A112249 -id:A112249 - cp's OEIS frontend

A336064 Numbers divisible by the maximal exponent in their prime factorization (A051903).

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

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is A336065 = 0.848957... (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051903(4) = 2 is a divisor of 4.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.

Cf. A051903, A124184, A336065.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336063.

H[1] = 0; H[n_] := Max[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, H[#]] &]

isok(m) = if (m>1, (m % vecmax(factor(m)[,2])) == 0); \\ Michel Marcus, Jul 08 2020

A112250 Numbers m such that m mod floor(log_2(m)) > 0.

Original entry on oeis.org

5, 7, 8, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Reinhard Zumkeller, Aug 30 2005

nonn

The asymptotic density of this sequence is 1 (Cooper and Kennedy, 1989). - Amiram Eldar, Jul 10 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Curtis N. Cooper and Robert E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly, Vol. 96, No. 2 (1989), pp. 118-124.

Complement of A112249.
A112251 is a subsequence.
Cf. A000523, A112248.

seq(op(select(t -> t mod d > 0, [$2^d .. 2^(d+1)-1])),d=1..6); # Robert Israel, Aug 27 2020

A112248(a(n)) > 0.

Name changed by Robert Israel, Aug 27 2020

A336066 Numbers k such that the exponent of the highest power of 2 dividing k (A007814) is a divisor of k.

Original entry on oeis.org

2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 90, 92, 94, 98, 100, 102, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 138, 140, 142, 144

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

All the terms are even by definition.
If m is a term then m*(2*k+1) is a term for all k>=1.
Šalát (1994) proved that the asymptotic density of this sequence is 0.435611... (A336067).

			2 is a term since A007814(2) = 1 is a divisor of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

A001146 and A039956 are subsequences.
Cf. A007814, A336067.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336068.

Select[Range[2, 150, 2], Divisible[#, IntegerExponent[#, 2]] &]

isok(m) = if (!(m%2), (m % valuation(m,2)) == 0); \\ Michel Marcus, Jul 08 2020

from itertools import count, islice
def A336066_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:n%(~n&n-1).bit_length()==0,count(max(startvalue+startvalue&1,2),2))
A336066_list = list(islice(A336066_gen(startvalue=3),30)) # Chai Wah Wu, Jul 10 2022

A112248 a(n) = n mod floor(log_2(n)).

Original entry on oeis.org

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

Offset: 2

Reinhard Zumkeller, Aug 30 2005

nonn

a(A112249(n)) = 0, a(A112250(n)) > 0, a(A112251(n)) = 1.

Cf. A000523, A112249, A112250.

seq(seq(n mod d, n=2^d .. 2^(d+1)-1),d=1..8); # Robert Israel, Aug 27 2020

Table[Mod[n, Floor @ Log2[n]], {n, 2, 100}] (* Amiram Eldar, Aug 27 2020 *)

a(n) = n % logint(n, 2); \\ Michel Marcus, Aug 27 2020

Name edited by Michel Marcus, Aug 27 2020

A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

Original entry on oeis.org

3, 6, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 51, 54, 57, 60, 66, 69, 72, 75, 78, 84, 87, 90, 93, 96, 102, 105, 108, 111, 114, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 156, 159, 165, 168, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

All the terms are divisible by 3 by definition.

Šalát (1994) proved that the asymptotic density of this sequence is 0.287106... (A336069).

			3 is a term since A007949(3) = 1 is a divisor of 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

A055777 is a subsequence.
Cf. A007949, A336069.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336066.

Select[Range[200], Mod[#, 3] == 0 && Divisible[#, IntegerExponent[#, 3]] &]

isok(m) = if (!(m%3), (m % valuation(m,3)) == 0); \\ Michel Marcus, Jul 08 2020

A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is 1 (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051904(4) = 2 is a divisor of 4.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.

Cf. A051904, A124184.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336064.

h[1] = 0; h[n_] := Min[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, h[#]] &]
Select[Range[2,100],Divisible[#,Min[FactorInteger[#][[All,2]]]]&] (* Harvey P. Dale, Aug 31 2020 *)

isok(m) = if (m>1, (m % vecmin(factor(m)[,2])) == 0); \\ Michel Marcus, Jul 08 2020

cp's OEIS Frontend

A336064 Numbers divisible by the maximal exponent in their prime factorization (A051903).

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A112250 Numbers m such that m mod floor(log_2(m)) > 0.

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A336066 Numbers k such that the exponent of the highest power of 2 dividing k (A007814) is a divisor of k.

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A112248 a(n) = n mod floor(log_2(n)).

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A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

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Mathematica

PARI

A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

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Author

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Comments

Examples

References

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Programs

Mathematica

PARI