n-1}(n+i+1/3))/n!. The denominators are powers of 3.

If 1/3 is everywhere changed to 1 we get the usual Catalan numbers A000108.

			The first few c_3(n) are 1, 1/3, 5/9, 104/81, 836/243, 7315/729, 202895/6561, 1949900/19683, 19284511/59049, 1754890501/1594323, 18058389349/4782969, ...

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.6.

Cf. A000108, A004128.

c := proc(n) 1/3 * 1/(n+1/3) * mul(n + i + 1/3, i = 0..(n-1))/n!: end;

c3[n_] := With[{k = 3}, Pochhammer[n+1+1/k, n-1]/(k*n!)];
Table[Numerator[c3[n]], {n, 1, 19}] (* Jean-François Alcover, Apr 14 2023 *)

a(n) = numerator((1/3)*(1/(n+1/3))*prod(i=0, n-1, n+i+1/3)/n!) \\ Rémy Sigrist, May 30 2022

More terms from Rémy Sigrist, May 30 2022

a(0)=1 prepended by Alois P. Heinz, Apr 14 2023

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A381974 Primes of the form Sum_{k >= 0} floor(m/3^k) for some number m.

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A306534 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Sum_{j=0..n} floor(n/k^j).

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A346502 a(n) = 3n - (sum of digits of 3n in base 3).

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A354157 Numerator of generalized Catalan number c_3(n) (see Comments).

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