A102448 -id:A102448 - 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]

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 &]

A102646 Numbers of form j k^2 with 1 <= j <= k, gcd(j,k) = 1.

Original entry on oeis.org

1, 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, Feb 25 2005

nonn

Numbers n such that A102448(n) >= 1.

All positive squares x^2 are in the sequence with j = 1, k = x.

			180 is in the sequence with j = 5, k = 6.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A102448.

is(n) = {my(f = factor(n), podd = 1, peven = 1); for(i = 1, #f~, if(bittest(f[i, 2], 0), podd *= f[i, 1]^f[i, 2] , peven *= f[i, 1]^(f[i, 2] >> 1) ) ); podd <= peven } \\ David A. Corneth, Nov 11 2019

Edited by David W. Wilson, Sep 03 2005

cp's OEIS Frontend

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

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

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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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Author

Keywords

Crossrefs

Programs

Mathematica

A102646 Numbers of form j k^2 with 1 <= j <= k, gcd(j,k) = 1.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions