A004432 -id:A004432 - cp's OEIS frontend

A024805 Positions of primes in A004432.

4, 8, 14, 17, 18, 29, 32, 38, 42, 43, 45, 57, 60, 62, 69, 75, 86, 89, 91, 101, 111, 124, 125, 127, 133, 142, 145, 154, 160, 163, 165, 171, 182, 187, 190, 200, 205, 213, 214, 216, 233, 238, 244, 250, 252, 258, 266, 268, 277, 286, 289, 296, 300, 305, 325, 330, 338, 346, 348, 362, 366

Offset: 1

Clark Kimberling

nonn

Data corrected and extended by Sean A. Irvine, Jul 23 2019

A024806 Positions of even numbers in A004432.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 16, 19, 21, 23, 24, 27, 30, 31, 33, 36, 37, 39, 41, 44, 46, 48, 50, 51, 53, 55, 59, 61, 63, 65, 67, 70, 71, 73, 76, 78, 79, 81, 82, 84, 87, 88, 90, 92, 93, 95, 97, 98, 100, 102, 103, 105, 108, 110, 112, 114, 115, 117, 120, 121, 123, 126, 128, 130, 132, 134, 136

Offset: 1

Clark Kimberling

nonn

a(58) onward corrected by Sean A. Irvine, Jul 23 2019

A024807 Positions of odd numbers in A004432.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 17, 18, 20, 22, 25, 26, 28, 29, 32, 34, 35, 38, 40, 42, 43, 45, 47, 49, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 72, 74, 75, 77, 80, 83, 85, 86, 89, 91, 94, 96, 99, 101, 104, 106, 107, 109, 111, 113, 116, 118, 119, 122, 124, 125, 127, 129, 131, 133, 135

Offset: 1

Clark Kimberling

nonn

a(56) onward corrected by Sean A. Irvine, Jul 23 2019

A024808 a(n) = position of 5 + n^2 in A004432.

Original entry on oeis.org

1, 2, 5, 8, 15, 22, 31, 40, 55, 69, 87, 104, 126, 149, 172, 198, 228, 256, 288, 323, 357, 394, 433, 473, 514, 557, 603, 651, 700, 751, 803, 857, 914, 972, 1031, 1092, 1156, 1218, 1285, 1353, 1423, 1496, 1569, 1641, 1718, 1796, 1877, 1958, 2041, 2126, 2211, 2299

Offset: 3

Clark Kimberling

nonn

Corrected and extended by David W. Wilson, May 15 1997

Offset corrected by Sean A. Irvine, Jul 24 2019

A024809 a(n) = position of n^2 + (n+1)^2 + (n+2)^2 in A004432.

Original entry on oeis.org

1, 4, 13, 26, 44, 69, 98, 137, 179, 227, 278, 338, 402, 470, 541, 622, 707, 796, 891, 991, 1097, 1206, 1320, 1443, 1569, 1697, 1834, 1974, 2120, 2269, 2423, 2585, 2752, 2923, 3099, 3280, 3467, 3658, 3853, 4056, 4261, 4473, 4689, 4911, 5136, 5366, 5602, 5844

Offset: 1

Clark Kimberling

nonn

Corrected and extended by David W. Wilson, May 15 1997

A003995 Sum of (any number of) distinct squares: of form r^2 + s^2 + t^2 + ... with 0 <= r < s < t < ...

Original entry on oeis.org

0, 1, 4, 5, 9, 10, 13, 14, 16, 17, 20, 21, 25, 26, 29, 30, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001983, A033461, A008935. Complement of A001422.

Cf. A000290; subsequences: A004431, A004432, A004433, A004434, A224981, A224982, A224983.

a003995 n = a003995_list !! (n-1)
a003995_list = filter (p a000290_list) [0..]
   where p (q:qs) m = m == 0 || q <= m && (p qs (m - q) || p qs m)
-- Reinhard Zumkeller, Apr 22 2013

lim = 10; s = {0}; Do[s = Union[s, s + n^2], {n, lim}]; Select[s, 0 <= # <= lim^2 &] (* T. D. Noe, Jul 10 2012 *)

a(n)=if(n<1,0,n=a(n-1); until(polcoeff(prod(k=1,sqrt(n),1+x^k^2), n), n++); n)

Exponents in expansion of (1+x)*(1+x^4)*(1+x^9)*(1+x^16)*(1+x^25)*(1+x^36)*(1+x^49)*(1+x^64)*(1+x^81)*(1+x^100)*(1+x^121)*(1+x^144)*...

For n > 98, a(n) = n + 30. - Charles R Greathouse IV, Sep 02 2011 (This implies a(n+2) = 2*a(n+1)-a(n) for n > 98.)

A025442 Number of partitions of n into 3 distinct nonzero squares.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 2, 2, 1, 0, 1, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 1, 2, 1, 1

Offset: 0

David W. Wilson

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A024803, A025339, A001974, A004432.

Column k=3 of A341040.

b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
      `if`(i<1 or t<1, 0, `if`(i=1, 0, b(n,i-1,t))+
      `if`(i^2>n, 0, b(n-i^2,i-1,t-1))))
    end:
a:= n-> b(n, isqrt(n), 3):
seq(a(n), n=0..120);  # Alois P. Heinz, Feb 07 2013

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[i<1 || t<1, 0, If[i==1, 0, b[n, i-1, t]] + If[i^2 > n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n, Sqrt[n] // Floor, 3]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Oct 10 2015, after Alois P. Heinz *)

A025442(n)={sum(x=1,sqrtint(n\3),sum(y=x+1,sqrtint((n-1-x^2)\2),issquare(n-x^2-y^2)))} \\ - M. F. Hasler, Feb 03 2013

a(n)>0 <=> n is in A004432. - M. F. Hasler, Feb 03 2013

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

A004433 Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0

Original entry on oeis.org

30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137

Offset: 1

N. J. A. Sloane

			30 = 1^2+2^2+3^2+4^2.

Cf. A001944, A001995, A003995, A004431, A004432, A004434, A224981, A224982, A224983, A000290.

a004433 n = a004433_list !! (n-1)
a004433_list = filter (p 4 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
-- Reinhard Zumkeller, Apr 22 2013

data = Flatten[ DeleteCases[ FindInstance[ w^2 + x^2 + y^2 + z^2 == # && 0 < w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[137], {}], 1]; w^2 + x^2 + y^2 + z^2 /. data (* Ant King, Oct 17 2010 *)
Select[Union[Total[#^2]&/@Subsets[Range[10],{4}]],#<=137&] (* Harvey P. Dale, Jul 03 2011 *)

list(lim)=my(v=List([30, 39, 46, 50, 51, 54, 57, 62, 63, 65, 66, 70, 71, 74, 75, 78, 79, 81, 84, 85, 86, 87, 90, 91, 93, 94, 95, 98, 99, 102, 105, 106, 107, 109, 110, 111, 113, 114, 116, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 129, 130, 131, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 145, 146, 147, 149, 150, 151, 153, 154, 155, 156]), u=[160, 168, 172, 176, 188, 192, 208, 220, 224, 232, 240, 256, 268, 272, 288, 292, 304, 320, 328, 352, 368, 384, 388, 400, 412, 416, 432, 448, 496, 512, 528, 544, 576, 592, 608], t=1); if(lim<156, return(select(k->k<=lim, Vec(v)))); for(n=158,lim\1, if(n#u, t=1)); Vec(v) \\ Charles R Greathouse IV, Jan 08 2025

{n: A025443(n) >=1}. Union of A025386 and A025376. - R. J. Mathar, Jun 15 2018

a(n) = n + O(log n). - Charles R Greathouse IV, Jan 08 2025

A004434 Numbers that are the sum of 5 distinct nonzero squares.

Original entry on oeis.org

55, 66, 75, 79, 82, 87, 88, 90, 94, 95, 99, 100, 103, 106, 110, 111, 114, 115, 118, 120, 121, 123, 126, 127, 129, 130, 131, 132, 134, 135, 138, 139, 142, 143, 144, 145, 146, 147, 148, 150, 151, 152, 154, 155, 156, 157, 158, 159, 160, 162, 163

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Franz Halter-Koch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), pp. 11-20.
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003995, A004431, A004432, A004433, A224981, A224982, A224983, A000290.

a004434 n = a004434_list !! (n-1)
a004434_list = filter (p 5 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
-- Reinhard Zumkeller, Apr 22 2013

upto(lim)=my(v=List(), tb, tc, td, te); for(a=5, sqrt(lim), for(b=4, min(a-1, sqrt(lim-a^2)), tb=a^2+b^2; for(c=3, min(b-1, sqrt(lim-tb)), tc=tb+c^2; for(d=2, min(c-1, sqrt(lim-tc)), td=tc+d^2; for(e=1, d-1, te=td+e^2; if(te>lim, break,listput(v, te))))))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jul 17 2011

a(n) = n + 124 for n > 121. - Charles R Greathouse IV, Jul 17 2011

A224981 Numbers that are the sum of exactly 6 distinct nonzero squares.

Original entry on oeis.org

91, 104, 115, 119, 124, 130, 131, 136, 139, 143, 146, 147, 151, 152, 154, 155, 156, 159, 160, 163, 164, 166, 167, 168, 169, 170, 171, 175, 176, 178, 179, 180, 181, 182, 184, 187, 188, 190, 191, 192, 194, 195, 196, 199, 200, 201, 202, 203, 204, 206, 207, 208

Offset: 1

Reinhard Zumkeller, Apr 22 2013

nonn

			a(1) = 1 + 4 + 9 + 16 + 25 + 36 = 91 = A000330(6);
a(2) = 1 + 4 + 9 + 16 + 25 + 49 = 104;
a(3) = 1 + 4 + 9 + 16 + 36 + 49 = 115;
a(4) = 1 + 4 + 9 + 16 + 25 + 64 = 119;
a(5) = 1 + 4 + 9 + 25 + 36 + 49 = 124.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Adolf J. Hildebrand, and George B. Purdy, Sums of distinct squares, Acta Arithmetica 67 (1994), pp. 349-380.
Index entries for sequences related to sums of squares

Cf. A003995, A004431, A004432, A004433, A004434, A224982, A224983, A000290.

a224981 n = a224981_list !! (n-1)
a224981_list = filter (p 6 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)

nmax = 1000;
S[n_] := S[n] = Union[Total /@ Subsets[
     Range[Floor[Sqrt[n]]]^2, {6}]][[1 ;; nmax]];
S[nmax];
S[n = nmax + 1];
While[S[n] != S[n - 1], n++];
S[n] (* Jean-François Alcover, Nov 20 2021 *)

cp's OEIS Frontend

A024805 Positions of primes in A004432.

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A024806 Positions of even numbers in A004432.

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A024807 Positions of odd numbers in A004432.

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A024808 a(n) = position of 5 + n^2 in A004432.

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A024809 a(n) = position of n^2 + (n+1)^2 + (n+2)^2 in A004432.

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A003995 Sum of (any number of) distinct squares: of form r^2 + s^2 + t^2 + ... with 0 <= r < s < t < ...

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A025442 Number of partitions of n into 3 distinct nonzero squares.

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A004433 Numbers that are the sum of 4 distinct nonzero squares: of form w^2+x^2+y^2+z^2 with 0

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A004434 Numbers that are the sum of 5 distinct nonzero squares.

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A224981 Numbers that are the sum of exactly 6 distinct nonzero squares.

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