A074946 -id:A074946 - cp's OEIS frontend

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

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

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

A175786 Numbers n such that the number of prime divisors of n (counted with multiplicity) is not a divisor of phi(n).

Original entry on oeis.org

8, 12, 20, 30, 32, 44, 48, 50, 54, 64, 66, 68, 72, 75, 80, 81, 92, 96, 102, 108, 110, 112, 116, 120, 125, 128, 138, 160, 162, 164, 165, 168, 170, 174, 180, 188, 192, 208, 212, 230, 236, 240, 242, 243, 246, 252, 255, 270, 272, 275, 280, 282, 284, 288, 290, 304

Offset: 1

Enrique Pérez Herrero, Sep 04 2010

nonn

a(n) is in the sequence if A001222(n) does not divides A000010(n).

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

Cf. A074946, A109421, A109423, A013929.

Select[Range[2,400],Mod[EulerPhi[#],PrimeOmega[#]]!=0&] (* Harvey P. Dale, Mar 14 2020 *)

isok(n) =  (eulerphi(n) % bigomega(n)); \\ Michel Marcus, Aug 27 2013

A224705 Composite numbers n divisible by Omega(n)^2 (the square of the number of their prime factors, counted with multiplicity).

Original entry on oeis.org

4, 16, 18, 27, 45, 63, 99, 117, 144, 153, 171, 200, 207, 216, 256, 261, 279, 300, 324, 333, 360, 369, 384, 387, 423, 450, 477, 500, 504, 531, 540, 549, 576, 603, 639, 640, 657, 675, 700, 711, 747, 750, 756, 792, 801, 873, 896, 900, 909, 927, 936, 960, 963, 981

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 16 2013

nonn

A number n is in the sequence if and only if mod(n, A001222(n)^2) == 0 and n is not prime.

Without the restriction that n must be composite, all prime numbers would trivially be included in the sequence.

			a(6)=63=3*3*7, and 63 is divisible by 9=3^2; a(9)=144, which has 6 prime factors and is divisible by 36.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A001222, A074946, A137230, A070003, A224703.

isA224705 := proc(n)
    if isprime(n) then
        return false;
    else
        if modp(n,numtheory[bigomega](n)^2) = 0 then
            true;
        else
            false;
        end if;
    end if;
end proc:
n := 1;
c := 4;
while n <= 10000 do
    if isA224705(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
    c := c+1 ;
end do: # R. J. Mathar, Mar 14 2016

Select[Range[2, 1000], ! PrimeQ[#] && Mod[#, PrimeOmega[#]^2] == 0 &] (* T. D. Noe, Apr 18 2013 *)

y=c(); i=2; isint<-function(x) x==as.integer(x)
while(length(y)<10000) {Omega=length(factorize(i)); if(Omega>1) if(isint(i/Omega^2)) y=c(y,i); i=i+1 }

A144147 A positive integer n is included if every exponent in the prime-factorization of n is coprime to n and if the sum of these prime-factorization exponents divides n.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 30, 31, 34, 37, 38, 40, 41, 42, 43, 45, 46, 47, 53, 56, 58, 59, 61, 62, 63, 66, 67, 71, 73, 74, 75, 78, 79, 82, 83, 86, 88, 89, 94, 96, 97, 99, 101, 102, 103, 104, 105, 106, 107, 109, 113, 114, 117, 118, 122, 127, 131

Offset: 1

Leroy Quet, Sep 11 2008

nonn

All primes are included in the sequence. Any integer = 2*(odd prime) is included in the sequence. This sequence contains those terms in both sequence A074946 and sequence A144146.

			40 has the prime factorization of 2^3 * 5^1. The exponents are therefore 3 and 1. Both 3 and 1 are coprime to 40. And 3+1 = 4 divides 40. So 40 is included in the sequence.

A074946, A144146

Extended by Ray Chandler, Nov 04 2008

cp's OEIS Frontend

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

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

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

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

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

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A175786 Numbers n such that the number of prime divisors of n (counted with multiplicity) is not a divisor of phi(n).

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A224705 Composite numbers n divisible by Omega(n)^2 (the square of the number of their prime factors, counted with multiplicity).

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A144147 A positive integer n is included if every exponent in the prime-factorization of n is coprime to n and if the sum of these prime-factorization exponents divides n.

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