A220218 -id:A220218 - cp's OEIS frontend

A055229 Greatest common divisor of largest square dividing n and squarefree part of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Labos Elemer, Jun 21 2000

Record values occur at cubes of squarefree numbers: a(A062838(n)) = A005117(n) and a(m) < A005117(n) for m < A062838(n). - Reinhard Zumkeller, Apr 09 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000188, A007913, A007947, A008833.

Cf. A027748, A005117, A062838, A060476, A124010, A220218.

a055229 n = product $ zipWith (^) ps (map (flip mod 2) es) where
   (ps, es) = unzip $
              filter ((> 1) . snd) $ zip (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Oct 27 2015

a[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]}& /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 05 2014 *)

a(n)=my(c=core(n));gcd(c,n/c) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = gcd[A008833(n), A007913(n)].
Multiplicative with a(p^e)=1 for even e, a(p)=1, a(p^e)=p for odd e>1. - Vladeta Jovovic, Apr 30 2002
A220218(a(n)) = 1; A060476(a(n)) > 1 for n > 1. - Reinhard Zumkeller, Nov 30 2015
a(n) = core(n)*rad(n/core(n))/rad(n), where core = A007913 and rad = A007947. - Conjecture by Velin Yanev, proof by David J. Seal, Sep 19 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 + p^2 + p - 1)/(p^2 * (p + 1))) = 1.2249749939341923764... . - Amiram Eldar, Oct 08 2022

A335275 Numbers k such that the largest square dividing k is a unitary divisor of k.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 06 2020

nonn

Numbers k such that gcd(A008833(k), k/A008833(k)) = 1.
Numbers whose prime factorization contains exponents that are either 1 or even.
Numbers whose powerful part (A057521) is a square.
First differs from A220218 at n = 227: a(227) = 256 is not a term of A220218.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465).
Complement of A295661. - Vaclav Kotesovec, Jul 07 2020
Differs from A096432 in having or not having 1, 256, 432, 648, 768, 1280, 1728, 1792, 2000, 2160, 2304,... - R. J. Mathar, Jul 22 2020
Equivalently, numbers k whose squarefree part (A007913) is a unitary divisor, or gcd(A007913(k), A008833(k)) = 1. - Amiram Eldar, Oct 09 2022

			12 is a term since the largest square dividing 12 is 4, and 4 and 12/4 = 3 are coprime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 3.1.2, p. 222.

A000290, A138302 and A220218 are subsequences.

Cf. A007913, A008833, A057521, A065465, A295661.

seqQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 1 || EvenQ[#] &];  Select[Range[100], seqQ]

isok(k) = my(d=k/core(k)); gcd(d, k/d) == 1; \\ Michel Marcus, Jul 07 2020

A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A138302 \ {1} at n = 378: a(378) = 432 = 2^4 * 3^3 is not a term of A138302.
First differs from A096432, A220218 \ {1}, A335275 \ {1} and A337052 \ {1} at n = 56, and from A270428 \ {1} at n = 113.
Numbers k such that A051903(k) is a power of 2.
The asymptotic density of this sequence is 1/zeta(3) + Sum_{k>=2} (1/zeta(2^k+1) - 1/zeta(2^k)) = 0.87442038669659566330... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000079, A002117, A051903.
Subsequences: A001146, A001248, A005117, A030514, A030635, A050376, A062503, A113849, A138302 \ {1}, A179645.
Similar sequences: A368714, A369937, A369939.
Cf. A096432, A220218, A270428, A335275, A337052.

pow2Q[n_] := n == 2^IntegerExponent[n, 2];
Select[Range[2, 100], pow2Q[Max[FactorInteger[#][[;; , 2]]]] &]
Select[Range[2,80],IntegerQ[Log2[Max[FactorInteger[#][[;;,2]]]]]&] (* Harvey P. Dale, Nov 06 2024 *)

ispow2(n) = n >> valuation(n, 2) == 1;
is(n) = n > 1 && ispow2(vecmax(factor(n)[, 2]));

A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 13 2024

First differs from A138302 and A270428 at n = 57: a(57) = 64 is not a term of A138302 and A270428.
First differs from A337052 at n = 193: A337052(193) = 216 is not a term of this sequence.
First differs from A335275 at n = 227: A335275(227) = 256 is not a term of this sequence.
First differs from A220218 at n = 903: A220218(903) = 1024 is not a term of this sequence.
Numbers k such that A376886(k) = A001221(k).
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^A051683(k))) = 0.87902453718626485582... .
a(n) = A096432(n-1) for 2<=n<380, but then the sequences start to differ: A096432 contains 432, 648, 1024, 1728, 2000, 2160,... which are not in this sequence. - R. J. Mathar, Oct 15 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A051683, A376886, A377021, A377022.
Subsequences: A005117, A004709, A377019.
Cf. A138302, A220218, A270428, A335275, A337052.

expQ[n_] := expQ[n] = Module[{m = n, k = 2}, While[Divisible[m, k], m /= k; k++]; m < k]; q[n_] := AllTrue[FactorInteger[n][[;;, 2]], expQ]; Select[Range[100], q]

isf(n) = {my(k = 2); while(!(n % k), n /= k; k++); n < k;}
is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isf(e[i]), return(0))); 1;}

A337052 Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 12 2020

nonn

Numbers k such that A001222(A057521(k)) == 0 (mod 2).
Numbers k such that A057521(k) is in A028260.
Differs from A096432 by having the additional terms 1 and 216, 256, 768, 864, ... and not having the terms 432, 648, ...
First differs from both A220218 and A335275 at n = 193: a(193) = 216 is not a term of these two sequences.
Cohen (1964) proved that this sequence has an asymptotic density, and gave the value 1/2 + (1/5) * Product_{p prime} (1 + (p^2 + p + 1)/(p^3 * (p + 1))) = 0.8172707179... But the numbers of terms not exceeding 10^k for k = 1, 2, ... are 9, 90, 885, 8849, 88499, 884993, 8849889, 88498711, 884987643, 8849876178, ... indicating that the asymptotic density is about 0.88498...

			2 is a term since the powerful part of 2 is 1, which has 0 prime divisors, and 0 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 4.2.2, pp. 226-227.

Cf. A001222, A028260, A057521, A096432.

Subsequences: A000290, A004709, A005117, A138302, A220218, A335275 and A336615.

Select[Range[100], EvenQ @ Total @ Select[FactorInteger[#][[;; , 2]], #1 > 1 &] &]

A220219 Products of primorials where all exponents in its prime factorization are one less than a prime.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 30, 36, 48, 60, 64, 144, 180, 192, 210, 240, 420, 576, 720, 900, 960, 1024, 1260, 1296, 1680, 2310, 2880, 3072, 3600, 4096, 4620, 5040, 5184, 6300, 6480, 6720, 9216, 12288, 13860, 14400

Offset: 1

Charles R Greathouse IV, Dec 07 2012

nonn

Erdős & Mirsky call these B-numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Erdős and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. 2 (1952), pp. 257-271.

Intersection of A025487 and A220218.

is(n)=my(e=valuation(n,2),e1); if(!isprime(e+1), return(n==1)); n>>=e; forprime(p=3,, if(n==1, return(1)); e1=valuation(n,p); if(!isprime(e1+1) || e1 > e, return(0)); n/=p^e1; e = e1)

Erdős & Mirsky show that there are exp((k + o(1)) sqrt(log x)/log log x) members of this sequence below x, where k = Pi * sqrt(8/3) = 5.130....

cp's OEIS Frontend

A055229 Greatest common divisor of largest square dividing n and squarefree part of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A335275 Numbers k such that the largest square dividing k is a unitary divisor of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A377020 Numbers whose prime factorization has exponents that are all numbers of the form m*k!, where 1 <= m <= k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A337052 Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A220219 Products of primorials where all exponents in its prime factorization are one less than a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula