A046755 -id:A046755 - cp's OEIS frontend

A063921 Quotients arising when A046755(n) is divided by the cube of the number of its divisors.

1, 5, 9, 1, 8, 1, 3, 343, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 16, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 16, 89, 97, 101, 103, 107, 109, 113, 6, 45, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 10, 211, 223, 227, 229, 64, 233, 239

Offset: 1

Labos Elemer, Sep 04 2001

nonn

			Since (2^15)^p is in A046755 when p is a prime > 2, then p appears here at least once. Several terms breaking this regularity come from entries of A046755 of other categories. E.g. x=(2^10)*p*(11^3), d(x)=88, d(x)^3=(2^9)*(11^3) divides x and the quotient is 2p (p not equal to 11). Similar subsequences arise if 11 is replaced with different suitable primes.

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

Cf. A046755, A046754, A046756, A033950.

[k/#Divisors(k)^3:k in [m:m in [1..9000000]|IsIntegral(m/#Divisors(m)^3)]]; // Marius A. Burtea, Aug 07 2019

f[n_] := n/DivisorSigma[0, n]^3; Select[f /@ Range[10^5], IntegerQ] (* Amiram Eldar, Aug 07 2019 *)

a(n)= A046755(n)/(A000005(A046755(n))^3).

A046754 Numbers k such that the square of d(k) (number of divisors) divides k.

Original entry on oeis.org

1, 9, 128, 625, 972, 1152, 2000, 2025, 5625, 6561, 7776, 8100, 10000, 10800, 18000, 21952, 26244, 30000, 32768, 35721, 50625, 55296, 56700, 64000, 64800, 65856, 70000, 80000, 84672, 89100, 90000, 97200, 98304, 99225, 105300, 109760, 110000

Offset: 1

Labos Elemer

nonn

Subset of A033950.

			If k = 972, d(k) = sigma(0,k) = 18. Its square is 324 which divides 972.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A033950, A046755, A046756.

Select[Range[110000], IntegerQ[#/DivisorSigma[0, #]^2] &] (* Jayanta Basu, Jun 28 2013 *)

A046756 Numbers k such that d(k)^4 divides k.

Original entry on oeis.org

1, 625, 6561, 117649, 4100625, 31250000, 37515625, 73530625, 771895089, 1000000000, 2147483648, 6442450944, 10737418240, 15032385536, 23622320128, 25937424601, 27917287424, 32212254720, 33059881728, 36507222016, 40802189312, 45097156608, 49392123904

Offset: 1

Labos Elemer

nonn

Proper subset of A033950, A046754 and A046755. Relatively prime terms are in the sequence together with their products like 73530625 or 771895089000000000.

2^31 is a term, as is every integer of the form 2^31*p, 2^31*p^3, and 2^31*p*q, where p and q are distinct odd primes; each of these has 32, 64, or 128 divisors. Of the first 10000 terms, 9609 are of one of those forms. Of the remaining 391 terms, 316 are of the form 2^8 * 3^17 * m, where m is 1, a prime > 3, or 5^4. - Jon E. Schoenfield, Aug 13 2022

			If k=625, d(k) = sigma(0,k) = 5. Its 4th power is 625, which divides k.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A033950, A046754, A046755.

Select[ Range[ 1, 14500000 ], IntegerQ[ #/(DivisorSigma[ 0, # ])^2 ]& ]

a(6)-a(20) from Donovan Johnson, Nov 30 2008

a(21)-a(23) from Donovan Johnson, Jun 08 2011

A048169 n is divisible by the cube of the number of unitary divisors of n (A034444).

Original entry on oeis.org

1, 8, 16, 32, 64, 128, 192, 256, 320, 384, 448, 512, 576, 640, 704, 768, 832, 896, 1024, 1088, 1152, 1216, 1280, 1408, 1472, 1536, 1600, 1664, 1728, 1792, 1856, 1984, 2048, 2176, 2304, 2368, 2432, 2560, 2624, 2752, 2816, 2944, 3008, 3072, 3136, 3200, 3328

Offset: 1

Labos Elemer

nonn

			a[ 54 ]=x=3968=128*31 has 4 unitary divisors: {1,3968,128,31} and the cube, 64, divides 3968: q=3968/64=62.

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

A034444, A046755, A048166-A048168.

Select[Range[3400], Divisible[#, 8^PrimeNu[#]] &] (* Amiram Eldar, Aug 05 2019 *)

cp's OEIS Frontend

A063921 Quotients arising when A046755(n) is divided by the cube of the number of its divisors.

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A046754 Numbers k such that the square of d(k) (number of divisors) divides k.

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A046756 Numbers k such that d(k)^4 divides k.

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A048169 n is divisible by the cube of the number of unitary divisors of n (A034444).

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