A379006 -id:A379006 - 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

A379005 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 7, 8, 9, 1, 10, 1, 2, 11, 12, 1, 13, 1, 14, 3, 2, 1, 15, 16, 2, 17, 4, 1, 18, 1, 19, 3, 2, 5, 20, 1, 2, 3, 21, 1, 6, 1, 4, 22, 2, 1, 23, 1, 24, 3, 4, 1, 25, 5, 7, 3, 2, 1, 26, 1, 2, 8, 27, 5, 6, 1, 4, 3, 9, 1, 28, 1, 2, 29, 4, 1, 6, 1, 30, 31, 2, 1, 10, 5, 2, 3, 7, 1, 32, 1, 4, 3, 2, 5, 33, 1, 2, 8, 34, 1, 6, 1, 7, 11

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of A355582.
For all i, j:
  A379001(i) = A379001(j) => a(i) = a(j),
  a(i) = a(j) => A322026(i) = A322026(j),
  a(i) = a(j) => A379004(i) = A379004(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007814, A007949, A112765, A355582, A379006 (ordinal transform).

Cf. A322026, A379001, A379004.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v379005 = rgs_transform(vector(up_to, n, [valuation(n,2), valuation(n,3), valuation(n,5)]));
A379005(n) = v379005[n];

A379003 Ordinal transform of A132741, where A132741 is the largest divisor of n having the form 2^i*5^j. a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 5, 2, 6, 3, 2, 1, 7, 4, 8, 1, 9, 5, 10, 2, 1, 6, 11, 3, 12, 2, 13, 1, 14, 7, 3, 4, 15, 8, 16, 1, 17, 9, 18, 5, 4, 10, 19, 2, 20, 1, 21, 6, 22, 11, 5, 3, 23, 12, 24, 2, 25, 13, 26, 1, 6, 14, 27, 7, 28, 3, 29, 4, 30, 15, 2, 8, 31, 16, 32, 1, 33, 17, 34, 9, 7, 18, 35, 5, 36, 4, 37, 10

Offset: 0

Antti Karttunen, Dec 15 2024

nonn

Ordinal transform of the ordered pair [A007814(n), A112765(n)].

This sequence and A379004 are ordinal transforms of each other (if the initial 0 is discarded).

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A007814, A112765, A132741, A379004 (ordinal transform of this sequence after the initial 0).

Cf. also A126760, A379006.

up_to = 20000;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v379003 = ordinal_transform(vector(up_to, n, [valuation(n,2), valuation(n,5)]));
A379003(n) = if(!n,n,v379003[n]);

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

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A379005 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

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A379003 Ordinal transform of A132741, where A132741 is the largest divisor of n having the form 2^i*5^j. a(0) = 0 by convention.

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