Jonas Wallgren - cp's OEIS frontend

A164098 Numbers of the form m * (k_1^2 + k_2^2 + ... + k_m^2).

1, 4, 9, 10, 16, 18, 20, 25, 26, 27, 28, 33, 34, 36, 40, 42, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 95, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 120, 121, 122, 123, 124, 125

Offset: 1

Jonas Wallgren, Aug 10 2009, Aug 17 2009

nonn

From Franklin T. Adams-Watters, Aug 29 2009: (Start)
The k_i must all be positive integers.
Note that every integer > 33 is the sum of 5 positive squares, and for n > 5, every integer > n+13 is the sum of n positive squares. (End)
The complement of this sequence includes: A000040, A037074, A143206, 2 * A002145, and 3 * A094712. - Robert Israel, Jan 27 2025

			34 = 2*(4^2 + 1^2), 42 = 3*(3^2 + 2^2 + 1^2), thus 34 and 42 are in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290, A000404, A000408, A000414, A047700, A111178. [From Franklin T. Adams-Watters, Aug 29 2009]

Cf. A000040, A002145, A037074, A094712, A143206.

g:= proc(y,m)
  # can we write y as sum of m positive squares?
   option remember;
   local x;
   if y < m then return false fi;
   if m = 1 then return issqr(y) fi;
   if issqr(y-m+1) then return true fi;
   for x from 1 while x^2 + m-1 < y do
     if procname(y-x^2,m-1) then return true fi
   od;
   false
end proc:
filter:= proc(n)
  ormap(t -> g(n/t, t), numtheory:-divisors(n))
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 26 2025

issumsqs(n,k) = if(n<=0||k<=0,return(k==0&&n==0)); forstep(j=sqrtint(n),max(sqrtint(n\k),1),-1,if(issumsqs(n-j^2,k-1),return(1)));0
isa(n)=local(ds);ds=divisors(n);for(k=1,(#ds+1)\2,if(issumsqs(n\ds[k],ds[k]),return(1)));0
for(n=1,200,if(isa(n),print1(n","))) \\ Franklin T. Adams-Watters, Aug 29 2009

More terms from Franklin T. Adams-Watters, Aug 29 2009

A138808 Number of integer pairs (x,y), x > 0, y > 0, such that x <= p, y <= q for any factorization n = p*q.

Original entry on oeis.org

1, 3, 5, 8, 9, 14, 13, 20, 21, 26, 21, 35, 25, 38, 41, 48, 33, 57, 37, 64, 61, 62, 45, 84, 65, 74, 81, 96, 57, 109, 61, 112, 101, 98, 101, 138, 73, 110, 121, 151, 81, 160, 85, 160, 161, 134, 93, 196, 133, 185, 161, 192, 105, 216, 173, 223, 181, 170, 117, 258

Offset: 1

Jonas Wallgren, May 16 2008

nonn

Conjecture: the row sums of the plane partitions A010766 are upper bounds. - R. J. Mathar, Aug 06 2008
a(n) is divisible by n iff n=1 or n belongs to A227993. - Rémy Sigrist, Mar 06 2017
a(n) >= 2*n - 1, with equality iff n is not composite. - Rémy Sigrist, Mar 12 2017

			a(8) = these 20 marked *'s:
-|12345678
-+--------
1|********
2|****
3|**
4|**
5|*
6|*
7|*
8|*

Paul Tek, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A227993.

a(n) = my(ar=0, pw=0); fordiv(n, w, ar=ar+(w-pw)*n/w; pw=w); return (ar) \\ Paul Tek, Mar 21 2015

a(n) = n*(m - Sum_{k=1..m-1} d(k)/d(k+1)), where d(1) < d(2) < ... < d(m) denote the divisors of n. - Rémy Sigrist, Mar 06 2017

More terms from Paul Tek, Mar 21 2015

Typo in name corrected by Rémy Sigrist, Mar 05 2017

A136436 Concatenation of subsequences: for each i the sequence of integers such that (1) they can be grouped into terms having the sums 1,2,3,...,i; (2) they can be grouped into terms having the sums i,...,3,2,1; (3) they are as large as possible.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 1, 2, 3, 4, 1, 4, 3, 2, 1, 1, 2, 3, 1, 3, 3, 2, 3, 3, 1, 3, 2, 1, 1, 2, 3, 2, 2, 5, 6, 5, 2, 2, 3, 2, 1, 1, 2, 3, 3, 1, 5, 2, 4, 3, 4, 2, 5, 1, 3, 3, 2, 1, 1, 2, 3, 4, 5, 4, 2, 6, 1, 6, 2, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 1, 4, 6, 7, 2, 6, 2, 7

Offset: 1

Jonas Wallgren, Apr 02 2008

			------------------------------------
....|1|2|.3.|..4..|
i=4: 1 2 1 2 1 2 1 is a subsequence
....|..4..|.3.|2|1|
------------------------------------
....|1|2|3|4|.5.|..6..|
i=6: 1 2 3 4 1 4 3 2 1 is a subsequence
....|..6..|.5.|4|3|2|1|
------------------------------------

Jonas Wallgren, Apr 02 2008, Table of n, a(n) for n = 1..212
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See pp. 65, 149, 155.

A131523 Composite numbers, not ending with 0, sharing a 3-digit sequence with some of its prime factors.

Original entry on oeis.org

2498, 8571, 9995, 9998, 11241, 11371, 11379, 11398, 11669, 11994, 12353, 14285, 14997, 14998, 15009, 17122, 19146, 19996, 21058, 21079, 21131, 21372, 22122, 22413, 22564, 22856, 23317, 24006, 24293, 24982, 24994, 24995, 25006, 26672

Offset: 1

Jonas Wallgren, Aug 24 2007

			62564=2*2*15641. Both 62564 and 15641 contain 564. Thus 62564 belongs to this sequence.

A128310 List of maximal breaks in generalized snooker.

Original entry on oeis.org

5, 9, 11, 14, 17, 20, 24, 27, 29, 32, 35, 39, 41, 44, 45, 47, 50, 51, 54, 59, 62, 65, 74, 75, 77, 84, 86, 87, 89, 90, 101, 104, 107, 110, 114, 116, 117, 119, 120, 125, 132, 135, 137, 140, 144, 147, 149, 152, 155, 164, 167, 170, 174, 182, 185, 186, 189, 194, 195, 200

Offset: 1

Jonas Wallgren, May 04 2007

nonn

Given A000217(R) red balls and C colored balls. For every red ball 1 point for it and C+1 points for the highest colored ball, followed by 2+3+4+... points for all colored balls. This sequence contains all numbers on the form A000217(R)*(C+2)+A000217(C+1)-1, R>0, C>0 (sorted, duplicates removed).

			Let R=2 (1 point each) and C=3 (2, 3 and 4 (call that one black) points). The sequence red-black-red-black-red-black-2-3-black gives 24 points. Thus 24 is an element in this sequence.

A129392 Triangle read by rows: T(r,c)=T(r,c-1)+T(r,c+1)+T(r-1,c-1).

Original entry on oeis.org

1, 1, 3, 1, 1, -5, -7, -5, 1, 1, 7, 5, 3, 5, 7, 1, 1, 3, 1, -9, -15, -9, 1, 3, 1, 1, -33, -35, -5, 29, 43, 29, -5, -35, -33, 1, 1, 91, 89, 31, -23, -49, -55, -49, -23, 31, 89, 91, 1, 1, -137, -139, -93, -43, 19, 85, 115, 85, 19, -43, -93, -139, -137, 1, 1, 51, 49, 135, 225, 183, 1, -201, -287, -201, 1, 183, 225, 135, 49, 51, 1, 1, 399

Offset: 0

Jonas Wallgren, Apr 13 2007

All edge elements are 1 and the triangle is symmetric about the central line. For all elements x inside the triangle
..a..
.bxc.
we have x=a+b+c. The triangle is symmetric. The top of the triangle thus is
........1
.....1..3..1
..1.-5.-7.-5..1
1..7..5..3..5..7..1

R. J. Mathar, Comments on A129392, A129394 and A129396

Cf. A129393, A129394, A129396, A129398.

More terms from R. J. Mathar, Jan 17 2008

A129394 Triangle read by rows: T(r,c)=T(r,c-1)+T(r,c+1)+T(r-1,c-1).

Original entry on oeis.org

0, 1, 2, 1, 2, -5, -8, -5, 2, 3, 9, 4, 0, 4, 9, 3, 4, 19, 12, -16, -32, -16, 12, 19, 4, 5, -107, -116, -28, 76, 120, 76, -28, -116, -107, 5, 6, 287, 276, 96, -64, -132, -144, -132, -64, 96, 276, 287, 6, 7, -367, -380, -300, -196, 8, 268, 392, 268, 8, -196, -300, -380, -367, 7, 8, -245, -260, 352, 992, 940, 144, -804, -1216

Offset: 0

Jonas Wallgren, Apr 13 2007

Like A129392, except edge elements=0,1,2,3,4,5,...

R. J. Mathar, Comments on A129392, A129394 and A129396

Cf. A129392, A129395.

Corrected and extended by R. J. Mathar, Jan 17 2008

A129396 Triangle read by rows: T(r,c)=T(r,c-1)+T(r,c+1)+T(r-1,c-1).

Original entry on oeis.org

1, 2, 5, 2, 3, -10, -15, -10, 3, 4, 16, 9, 3, 9, 16, 4, 5, 22, 13, -25, -47, -25, 13, 22, 5, 6, -140, -151, -33, 105, 163, 105, -33, -151, -140, 6, 7, 378, 365, 127, -87, -181, -199, -181, -87, 127, 365, 378, 7, 8, -504, -519, -393, -239, 27, 353, 507, 353, 27, -239, -393, -519, -504, 8, 9, -194, -211, 487, 1217, 1123, 145

Offset: 0

Jonas Wallgren, Apr 13 2007

Like A129392, except edge elements=1,2,3,4,5...

R. J. Mathar, Comments on A129392, A129394 and A129396

Cf. A129392, A129397.

More terms from R. J. Mathar, Jan 17 2008

A129399 Level sums of A129398.

Original entry on oeis.org

1, 9, -11, 33

Offset: 0

Jonas Wallgren, Apr 13 2007

sign

			a(2)=-11 because it is the sum of all elements for h=2 in A129398:
.....1
..1.-3..1
1.-3.-7.-3..1
..1..3..1
.....1

Cf. A129398.

A129398 Pyramid P(h,x,y)=P(h,x,y-1)+P(h,x,y+1)+P(h,x-1,y)+P(h,x+1,y)+P(h-1,x,y), read level by level, x by x.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 1, -3, 1, 1, -3, -7, -3, 1, 1, -3, 1, 1, 1, 1, 4, 1, 1, 3, 0, 3, 1, 1, 4, 0, -7, 0, 4, 1, 1, 3, 0, 3, 1, 1, 4, 1, 1

Offset: 0

Jonas Wallgren, Apr 13 2007

sign

All face elements=1. For an element x on a level inside the pyramid
..b
.cxd
..e
with a above x, x=a+b+c+d+e. Every level of the pyramid is symmetric. Thus the top (h=0) of the pyramid is 1. The level h=1 is
..1
.1.5.1
..1
The level h=2 is
.....1
..1.-3..1
1.-3.-7.-3..1
..1.-3..1
.....1
The level h=3 is
........1
.....1..4..1
..1..3..0..3..1
1..4..0.-7..0..4..1
..1..3..0..3..1
.....1..4..1
........1

This is a three-dimensional analog of A129392.

Cf. A129392, A129399.

cp's OEIS Frontend

User: Jonas Wallgren

A164098 Numbers of the form m * (k_1^2 + k_2^2 + ... + k_m^2).

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A138808 Number of integer pairs (x,y), x > 0, y > 0, such that x <= p, y <= q for any factorization n = p*q.

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PARI

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A136436 Concatenation of subsequences: for each i the sequence of integers such that (1) they can be grouped into terms having the sums 1,2,3,...,i; (2) they can be grouped into terms having the sums i,...,3,2,1; (3) they are as large as possible.

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A131523 Composite numbers, not ending with 0, sharing a 3-digit sequence with some of its prime factors.

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Author

Keywords

Examples

A128310 List of maximal breaks in generalized snooker.

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Comments

Examples

A129392 Triangle read by rows: T(r,c)=T(r,c-1)+T(r,c+1)+T(r-1,c-1).

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Extensions

A129394 Triangle read by rows: T(r,c)=T(r,c-1)+T(r,c+1)+T(r-1,c-1).

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A129396 Triangle read by rows: T(r,c)=T(r,c-1)+T(r,c+1)+T(r-1,c-1).

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Extensions

A129399 Level sums of A129398.

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A129398 Pyramid P(h,x,y)=P(h,x,y-1)+P(h,x,y+1)+P(h,x-1,y)+P(h,x+1,y)+P(h-1,x,y), read level by level, x by x.

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