A104021 -id:A104021 - cp's OEIS frontend

A102448 a(n) is the number of ways to write n = k^2 * j, j <= k, gcd(k,j) = 1, where j and k are positive integers.

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 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 23 2005

nonn

Sum_{n>0} a(n)/n = 2.

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

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

Cf. A102354, A104021, A104023, A104025.

t = Sort[ Flatten[ Table[ If[ GCD[j, k] == 1, k^2*j, {}], {k, 11}, {j, k}]]]; Table[ Count[t, n], {n, 105}] (* Robert G. Wilson v, Feb 25 2005 *)

A102448(n) = sumdiv(n,d,((1==gcd(d,(n/d))) && issquare(d) && (sqrtint(d) >= (n/d)))); \\ Antti Karttunen, Aug 27 2017

a(n) <= A102354(n). - Antti Karttunen, Aug 27 2017

More terms from Robert G. Wilson v, Feb 24 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 &]

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]

A104025 Least number k that can be written as k^2 * j with 0 < j <= k and gcd(k, j) = 1 in n ways.

Original entry on oeis.org

2, 1, 100, 900, 44100, 108900, 1232100, 11492100, 5336100, 12744900, 97416900

Offset: 0

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

nonn

Cf. A102354, A104021, A104023.

t = Sort[ Flatten[ Table[n = k^2*j; If[ GCD[k, j] == 1 && 10^5 < n < 10^8, n, {}], {k, 15000}, {j, k}]]]; l = Length[t]; t[[Select[ Range[l - 11], t[[ # ]] == t[[ # + 9]] &]]]

cp's OEIS Frontend

A102448 a(n) is the number of ways to write n = k^2 * j, j <= k, gcd(k,j) = 1, where j and k are positive integers.

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

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A104025 Least number k that can be written as k^2 * j with 0 < j <= k and gcd(k, j) = 1 in n ways.

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