A372401 -id:A372401 - cp's OEIS frontend

A202821 Position of 6^n among 3-smooth numbers A003586.

1, 5, 14, 26, 43, 64, 89, 119, 153, 191, 233, 279, 330, 385, 444, 507, 575, 646, 722, 802, 886, 975, 1067, 1164, 1266, 1371, 1481, 1595, 1713, 1835, 1961, 2092, 2227, 2366, 2509, 2657, 2809, 2965, 3125, 3289, 3458, 3630, 3807, 3989, 4174, 4364, 4558, 4756

Offset: 0

Zak Seidov, Dec 25 2011

nonn

			a(0) = 1 because A003586(1) = 6^0 = 1.
a(1) = 5 because A003586(5) = 6^1 = 6.
a(2) = 14 because A003586(14) = 6^2 = 36.

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)

Cf. A000400, A003586, A022330, A022331.

a[n_] := Sum[Floor[Log[3, 6^n/2^i]] + 1, {i, 0, Log2[6^n]}]; Array[a, 50, 0] (* Amiram Eldar, Jul 15 2023 *)

# uses imports/function in A372401
print(list(islice(A372401gen(p=3), 1000))) # Michael S. Branicky, Jun 06 2024

from sympy import integer_log
def A202821(n): return 1+n*(n+1)+sum((m:=3**i).bit_length()+((1<Chai Wah Wu, Oct 22 2024

A003586(a(n)) = 6^n, for n >= 0.

a(n) ~ (log(6))^2/(log(3)*log(4))*n^2 = 2.1079...*n^2.

A372400 Position of 30^n among 5-smooth numbers A051037.

Original entry on oeis.org

1, 18, 83, 228, 486, 888, 1466, 2255, 3283, 4583, 6189, 8134, 10445, 13158, 16305, 19916, 24027, 28667, 33870, 39665, 46086, 53166, 60937, 69429, 78675, 88709, 99561, 111263, 123849, 137347, 151793, 167219, 183658, 201139, 219695, 239359, 260165, 282141, 305320

Offset: 0

Michael De Vlieger, Jun 03 2024

nonn

Also position of 30^(n+1) in A143207.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Cf. A002110, A051037, A143207, A202821, A372401, A372402.

Table[Sum[Floor@ Log[5, 30^n/(2^i*3^j)] + 1, {i, 0, Log[2, 30^n]}, {j, 0, Log[3, 30^n/2^i]}], {n, 0, 38}]

a(n)=my(t=30^n,u=5*t); sum(a=0,logint(t,5), u\=5; sum(b=0,logint(u,3), logint(u\3^b,2)+1)) \\ Charles R Greathouse IV, Sep 18 2024

# uses imports/function in A372401
print(list(islice(A372401gen(p=5), 40))) # Michael S. Branicky, Jun 05 2024

from sympy import integer_log
def A372400(n):
    c, x = 0, 30**n
    for i in range(integer_log(x,5)[0]+1):
        for j in range(integer_log(y:=x//5**i,3)[0]+1):
            c += (y//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = k*n^3 + (3k/2)*n^2 + O(n) where k = (log 30)^3/(6 log 2 log 3 log 5) = 5.35057081984.... - Charles R Greathouse IV, Sep 19 2024

A372402 Position of 2310^n among 11-smooth numbers A051038.

Original entry on oeis.org

1, 283, 3847, 20996, 74228, 203084, 469053, 960396, 1797086, 3135610, 5173909, 8156188, 12377846, 18190320, 26005929, 36302854, 49629820, 66611231, 87951744, 114441450, 146960432, 186483973, 234087084, 290949702, 358361266, 437725888, 530566933, 638532124, 763398291, 907076258

Offset: 0

Michael De Vlieger, Jun 03 2024

nonn

Also position of 2310^(n+1) in A147572.

Cf. A022330, A051038, A147572, A372400, A372401.

Table[
  Sum[Floor@ Log[11, 2310^n/(2^i*3^j*5^k*7^m)] + 1,
    {i, 0, Log[2, 2310^n]},
    {j, 0, Log[3, 2310^n/2^i]},
    {k, 0, Log[5, 2310^n/(2^i*3^j)]},
    {m, 0, Log[7, 2310^n/(2^i*3^j*5^k)]}],
  {n, 0, 8}]

# uses imports/function in A372401
print(list(islice(A372401gen(p=11), 7))) # Michael S. Branicky, Jun 05 2024

from sympy import integer_log, prevprime
def A372402(n):
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    return g(2310**n,11) # Chai Wah Wu, Sep 16 2024

a(14)-a(18) from Michael S. Branicky, Jun 05 2024

More terms from David A. Corneth, Jun 05 2024

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A202821 Position of 6^n among 3-smooth numbers A003586.

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A372400 Position of 30^n among 5-smooth numbers A051037.

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A372402 Position of 2310^n among 11-smooth numbers A051038.

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