cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 30 results. Next

A024805 Positions of primes in A004432.

Original entry on oeis.org

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

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Extensions

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

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Extensions

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

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Extensions

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

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Extensions

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

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Extensions

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

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Keywords

Crossrefs

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

Programs

  • Haskell
    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
  • Mathematica
    lim = 10; s = {0}; Do[s = Union[s, s + n^2], {n, lim}]; Select[s, 0 <= # <= lim^2 &] (* T. D. Noe, Jul 10 2012 *)
  • PARI
    a(n)=if(n<1,0,n=a(n-1); until(polcoeff(prod(k=1,sqrt(n),1+x^k^2), n), n++); n)
    

Formula

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

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Keywords

Crossrefs

Column k=3 of A341040.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

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Examples

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

Crossrefs

Programs

  • Haskell
    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
    
  • Mathematica
    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 *)
  • PARI
    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

Formula

{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

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Crossrefs

Programs

  • Haskell
    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
  • PARI
    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
    

Formula

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

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Author

Reinhard Zumkeller, Apr 22 2013

Keywords

Examples

			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.
		

Crossrefs

Programs

  • Haskell
    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)
  • Mathematica
    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 *)
Showing 1-10 of 30 results. Next