A104022 -id:A104022 - 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

A104020 Numbers that can be represented as k^2*j, 0 < j <= k.

Original entry on oeis.org

4, 9, 16, 18, 25, 32, 36, 48, 49, 50, 64, 72, 75, 81, 98, 100, 108, 121, 128, 144, 147, 162, 169, 180, 192, 196, 200, 225, 242, 243, 245, 256, 288, 289, 294, 300, 320, 324, 338, 361, 363, 384, 392, 400, 405, 432, 441, 448, 450, 484, 486, 500, 507, 512, 529, 567

Offset: 1

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

nonn

Cf. A104021, A102354, A104022, A104024, A104026.

Take[ Union[ Flatten[ Table[ k^2*j, {k, 25}, {j, k - 1}]]], 56]

A104023 Numbers that have more than one way of being written as k^2*j with 0 < j <= k and gcd(k, j) = 1.

Original entry on oeis.org

100, 196, 324, 484, 676, 900, 1089, 1156, 1444, 1521, 1764, 2028, 2116, 2304, 2500, 2601, 2916, 3249, 3364, 3468, 3600, 3844, 4332, 4356, 4624, 4761, 4900, 5476, 5625, 5776, 6084, 6348, 6498, 6724, 7056, 7396, 7500, 7569, 8100, 8464, 8649, 8820, 8836

Offset: 1

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

nonn

Cf. A104022, A102448, A104021.

t = Sort[ Flatten[ Table[ If[ GCD[k, j] == 1, k^2*j, {}], {k, 1000}, {j, k}]]]; u = Table[ Count[t, n], {n, 9215}]; Select[ Range[9215], 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]] &]]]

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

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A104020 Numbers that can be represented as k^2*j, 0 < j <= k.

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A104023 Numbers that have more than one way of being written as k^2*j with 0 < j <= k and gcd(k, j) = 1.

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Mathematica

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

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Mathematica