A097563 -id:A097563 - cp's OEIS frontend

A097759 Number of integers that can be written as a sum of distinct squares in exactly n ways.

31, 38, 25, 30, 19, 14, 13, 10, 17, 15, 6, 11, 7, 6, 10, 5, 8, 6, 8, 5, 7, 6, 1, 4, 4, 10, 3, 7, 7, 3, 4, 5, 3, 6, 5, 2, 5, 3, 2, 1, 2, 8, 1, 3, 5, 1, 4, 1, 4, 2, 4, 7, 2, 2, 2, 4, 1, 1, 5, 1, 2, 2, 5, 2, 3, 2, 2, 2, 0, 1, 5, 2, 3, 1, 1, 1, 2, 2, 6, 1, 4, 0, 2, 1, 3, 1, 3, 4, 0, 2, 4, 5, 1, 1, 1, 0, 2, 1

Offset: 0

Ray Chandler, Sep 06 2004

nonn

Number of terms in row n of table described in A097757.

Eric Weisstein's World of Mathematics, Square Number.

Cf. A001422, A003995, A033461, A097563, A097757, A097758, A097760.

A350195 a(n) is the smallest positive integer which can be represented as the sum of distinct odd squares in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 130, 251, 299, 525, 420, 588, 645, 790, 660, 741, 900, 909, 924, 885, 1006, 1054, 1005, 1173, 1125, 1101, 1045, 1174, 1221, 1261, 1245, 1285, 1365, 1294, 1342, 1270, 1366, 1438, 1414, 1390, 1550, 1485, 1621, 1710, 1574, 1486, 1725, 1631, 1686, 1534, 1510, 1655, 1751, 1702, 1845

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Index entries for sequences related to sums of squares

Cf. A016754, A097563, A167700.

A350206 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero square pyramidal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 91, 300, 490, 580, 525, 810, 910, 1100, 1050, 1330, 1340, 1430, 1400, 1625, 1750, 1540, 1700, 1820, 1955, 2010, 1995, 1980, 1925, 2185, 2280, 2190, 2275, 2360, 2535, 2445, 2415, 2505, 2500, 2605, 2645, 2640, 2785, 2780, 2865, 2870, 2925, 2800, 2835, 2940

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Square Pyramidal Number

Cf. A000330, A097563, A298246, A350205.

A287965 Smallest number which can be represented as the sum of distinct squares of primes in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

4, 410, 1014, 1494, 1685, 2188, 2335, 2573, 2717, 2863, 3054, 3389, 3224, 3654, 3534, 4014, 4232, 4183, 4254, 4064, 4589, 4618, 4544, 4593, 4903, 5193, 5503, 5215, 5579, 5433, 5455, 5673, 5962, 5983, 6158, 6178, 5744, 5864, 5984, 5913, 6223, 6273, 6678, 6393, 6442, 6513, 6870, 6535, 7038, 7015

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

nonn

It appears that 1275 is the first k for which a(k) = 0. - Robert Israel, Oct 14 2024

			a(2) = 410 because 410 = 7^2 + 19^2 = 11^2 + 17^2 and this is the smallest number that can be written as the sum of distinct squares of primes in 2 different ways.

Robert Israel, Table of n, a(n) for n = 1..1274
Index entries for sequences related to sums of squares

Cf. A001248, A048261, A097563, A111900, A121518.

N:= 100: # to try with primes up to N
P:= select(isprime, [2,seq(i,i=3..N,2)]):
nP:= nops(P):
S:= mul(1+x^(P[i]^2), i=1..nP):
M:= 100: # for a(1) .. a(M)
V:= Vector(M): count:= 0:
for i from 4 to N^2 while count < M do
  r:= coeff(S,x,i);
  if r >= 1 and r <= M and V[r] = 0 then count:= count+1; V[r]:= i; fi
od:
convert(V,list); # Robert Israel, Oct 14 2024

A111900(a(n)) = n.

A350204 a(n) is the smallest positive integer which can be represented as the sum of distinct centered square numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 66, 127, 151, 277, 212, 296, 325, 404, 332, 373, 440, 452, 458, 445, 464, 530, 505, 586, 572, 553, 578, 525, 637, 613, 632, 625, 626, 650, 692, 674, 638, 705, 686, 734, 710, 698, 789, 777, 745, 771, 817, 794, 746, 850, 770, 758, 847, 972, 908

Offset: 1

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Centered Square Number

Cf. A001844, A097563, A281082, A350195, A350203.

A364150 a(n) is the smallest positive integer which can be represented as the sum of distinct positive quarter-squares in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 6, 12, 16, 21, 22, 27, 33, 31, 32, 36, 37, 41, -1, 42, 43, 47, -1, 49, 48, -1, 54, 52, -1, 60, 59, 57, -1, 58, 61, 62, 63, 65, 64, -1, -1, 69, 67, 70, -1, 68, 72, -1, 75, -1, 73, 76, 74, -1, -1, -1, 77, 80, 78, 79, 81, -1, 82, -1, -1

Offset: 1

Ilya Gutkovskiy, Jul 10 2023

sign

Cf. A002620, A097563, A197081.

cp's OEIS Frontend

A097759 Number of integers that can be written as a sum of distinct squares in exactly n ways.

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A350195 a(n) is the smallest positive integer which can be represented as the sum of distinct odd squares in exactly n ways, or 0 if no such integer exists.

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A350206 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero square pyramidal numbers in exactly n ways, or 0 if no such integer exists.

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A287965 Smallest number which can be represented as the sum of distinct squares of primes in exactly n ways, or 0 if no such integer exists.

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A350204 a(n) is the smallest positive integer which can be represented as the sum of distinct centered square numbers in exactly n ways, or 0 if no such integer exists.

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A364150 a(n) is the smallest positive integer which can be represented as the sum of distinct positive quarter-squares in exactly n ways, or -1 if no such integer exists.

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