cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Labos Elemer

Keywords

Comments

Subset of A033950.

Examples

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

Links

Crossrefs

Cf. A033950, A046755, A046756.

Programs

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

A046755 Numbers k such that d(k)^3 divides k.

Original entry on oeis.org

1, 625, 6561, 21952, 32768, 64000, 98304, 117649, 163840, 229376, 360448, 425984, 557056, 622592, 753664, 950272, 1015808, 1212416, 1343488, 1362944, 1409024, 1540096, 1736704, 1933312, 1998848, 2195456, 2326528, 2392064, 2588672
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Proper subset of both A033950 and A046754. If two terms in the sequence are coprime then their product is also in the sequence.

Examples

			If k = 21952, d(k) = sigma(0,k) = 28. Its 3rd power is 21952, which divides k.
a(103) = 14385152 = (2^15)*439 with 32 divisors and 14385152/(32^3) = 439; 2^15*prime is a typical form of terms in the sequence that have 32 divisors.

Links

Crossrefs

Cf. A033950, A046754, A046756.

Programs

  • Mathematica
    Select[ Range[ 1, 14500000 ], IntegerQ[ #/(DivisorSigma[ 0, # ])^3 ]& ]

A048170 n is divisible by the 4th power of the number of unitary divisors of n (A034444).

Original entry on oeis.org

1, 16, 32, 64, 128, 256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 3584, 4096, 4352, 4608, 4864, 5120, 5632, 5888, 6144, 6400, 6656, 6912, 7168, 7424, 7936, 8192, 8704, 9216, 9472, 9728, 10240, 10496, 11008, 11264, 11776
Offset: 1

Views

Author

Labos Elemer

Keywords

Links

Crossrefs

A034444, A046756, A048166-A048169.

Programs

  • Mathematica
    Select[Range[12000], Divisible[#, 16^PrimeNu[#]] &] (* Amiram Eldar, Aug 05 2019 *)

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Sep 04 2001

Keywords

Examples

			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.

Links

Crossrefs

Cf. A046755, A046754, A046756, A033950.

Programs

  • Magma
    [k/#Divisors(k)^3:k in [m:m in [1..9000000]|IsIntegral(m/#Divisors(m)^3)]]; // Marius A. Burtea, Aug 07 2019
  • Mathematica
    f[n_] := n/DivisorSigma[0, n]^3; Select[f /@ Range[10^5], IntegerQ] (* Amiram Eldar, Aug 07 2019 *)

Formula

a(n)= A046755(n)/(A000005(A046755(n))^3).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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