A355584 -id:A355584 - cp's OEIS frontend

A355582 a(n) is the largest 5-smooth divisor of n.

1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 20, 3, 2, 1, 24, 25, 2, 27, 4, 1, 30, 1, 32, 3, 2, 5, 36, 1, 2, 3, 40, 1, 6, 1, 4, 45, 2, 1, 48, 1, 50, 3, 4, 1, 54, 5, 8, 3, 2, 1, 60, 1, 2, 9, 64, 5, 6, 1, 4, 3, 10, 1, 72, 1, 2, 75, 4, 1, 6, 1, 80

Offset: 1

Amiram Eldar, Jul 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A006519, A007814, A007949, A038500, A051037, A060904, A065331, A112765, A132741, A165725, A355583, A355584.

Cf. A379005 (rgs-transform), A379006 (ordinal transform).

a[n_] := Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

a(n) = 3^valuation(n, 3) * 5^valuation(n, 5) << valuation(n, 2);

from sympy import multiplicity as v
def a(n): return 2**v(2, n) * 3**v(3, n) * 5**v(5, n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = p^e if p <= 5 and 1 otherwise.
a(n) = A006519(n) * A038500(n) * A060904(n).
a(n) = 2^A007814(n) * 3^A007949(n) * 5^A112765(n).
a(n) = n / A165725(n).
Dirichlet g.f.: zeta(s)*(2^s-1)*(3^s-1)*(5^s-1)/((2^s-2)*(3^s-3)*(5^s-5)). - Amiram Eldar, Dec 25 2022
Sum_{k=1..n} a(k) ~ 2*n*log(n)^3 / (45*log(2)*log(3)*log(5)) + O(n*log(n)^2). - Vaclav Kotesovec, Apr 20 2025

A355583 a(n) is the number of the 5-smooth divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 1, 4, 3, 4, 1, 6, 1, 2, 4, 5, 1, 6, 1, 6, 2, 2, 1, 8, 3, 2, 4, 3, 1, 8, 1, 6, 2, 2, 2, 9, 1, 2, 2, 8, 1, 4, 1, 3, 6, 2, 1, 10, 1, 6, 2, 3, 1, 8, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 6, 3, 1, 4, 1, 10, 5, 2, 1, 6, 2, 2

Offset: 1

Amiram Eldar, Jul 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A007814, A007949, A051037, A072078, A112765, A355582, A355584.

a[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

a(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);

from sympy import multiplicity as v
def a(n): return (v(2, n)+1)*(v(3, n)+1)*(v(5, n)+1)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = e+1 if p <= 5 and 1 otherwise.
a(n) = (A007814(n) + 1)*(A007949(n) + 1)*(A112765(n) + 1).
a(n) = A000005(A355582(n)).
a(n) <= A000005(n), with equality if and only if n is in A051037.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/4.
Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)*(1-1/5^s)). - Amiram Eldar, Dec 25 2022

A356006 The number of prime divisors of n that are not greater than 5, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 0, 3, 0, 1, 2, 4, 0, 3, 0, 3, 1, 1, 0, 4, 2, 1, 3, 2, 0, 3, 0, 5, 1, 1, 1, 4, 0, 1, 1, 4, 0, 2, 0, 2, 3, 1, 0, 5, 0, 3, 1, 2, 0, 4, 1, 3, 1, 1, 0, 4, 0, 1, 2, 6, 1, 2, 0, 2, 1, 2, 0, 5, 0, 1, 3, 2, 0, 2, 0, 5, 4, 1, 0, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 23 2022

Equivalently, the number of prime divisors, counted with multiplicity, of the largest 5-smooth divisor of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007814, A007949, A112765.

Cf. A001222, A112759, A169611, A355582, A355583, A355584.

a[n_] := Plus @@ IntegerExponent[n, {2, 3, 5}]; Array[a, 100]

a(n) = valuation(n, 2) + valuation(n, 3) + valuation(n, 5);

from sympy import multiplicity as v
def a(n): return v(2, n) + v(3, n) + v(5, n)
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 25 2022

Totally additive with a(p) = 1 if p <= 5, and 0 otherwise.
a(n) = A007814(n) + A007949(n) + A112765(n).
a(n) = A001222(A355582(n)).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 7/4.

A355713 Numbers k such that k and k+1 have the same sum of 5-smooth divisors.

Original entry on oeis.org

175, 2224, 2575, 4975, 7024, 9424, 9775, 11824, 12175, 14224, 14575, 16975, 19024, 21424, 21775, 23824, 24175, 26224, 26575, 28975, 31024, 33424, 33775, 35824, 36175, 38224, 38575, 40975, 43024, 45424, 45775, 47824, 48175, 50224, 50575, 52975, 55024, 57424, 57775

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355584(k) = A355584(k+1).
Equivalently, numbers k such that the largest 5-smooth divisors of k and k+1, A355582(k) and A355582(k+1), have the same sum of divisors (A000203).
For all the terms k, both k and k+1 are not squarefree: each of the two largest 5-smooth divisors, of k and k+1, cannot be squarefree, since the squarefree 5-smooth numbers are the divisors of 30 = 2*3*5 (A018255) whose values of sigma (A000203), {1, 3, 4, 6, 12, 18, 24, 72}, are not shared with sigma of any other 5-smooth number.
Apparently, all the terms are of only two types: numbers k such that A355582(k) = 16 and A355582(k+1) = 25, or numbers k such that A355582(k) = 25 and A355582(k+1) = 16. Both types are infinite sequences: The first type is the sequence of numbers of the form 2224 + 2400*m, where m is not congruent to 1 (mod 5), and the second type is the sequence of numbers of the form 175 + 2400*m, where m is not congruent to 3 (mod 5). If there are no other terms, then this sequence is a linear recurrence with a signature (1,0,0,0,0,0,0,1,-1). The question of the existence of other types is equivalent to the question of the existence of two coprime 5-smooth numbers other than 16 and 25 whose sums of divisors are equal.
Are there runs of 3 consecutive numbers with the same sum of 5-smooth divisors? There are no such runs below 5*10^10.

			175 is a term since A355584(175) = A355584(176) = 31.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A002961, A018255, A355584.
Subsequence of A013929 and A068781.
Similar sequences: A002961, A064115, A064125, A293183, A306985, A333949.

f[p_, e_] := If[p > 5, 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] == s[# + 1] &]

s(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;
s1 = s(1); for(k = 2, 6e4, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);

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A355582 a(n) is the largest 5-smooth divisor of n.

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A355583 a(n) is the number of the 5-smooth divisors of n.

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A356006 The number of prime divisors of n that are not greater than 5, counted with multiplicity.

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A355713 Numbers k such that k and k+1 have the same sum of 5-smooth divisors.

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