A379005 -id:A379005 - 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

A379001 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(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, 7, 8, 9, 10, 7, 11, 7, 12, 13, 14, 7, 15, 7, 16, 17, 12, 7, 18, 19, 12, 20, 21, 7, 22, 7, 23, 17, 12, 24, 25, 7, 12, 17, 26, 7, 27, 7, 21, 28, 12, 7, 29, 30, 31, 17, 21, 7, 32, 24, 33, 17, 12, 7, 34, 7, 12, 35, 36, 24, 27, 7, 21, 17, 37, 7, 38, 7, 12, 39, 21, 40, 27, 7, 41, 42, 12, 7, 43, 24, 12, 17, 33, 7, 44, 40, 21, 17, 12, 24, 45, 7, 46, 35

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of ordered 4-tuple [A046523(n), A007814(n), A007949(n), A112765(n)].
For all i, j:
  A379000(i) = A379000(j) => a(i) = a(j),
  a(i) = a(j) => A358230(i) = A358230(j),
  a(i) = a(j) => A379002(i) = A379002(j),
  a(i) = a(j) => A379005(i) = A379005(j).

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

Cf. A046523, A007814, A007949, A112765.

Cf. A358230, A379000, A379002, A379005.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v379001 = rgs_transform(vector(up_to, n, [A046523(n), valuation(n,2), valuation(n,3), valuation(n,5)]));
A379001(n) = v379001[n];

A379004 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814) and v_5 (A112765) give the 2- and 5-adic valuations of n respectively.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 1, 5, 1, 6, 1, 3, 1, 2, 4, 7, 1, 2, 1, 8, 1, 2, 1, 5, 9, 2, 1, 3, 1, 6, 1, 10, 1, 2, 4, 3, 1, 2, 1, 11, 1, 2, 1, 3, 4, 2, 1, 7, 1, 12, 1, 3, 1, 2, 4, 5, 1, 2, 1, 8, 1, 2, 1, 13, 4, 2, 1, 3, 1, 6, 1, 5, 1, 2, 9, 3, 1, 2, 1, 14, 1, 2, 1, 3, 4, 2, 1, 5, 1, 6, 1, 3, 1, 2, 4, 10, 1, 2, 1, 15, 1, 2, 1, 5, 4

Offset: 1

Antti Karttunen, Dec 15 2024

Restricted growth sequence transform of A132741, or equally, of the ordered pair [A007814(n), A112765(n)].
For all i, j:
  A379005(i) = A379005(j) => a(i) = a(j).
A379003 (after its initial 0) and this sequence are ordinal transforms of each other.

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

Cf. A007814, A112765, A132741, A379003 (ordinal transform), A379005.

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; };
v379004 = rgs_transform(vector(up_to, n, [valuation(n,2), valuation(n,5)]));
A379004(n) = v379004[n];

A379006 Ordinal transform of A355582, where A355582 is the largest 5-smooth divisor of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A355582.
Cf. A379005 (ordinal transform).
Cf. also A003602, A126760.

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; };
is_A051037(n) = (n<7||vecmax(factor(n, 6)[, 1])<7); \\ From A051037
A355582(n) = fordiv(n,d,if(is_A051037(n/d),return(n/d)));
v379006 = ordinal_transform(vector(up_to, n, A355582(n)));
A379006(n) = v379006[n];

A379000 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is prime > 5, with f(n) = n for all other n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 7, 11, 7, 12, 13, 14, 7, 15, 7, 16, 17, 18, 7, 19, 20, 21, 22, 23, 7, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 7, 33, 7, 34, 35, 36, 7, 37, 38, 39, 40, 41, 7, 42, 43, 44, 45, 46, 7, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 7, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 7, 65, 66, 67, 68, 69, 7, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 7

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

For all i, j:
  a(i) = a(j) => A305900(i) = A305900(j) => A305801(i) = A305801(j) => A305800(i) = A305800(j),
  a(i) = a(j) => A379001(i) = A379001(j) => A379002(i) = A379002(j),
  a(i) = a(j) => A379005(i) = A379005(j).

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

Cf. A000720.

Cf. A112765, A305800, A305801, A305900, A379001, A379002, A379005.

A379000(n) = if(n<=7, n, if(isprime(n), 7, 4+n-primepi(n)));

For n < 7, a(n) = n, for primes > 5, a(n) = 7, and for composite n > 7, a(n) = 4+n-A000720(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A379001 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A379004 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814) and v_5 (A112765) give the 2- and 5-adic valuations of n respectively.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A379006 Ordinal transform of A355582, where A355582 is the largest 5-smooth divisor of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A379000 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is prime > 5, with f(n) = n for all other n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula