A104020 -id:A104020 - cp's OEIS frontend

A102354 a(n) is the number of ways n can be written as k^2 * j, 0 < j <= k.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0

Offset: 1

Leroy Quet, Feb 21 2005

nonn

Sum_{n>0} a(n)/n = 2*zeta(3). See A152648.

			a(18) = 1 because 18 = k^2 * j, j <= k, in one way: k=3, j=2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A102448, A104020, A104022, A104024, A152648.

t = Sort[ Flatten[ Table[k^2*j, {k, 11}, {j, k}]]]; Table[ Count[t, n], {n, 105}] (* Robert G. Wilson v, Feb 22 2005 *)

A102354(n) = sumdiv(n,d,(issquare(d) && (sqrtint(d) >= (n/d)))); \\ Antti Karttunen, Aug 27 2017

a(n) >= A102448(n). - Antti Karttunen, Aug 27 2017

More terms from Robert G. Wilson v, Feb 22 2005

A104021 Numbers that can be represented as k^2*j with 0 < j <= k and gcd(k, j) = 1.

Original entry on oeis.org

4, 9, 16, 18, 25, 36, 48, 49, 50, 64, 75, 81, 98, 100, 121, 144, 147, 162, 169, 180, 192, 196, 225, 242, 245, 256, 289, 294, 300, 320, 324, 338, 361, 363, 400, 405, 441, 448, 450, 484, 507, 529, 567, 576, 578, 588, 605, 625, 648, 676, 700, 720, 722, 726, 729

Offset: 1

Leroy Quet and Robert G. Wilson v, Feb 24 2005

nonn

Essentially the same as A102646. - Georg Fischer, Oct 07 2018

Cf. A102448, A104020, A104023, A104026.

Take[ Union[ Flatten[ Table[ If[ GCD[k, j] == 1, k^2*j, {}], {k, 27}, {j, k - 1}]]], 55]

A104022 Numbers that have more than one way of being written as k^2*j, 0 < j <= k.

Original entry on oeis.org

64, 100, 144, 196, 256, 324, 400, 484, 512, 576, 648, 676, 729, 784, 800, 900, 968, 1024, 1089, 1152, 1156, 1296, 1352, 1444, 1521, 1568, 1600, 1728, 1764, 1800, 1936, 2025, 2028, 2048, 2116, 2304, 2312, 2352, 2500, 2592, 2601, 2700, 2704, 2888, 2916

Offset: 1

Leroy Quet and Robert G. Wilson v, Feb 23 2005

nonn

Cf. A104023, A102354, A104020, A104024.

t = Sort[ Flatten[ Table[k^2*j, {k, 55}, {j, k}]]]; u = Table[ Count[t, n], {n, 3000}]; Select[ Range[3000], u[[ # ]] > 1 &]

A104024 Least number k which can be written as k^2 * j, 0 < j <= k in n ways.

Original entry on oeis.org

2, 1, 64, 900, 5184, 32400, 57600, 176400, 705600, 1166400, 3240000, 6350400, 14288400, 37454400, 25401600, 87609600

Offset: 0

Leroy Quet and Robert G. Wilson v, Feb 23 2005

nonn

Cf. A102354, A104020, A104022.

t = Sort[ Flatten[ Table[n = k^2*j; If[n < 10^7, n, {}], {k, 10000}, {j, k}]]]; l = Length[t]; t[[Select[ Range[l - 11], t[[ # ]] == t[[ # + 11]] &]]]

A104026 Numbers that can be represented as k^2*j, 0 < j <= k but not if gcd(k, j) = 1.

Original entry on oeis.org

32, 72, 108, 128, 200, 243, 288, 384, 392, 432, 486, 500, 512, 600, 675, 800, 864, 972, 1125, 1152, 1176, 1323, 1350, 1372, 1440, 1536, 1568, 1728, 1944, 1960, 2000, 2048, 2187, 2250, 2400, 2560, 2592, 2646, 2700, 2904, 3087, 3125, 3200, 3240, 3267, 3380

Offset: 1

Leroy Quet and Robert G. Wilson v, Feb 25 2005

nonn

The complement of A104020 and A104021.

			32 is in the list because 32 = 4^2*2 but gcd(4,2) = 2.

Cf. A104020, A104021.

A104020 = Take[Union[Flatten[Table[k^2 * j, {k, 70}, {j, k - 1}]]], 200]; A104021 = Take[Union[Flatten[Table[If[GCD[k, j] == 1, k^2 * j, {}], {k, 70}, {j, k - 1}]]], 150]; Complement[A104020, A104021]

cp's OEIS Frontend

A102354 a(n) is the number of ways n can be written as k^2 * j, 0 < j <= k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A104021 Numbers that can be represented as k^2*j with 0 < j <= k and gcd(k, j) = 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A104022 Numbers that have more than one way of being written as k^2*j, 0 < j <= k.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A104024 Least number k which can be written as k^2 * j, 0 < j <= k in n ways.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A104026 Numbers that can be represented as k^2*j, 0 < j <= k but not if gcd(k, j) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica