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 1-7 of 7 results.

A018936 a(n) is the smallest square that is the sum of n distinct positive squares.

Original entry on oeis.org

1, 25, 49, 81, 100, 169, 196, 289, 361, 529, 529, 900, 900, 1156, 1444, 1681, 2025, 2304, 2704, 3481, 3600, 4096, 4624, 4900, 5625, 6889, 7225, 8281, 9409, 10404, 11236, 11881, 12996, 14400, 15129, 16900, 18769, 19600, 21316, 23104, 24964, 25921, 27889
Offset: 1

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Author

N. J. A. Sloane

Keywords

Examples

			a(3) = 49 = 36 + 9 + 4 is the smallest square which is the sum of three distinct positive squares.

Crossrefs

Cf. A018935, A018937.

Formula

a(n) = A018935(n)^2. - Hugo Pfoertner, Sep 30 2023

Extensions

Corrected and extended by David W. Wilson
Name simplified by Jon E. Schoenfield, Sep 29 2023

A018937 Let S be the smallest square that is the sum of n distinct positive integers. Then a(n) is the smallest k such that there exist n distinct positive integers <= k whose squares sum to S.

Original entry on oeis.org

1, 4, 6, 6, 7, 9, 9, 11, 11, 13, 12, 16, 15, 16, 20, 21, 19, 22, 22, 22, 23, 25, 28, 24, 26, 29, 32, 36, 34, 33, 34, 35, 36, 38, 38, 39, 40, 45, 42, 44, 44, 44, 48, 48, 49, 50, 49, 51, 52, 54, 57, 56, 57, 56, 61, 63, 59, 61, 64, 64, 65, 65, 69, 67, 69, 76, 76, 71, 75, 73, 80, 73
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A018935, A018936, A103411, A103412, A103413, A103414.

Extensions

Corrected and extended by David W. Wilson
Name edited by Jon E. Schoenfield, Sep 30 2023

A103411 Consider smallest m such that m^2 = x_1^2 + ... + x_n^2 with 0 < x_1 < ... < x_n. Sequence gives greatest value of x_n.

Original entry on oeis.org

1, 4, 6, 6, 7, 9, 9, 11, 11, 15, 12, 17, 15, 16, 20, 21, 23, 22, 22, 30, 23, 25, 28, 24, 26, 34, 32, 36, 36, 43, 35, 35, 36, 38, 38, 39, 44, 45, 42, 45, 50, 44, 48, 48, 49, 53, 49, 51, 53, 64, 58, 56, 57, 65, 62, 63, 65, 64, 64, 68, 65, 65, 70, 69, 69, 76, 76, 79, 75, 75, 80, 73
Offset: 1

Views

Author

Ray Chandler, Feb 04 2005

Keywords

Crossrefs

Cf. A018935, A018936, A018937, A103412, A103413, A103414.

A103412 Consider smallest m such that m^2 = x_1^2 + ... + x_n^2 with 0 < x_1 < ... < x_n. Sequence gives value of x_n for lexicographically least {x_1,...,x_n}.

Original entry on oeis.org

1, 4, 6, 6, 7, 9, 9, 11, 11, 15, 12, 16, 15, 16, 20, 21, 23, 22, 22, 30, 23, 25, 28, 24, 26, 33, 32, 36, 35, 43, 34, 35, 36, 38, 38, 39, 42, 45, 42, 44, 50, 44, 48, 48, 49, 50, 49, 51, 52, 64, 57, 56, 57, 65, 61, 63, 65, 61, 64, 64, 65, 65, 69, 67, 69, 76, 76, 79, 75, 73, 80, 73
Offset: 1

Views

Author

Ray Chandler, Feb 04 2005

Keywords

Crossrefs

Cf. A018935, A018936, A018937, A103411, A103413, A103414.

A103413 Consider smallest m such that m^2 = x_1^2 + ... + x_n^2 with 0 < x_1 < ... < x_n. Sequence gives value of x_n for lexicographically greatest {x_1,...,x_n}.

Original entry on oeis.org

1, 4, 6, 6, 7, 9, 9, 11, 11, 13, 12, 17, 15, 16, 20, 21, 19, 22, 22, 22, 23, 25, 28, 24, 26, 29, 32, 36, 34, 35, 35, 35, 36, 38, 38, 39, 40, 45, 42, 45, 44, 44, 48, 48, 49, 53, 49, 51, 53, 56, 58, 56, 57, 59, 62, 63, 59, 64, 64, 68, 65, 65, 70, 69, 69, 76, 76, 71, 75, 75, 80, 73
Offset: 1

Views

Author

Ray Chandler, Feb 04 2005

Keywords

Crossrefs

Cf. A018935, A018936, A018937, A103411, A103412, A103414.

A103414 Consider smallest m such that m^2 = x_1^2 + ... + x_n^2 with 0 < x_1 < ... < x_n. Sequence gives number of solutions {x_1,...,x_n}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 2, 5, 3, 1, 1, 3, 2, 1, 3, 5, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 5, 1, 3, 10, 4, 9, 1, 1, 1, 1, 5, 5, 4, 6, 9, 1, 4, 1, 1, 2, 1, 1, 2, 1, 9
Offset: 1

Views

Author

Ray Chandler, Feb 04 2005

Keywords

Crossrefs

Cf. A018935, A018936, A018937, A103411, A103412, A103413.

A385967 Smallest nonnegative integer whose square is the sum of the squares of A047432(n) distinct primes.

Original entry on oeis.org

0, 2, 18, 16, 27, 52, 54, 102, 96, 103, 152, 142, 218, 216, 225, 288, 282, 366, 352, 387, 440, 474, 558, 528, 559, 648, 626, 758, 780, 783, 900, 858, 978, 976, 1047, 1112, 1146, 1290, 1248, 1285, 1404, 1394, 1550, 1584, 1587, 1764, 1710, 1866, 1868, 1959, 2048
Offset: 1

Views

Author

Charles L. Hohn, Jul 13 2025

Keywords

Comments

Terms are odd when n is a multiple of 5, and even otherwise.
a(10) = 103 is also the smallest number whose square is the sum of the squares of at least 2 distinct primes and which itself is also a prime.
From David A. Corneth, Jul 14 2025: (Start)
If A047432(n) is a multiple of 4 then 4 cannot be one of the squares of primes in the sum. Proof: If 4 is there the sum of squares mod 4 will be 3 (mod 4) in that case. No square is 3 (mod 4). A contradiction.
If A047432(n) is a multiple of 3 then 9 cannot be one of the squares of primes in the sum. Proof: If 9 is there then the sum of squares will be 2 (mod 3) in that case. No square is 2 (mod 3). A contradiction. (End)

Examples

			a(5) = 27 because prime count A047432(5) = 6 and the smallest sum of squares of 6 distinct primes that is a square is 19^2 + 13^2 + 11^2 + 7^2 + 5^2 + 2^2 = 27^2.

Crossrefs

Cf. A047432, A018935.

Programs

  • PARI
    a(n, c1=0, c2=0, c3=0, ~r, ~pc)={if(c1==0, n--; my(n5=n%5); n=(n-n5)/5*8+n5+if(n5>=2, 2, 0); r=[oo]; pc=vector(max(n-1, 0)); for(i=1, #pc, pc[i]=if(i>1, pc[i-1], 0)+prime(i)^2)); if(c1==n, return(if(issquare(c3), c3, oo))); for(i=n-c1, if(c1, c2-1, oo), my(p2=prime(i)^2); if(c3+p2+if(n-c1-1>0, pc[n-c1-1], 0)>=r[1], break); r[1]=min(r[1], a(n, c1+1, i, c3+p2, ~r, ~pc))); if(c1, r[1], sqrtint(r[1]))}
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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