Ralph-Joseph Tatt - cp's OEIS frontend

User: Ralph-Joseph Tatt

Ralph-Joseph Tatt has authored 3 sequences.

A301506 Number of integers less than or equal to n whose largest prime factor is 5.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11

Offset: 0

Ralph-Joseph Tatt, Mar 22 2018

nonn

a(n) increases when n has the form 2^a*3^b*5^c, with a,b >= 0 and c > 0.

A distinct sequence can be generated for each prime number; this sequence is for the prime number 5. For an example using another prime number see A301461.

			a(15) = a(2^0 * 3^1 * 5^1); 5 is the largest prime factor, so a(15) exceeds the previous term by 1. For a(16) = a(2^4), there is no increase from the previous term.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A080193.

Cf. A301461.

clear;clc;
prime = 5;
limit = 10000;
largest_divisor = ones(1,limit+1);
for k = 0:limit
    f = factor(k);
    largest_divisor(k+1) = f(end);
end
for i = 1:limit+1
    FQN(i) = sum(largest_divisor(1:i)==prime);
end
output = [0:limit;FQN]'

N:= 100: # for a(0)..a(N)
L:= sort([seq(seq(seq(2^a*3^b*5^c, c=1..floor(log[5](N/(2^a*3^b)))),
  b = 0..floor(log[3](N/2^a))), a = 0 .. floor(log[2](N)))]):
V:= Array(0..N):
V[L]:= 1:
ListTools:-PartialSums(convert(V,list)); # Robert Israel, Sep 22 2020

Accumulate@ Array[Boole[FactorInteger[#][[-1, 1]] == 5] &, 80, 0] (* Michael De Vlieger, Apr 21 2018 *)

A301461 Number of integers less than or equal to n whose largest prime factor is 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Ralph-Joseph Tatt, Mar 21 2018

nonn

a(n) increases when n has the form 2^a*3^b, with a >= 0 and b > 0.

A distinct sequence can be generated for each prime number; this sequence is for the prime number 3. For an example using another prime number see A301506.

			a(12) = a(2^2 * 3^1); 3 is the largest prime factor, so a(12) exceeds the previous term by 1. For a(13), 13 is a prime, so there is no increase from the previous term.

Cf. A003586, A065119, A071521.

Cf. A301506, A071521, A070939.

clear;clc;
prime = 3;
limit = 10000;
largest_divisor = ones(1,limit+1);
for k = 0:limit
    f = factor(k);
    largest_divisor(k+1) = f(end);
end
for i = 1:limit+1
    FQN(i) = sum(largest_divisor(1:i)==prime);
end
output = [0:limit;FQN]'

Accumulate@ Array[Boole[FactorInteger[#][[-1, 1]] == 3] &, 80, 0] (* Michael De Vlieger, Apr 21 2018 *)

gpf(n) = if (n<=1, n, vecmax(factor(n)[,1]));
a(n) = sum(k=1, n, gpf(k)==3); \\ Michel Marcus, Mar 27 2018

From David A. Corneth, Mar 27 2018: (Start)
a(n) - a(n - 1) = 1 if and only if n is in 3 * A003586. If n isn't in that sequence then a(n) = a(n - 1).
a(3 * n + b) = A071521(n), n > 0, 0 <= b < 3. (End)
a(n) = A071521(n) - A070939(n). - Ridouane Oudra, Mar 24 2025

A303748 a(n) is the number of distinct terms of the form i^j where 0 <= i,j <= n.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 29, 41, 51, 61, 77, 97, 116, 140, 164, 190, 208, 240, 271, 307, 341, 379, 418, 462, 504, 540, 586, 622, 671, 727, 780, 840, 882, 942, 1004, 1068, 1114, 1186, 1255, 1327, 1398, 1478, 1554, 1638, 1718, 1800, 1885, 1977, 2064, 2136, 2226, 2322

Offset: 0

Ralph-Joseph Tatt, Apr 30 2018

nonn

			For n=3 the distinct terms are 0,1,2,3,4,8,9,27 so a(3) = 8.

Michael De Vlieger, Table of n, a(n) for n = 0..500
Ovidiu Bagdasar and Ralph Tatt, On some new arithmetic functions involving prime divisors and perfect powers, Electronic Notes in Discrete Mathematics (2018) Vol. 70, 9-15.

Cf. A126254.

{1}~Join~Array[Length@ Union@ Map[#1^#2 & @@ # &, Rest@ Tuples[Range[0, #], {2}]] &, 51] (* Michael De Vlieger, Jan 31 2019 *)

def distinct(limit):
    unique = set()
    for i in range(limit+1):
        for j in range(limit+1):
            if i**j not in unique:
                unique.add(i**j)
    return len(unique)
print([distinct(i) for i in range(40)])

a(n) = A126254(n) + 1.

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User: Ralph-Joseph Tatt

A301506 Number of integers less than or equal to n whose largest prime factor is 5.

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A301461 Number of integers less than or equal to n whose largest prime factor is 3.

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A303748 a(n) is the number of distinct terms of the form i^j where 0 <= i,j <= n.

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Python

Formula