A167662 -id:A167662 - cp's OEIS frontend

A038838 Numbers that are divisible by the square of an odd prime.

9, 18, 25, 27, 36, 45, 49, 50, 54, 63, 72, 75, 81, 90, 98, 99, 100, 108, 117, 121, 125, 126, 135, 144, 147, 150, 153, 162, 169, 171, 175, 180, 189, 196, 198, 200, 207, 216, 225, 234, 242, 243, 245, 250, 252, 261, 270, 275, 279, 288, 289, 294, 297, 300, 306

Offset: 1

David W. Wilson

nonn

Condition 1 of Theorem 7.5 (Robinson, 1979) includes: "k is a multiple of a square of an odd prime." - Jonathan Vos Post, Aug 06 2007
If m is a term, every k*m with k > 1 is another term and the primitive terms are the square of odd primes. The subsequence of odd terms is A053850 while the even terms 18, 36, 50, 54, 72, 90, 98, ... are exactly twice the terms of this sequence. - Bernard Schott, Nov 20 2020
The asymptotic density of this sequence is 1 - 8/Pi^2 = 0.189430... - Amiram Eldar, Nov 21 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM ITEM 45.
Raphael M. Robinson, Multiple tiling of n-dimensional space by unit cubes, Math. Z., Vol. 166 (1979), pp. 225-264.
Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., Vol. 42, No. 2 (2005), pp. 181-211; Robinson theorem cited on p. 204.

Cf. A000040, A065091, A122132 (complement).
Cf. A013929 (supersequence of nonsquarefrees).
Subsequences: A001248 \ {2} (primitives), A053850 (odds), A036785 (divisible by the squares of two distinct primes).
Subsequence of A167662. - Reinhard Zumkeller, Nov 08 2009

{a(n) = my(m, c); if( n<1, 0, c=0; m=0; while( cMichael Somos, Aug 22 2006 */

list(lim)=my(v=List(),n,e,t); forfactored(k=9,lim\1, e=k[2][,2]; t=#e; n=k[1]; if(if(n%2 && t, vecmax(e)>1, t>1, vecmax(e[2..t])>1, 0), listput(v, k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018

{a(n)} = {j such that for some k>1 A001248(k)|j} = {j such that for some k>0 (A065091(k)^2)|j}. - Jonathan Vos Post, Aug 06 2007

A008966(A000265(a(n))) = 0. - Reinhard Zumkeller, Nov 08 2009

A167661 Number of partitions of n into odd squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 15, 15, 16, 16, 17, 17, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 24, 25, 25, 26, 26, 28, 28, 29, 30, 31

Offset: 0

Reinhard Zumkeller, Nov 08 2009

nonn

A167662 and A167663 give record values and where they occur: A167662(n)=a(A167663(n)) and a(m) < A167662(n) for m < A167663(n).

			a(10)=#{9+1,1+1+1+1+1+1+1+1+1+1}=2;
a(20)=#{9+9+1+1,9+1+1+1+1+1+1+1+1+1+1+1,20x1}=3;
a(30)=#{25+1+1+1+1+1,9+9+9+1+1+1,9+9+12x1,9+21x1,30x1}=5.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from R. Zumkeller)
Index entries for sequences related to sums of squares.

Cf. A001156, A000009, A101412, A016754, A167700.

g := 1/mul(1-x^((2*i-1)^2), i = 1 .. 150): gser := series(g, x = 0, 105): seq(coeff(gser, x, n), n = 0 .. 100);

nmax = 100; CoefficientList[Series[Product[1/(1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)

a(n) = f(n,1,8) with f(x,y,z) = if x

G.f.: G = 1/Product_{i>=1}(1-x^{(2i-1)^2}). - Emeric Deutsch , Jan 26 2016

a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 4) * Zeta(3/2)^(1/3) / (4 * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 18 2017

A167663 Where records occur for partitions into odd squares, cf. A167661.

Original entry on oeis.org

0, 9, 18, 25, 27, 34, 36, 43, 45, 49, 50, 52, 54, 58, 59, 61, 63, 67, 68, 70, 72, 74, 75, 76, 77, 79, 81, 83, 84, 85, 86, 88, 90, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123

Offset: 1

Reinhard Zumkeller, Nov 08 2009

nonn

A167662(n)=A167661(a(n)) and A167661(m) < A167662(n) for m

A038838 is a subsequence. [From Reinhard Zumkeller, Nov 09 2009]

Cf. A167702. [From Reinhard Zumkeller, Nov 09 2009]

A167701 Records for partitions into distinct odd squares, cf. A167700.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 15, 18, 22, 23, 24, 26, 31, 32, 35, 37, 41, 46, 51, 53, 57, 60, 66, 75, 84, 89, 104, 113, 119, 122, 130, 142, 150, 162, 173, 176, 193, 202, 203, 223, 229, 236, 256, 272, 304, 305, 332, 341, 350, 372, 394, 404, 409, 428, 440, 461, 464, 467

Offset: 1

Reinhard Zumkeller, Nov 09 2009

nonn

a(n)=A167700(A167702(n)) and A167700(m)A167702(n).

Cf. A167662.

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A038838 Numbers that are divisible by the square of an odd prime.

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A167661 Number of partitions of n into odd squares.

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A167663 Where records occur for partitions into odd squares, cf. A167661.

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A167701 Records for partitions into distinct odd squares, cf. A167700.

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