A061701 -id:A061701 - cp's OEIS frontend

A069782 Numbers k such that gcd(d(k^3), d(k)) = 2^w for some w.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107

Offset: 1

Labos Elemer, Apr 08 2002

The first missing integer is 432 (see in A069781).

			Below 100000 only 314 integers are missing, collected in A069781.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Tanya Khovanova, Non Recursions

Cf. A000005, A069780, A069781, A037992, A061701.

f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]]; Do[s=f[n]; If[IntegerQ[Log[2, s]], Print[{n, s}]], {n, 1, 100000}]

is(n)=my(f=factor(n)[, 2], g=gcd(prod(i=1, #f, 3*f[i]+1), prod(i=1, #f, f[i]+1))); g>>valuation(g, 2)==1 \\ Charles R Greathouse IV, Oct 16 2015

A069781 Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

Original entry on oeis.org

432, 576, 648, 1600, 2000, 2160, 2880, 2916, 3024, 3136, 3240, 4032, 4536, 4752, 4800, 5000, 5488, 5616, 6000, 6336, 7128, 7344, 7488, 7744, 8208, 8424, 9408, 9792, 9936, 10125, 10800, 10816, 10944, 11016, 11200, 12312, 12528, 13248, 13392

Offset: 1

Labos Elemer, Apr 08 2002

nonn

The complement of this sequence in the positive integers A000027 is A069782. - M. F. Hasler, Jan 18 2015

The numbers of the form 4*3^(7*m - 1), m >= 1, are terms. - Marius A. Burtea, Oct 18 2019

			For n<100000, gcd[d(n^3),d[n]] = {5,7,10,14,20,28,40,80} which is obtained for n={20736,576,432,2880,54000,20160,2160,15120} respectively.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A061701, A037992, A069780, A069782.

f:=func; [k:k in [1..14000]| not IsIntegral(Log(2,f(k)))]; // Marius A. Burtea, Oct 18 2019

f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]] Do[s=f[n]; If[ !IntegerQ[Log[2, s]], Print[n]], {n, 1, 100000}]
Select[Range[14000],!IntegerQ[Log[2,GCD[DivisorSigma[0,#^3], DivisorSigma[ 0,#]]]]&] (* Harvey P. Dale, Mar 20 2018 *)

is(n)=my(f=factor(n)[,2], g=gcd(prod(i=1,#f,3*f[i]+1), prod(i=1,#f,f[i]+1))); g!=1<Charles R Greathouse IV, Oct 16 2015

log_2(gcd(A000005(n^3), A000005(n))) is nonintegral.

A069785 a(n) = A061680(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 15, 15, 3, 5, 135, 135, 99, 99, 9, 63, 21, 21, 459, 459, 135, 19, 15, 15, 15, 21, 189, 189, 585, 585, 18225, 18225, 675, 15, 135, 891, 8505, 25515, 81, 81, 7695, 7695, 1575, 1575, 4725, 6615, 40635, 40635, 945, 1215, 3645, 3645, 151875, 151875

Offset: 1

Labos Elemer, Apr 09 2002

			Observe cases when consecutive terms are equal: n={1,2,3,4,6,10,...,78,80,82,88,96}.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000142, A061680, A061701, A048691, A069780, A069781, A069782, A069783, A069784.

a[n_] := Module[{e = FactorInteger[n!][[;;, 2]]}, GCD[Times @@ (2*e+1), Times @@ (e+1)]]; Array[a, 100] (* Amiram Eldar, Dec 02 2023 *)

a(n) = {my(e = factor(n!)[,2]); gcd(vecprod(apply(x -> 2*x+1, e)), vecprod(apply(x -> x+1, e)));} \\ Amiram Eldar, Dec 02 2023

a(n) = A061680(A000142(n)). - Amiram Eldar, Dec 02 2023

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A069782 Numbers k such that gcd(d(k^3), d(k)) = 2^w for some w.

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A069781 Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

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A069785 a(n) = A061680(n!).

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