A075592 -id:A075592 - 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

A185307 Numbers not divisible by the number of their distinct prime factors.

Original entry on oeis.org

1, 15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 70, 75, 77, 85, 87, 91, 93, 95, 99, 110, 111, 115, 117, 119, 123, 129, 130, 133, 135, 140, 141, 143, 145, 147, 153, 154, 155, 159, 161, 170, 171, 175, 177, 182, 183, 185, 187, 189, 190, 201, 203, 205, 207, 209

Offset: 1

Christian N. K. Anderson and Kevin L. Schwartz, Apr 23 2013

nonn

The complement of A075592 (omega(n) divides n).
Though initially sparse, the sequence increases in density. There are more numbers divisible by omega(n) than not from [3,9265], but there are always more indivisible numbers thereafter.
There are 308 more numbers divisible than indivisible in the range from 1 to 2754, 2778, and 2880. This three values are the global maxima.
The asymptotic density of this sequence is 1 (Cooper and Kennedy, 1989). - Amiram Eldar, Jul 10 2020

			The distinct prime factors of 45 are 3 and 5, but 45 is not divisible by 2.

Christian N. K. Anderson, 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.

Cf. A075592 (complement), A001221, A001222, A074946, A134334.

Join[{1},Select[Range[2,300],Mod[#,PrimeNu[#]]!=0&]] (* Harvey P. Dale, Jun 05 2023 *)

isok(n) = iferr(n % omega(n), E, 1); \\ Michel Marcus, Jul 10 2020

library(numbers); isint<-function(x) x==as.integer(x); which(!vapply(1:500,function(n) isint(n/omega(n)),T))

A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664, 12288, 13122, 13824, 15552, 17496, 18432, 20736, 23328

Offset: 1

Enrique Pérez Herrero, Jan 05 2013

nonn

This sequence is closed under multiplication.
Also 1 and the numbers where psi(n)/n = 2, or n/phi(n)=3, or psi(n)/phi(n)=6.
Also 1 and the numbers of the form 2^i*3^j with i, j >= 1 (A033845).
If M(n) is the n X n matrix whose elements m(i,j) = 2^i*3^j, with i, j >= 1, then det(M(n))=0.
Numbers n such that Product_{i=1..q} (1 + 1/d(i)) is an integer where q is the number of the distinct prime divisors d(i) of n. - Michel Lagneau, Jun 17 2016

			psi(48) = 96 and 96/48 = 2 so 48 is in this sequence.

S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..191 from Vincenzo Librandi)
R. Blecksmith, M. McCallum and J. L. Selfridge, 3-smooth representations of integers, Amer. Math. Monthly, 105 (1998), 529-543.
E. Deutsch, Tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Smooth Number
Wikipedia, Closure

Cf. A003586, A001615, A007694, A033950, A074946, A075592.

[6*n: n in [1..3000] | PrimeDivisors(n) subset [2, 3]]; // Vincenzo Librandi, Jan 11 2019

Select[Range[10^4],#/EulerPhi[#]==3 || #==1&]
Join[{1}, 6 Select[Range@4000, Last@Map[First, FactorInteger@#]<=3 &]] (* Vincenzo Librandi, Jan 11 2019 *)

dedekindpsi(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n]);
k=0; n=0; while(k<10000,n++; if( dedekindpsi(n) % n== 0, k++; print1(n, ", ")));

For n > 1, a(n) = 6 * A003586(n).

Sum_{n>0} 1/a(n)^k = 1 + Sum_{i>0} Sum_{j>0} 1/(2^i * 3^j)^k = 1 + 1/((2^k-1)*(3^k-1)).

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

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

A175785 Numbers n such that the number of distinct prime divisors of n does not divide phi(n).

Original entry on oeis.org

30, 60, 66, 102, 110, 120, 132, 138, 150, 165, 170, 174, 204, 220, 230, 240, 246, 255, 264, 276, 282, 290, 300, 318, 340, 345, 348, 354, 374, 408, 410, 426, 435, 440, 460, 470, 480, 492, 498, 506, 528, 530, 534, 550, 552, 561, 564, 580, 590, 600, 606, 615

Offset: 1

Enrique Pérez Herrero, Sep 04 2010

nonn

a(n) gives the integers where omega(n) = A001221(n) does not divide phi(n) = A000010(n).

This sequence does not contain any prime powers (A000961), nor any numbers with only two distinct prime divisors (A007774); so it is a subsequence of A000977.

			30 is in this sequence because omega(30)=3 does not divide phi(30)=8.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Cf. A001221, A000010, A075592, A109425, A109427, A192218, A000977, A000961, A007774.

Select[Range[2,700],Mod[EulerPhi[#],PrimeNu[#]]!=0&] (* Harvey P. Dale, Dec 29 2019 *)

isok(n) = (eulerphi(n) % omega(n) != 0) \\ Michel Marcus, Jun 12 2013

A231876 Numbers n such that omega(n)^2 (cf. A001221) divides n.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 47, 48, 49, 52, 53, 56, 59, 61, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 88, 89, 90, 92, 96, 97, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 121, 124, 125, 126, 127, 128, 131, 136, 137, 139, 144, 148, 149

Offset: 1

N. J. A. Sloane, Nov 17 2013

nonn

Includes all prime powers (A246655), as well as 4*A246655. - Robert Israel, Apr 25 2017

G. C. Greubel, Table of n, a(n) for n = 1..1100

Cf. A001221, A075592, A185307, A001222, A074946, A134334, A231877 (complement).

Cf. A246655.

select(n -> n mod nops(numtheory:-factorset(n))^2 = 0, [$2..1000]); # Robert Israel, Apr 25 2017

Select[Range[2, 500], Mod[#, PrimeNu[#]^2] == 0  &] (* G. C. Greubel, Apr 24 2017 *)

isok(n) = !(n % omega(n)^2); \\ Michel Marcus, Apr 25 2017

A231877 Numbers n such that omega(n)^2 (cf. A001221) does not divide n.

Original entry on oeis.org

1, 6, 10, 14, 15, 18, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 45, 46, 50, 51, 54, 55, 57, 58, 60, 62, 63, 65, 66, 69, 70, 74, 75, 77, 78, 82, 84, 85, 86, 87, 91, 93, 94, 95, 98, 99, 102, 105, 106, 110, 111, 114, 115, 117, 118, 119, 120, 122, 123, 129, 130, 132, 133, 134, 135, 138, 140, 141, 142, 143, 145, 146, 147, 150

Offset: 1

N. J. A. Sloane, Nov 17 2013

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1300

Cf. A001221, A075592, A185307, A001222, A074946, A134334, A231876 (complement).

Join[{1}, Select[Range[2, 500], Mod[#, PrimeNu[#]^2] != 0  &]] (* G. C. Greubel, Apr 24 2017 *)

A231878 Numbers k such that bigomega(k)^2 (cf. A001222) divides k.

Original entry on oeis.org

2, 3, 4, 5, 7, 11, 13, 16, 17, 18, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 53, 59, 61, 63, 67, 71, 73, 79, 83, 89, 97, 99, 101, 103, 107, 109, 113, 117, 127, 131, 137, 139, 144, 149, 151, 153, 157, 163, 167, 171, 173, 179, 181, 191, 193, 197, 199, 200, 207, 211, 216, 223, 227, 229, 233, 239, 241, 251, 256, 257, 261, 263

Offset: 1

N. J. A. Sloane, Nov 17 2013

nonn

Contains all primes. - Ivan Neretin, Apr 05 2016

Ivan Neretin, Table of n, a(n) for n = 1..13517 (all terms up to 10^5)

Cf. A001221, A075592, A185307, A001222, A074946, A134334, A231879 (complement).

Select[Range[2, 265], Divisible[#, PrimeOmega[#]^2] &] (* Ivan Neretin, Apr 05 2016 *)

isok(n) = !(n % bigomega(n)^2); \\ Michel Marcus, Apr 05 2016

A231879 Numbers n such that bigomega(n)^2 (cf. A001222) does not divide n.

Original entry on oeis.org

1, 6, 8, 9, 10, 12, 14, 15, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115

Offset: 1

N. J. A. Sloane, Nov 17 2013

nonn

Contains all semiprimes (A001358) except 4. - Ivan Neretin, Apr 05 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001221, A075592, A185307, A001222, A074946, A134334, A231878 (complement).

Join[{1}, Select[Range[2, 115], ! Divisible[#, PrimeOmega[#]^2] &]] (* Ivan Neretin, Apr 05 2016 *)

lista(nn) = {print1(1, ", "); for(n=2, nn, if(n % bigomega(n)^2 != 0, print1(n, ", ")));} \\ Altug Alkan, Apr 05 2016

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A185307 Numbers not divisible by the number of their distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

R

A187778 Numbers k dividing psi(k), where psi is the Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A175785 Numbers n such that the number of distinct prime divisors of n does not divide phi(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A231876 Numbers n such that omega(n)^2 (cf. A001221) divides n.

Views

Author

Keywords

Comments

Links

Crossrefs