A130111 Rearrangement of positive integers such that each five terms sum up to a perfect square.

1, 2, 3, 4, 6, 5, 7, 8, 9, 20, 10, 11, 12, 13, 18, 14, 15, 16, 17, 19, 21, 22, 23, 24, 31, 25, 26, 27, 28, 38, 29, 30, 32, 33, 45, 34, 35, 36, 37, 54, 39, 40, 41, 42, 63, 43, 44, 46, 47, 76, 48, 49, 50, 51, 58, 52, 53, 55, 56, 73, 57, 59, 60, 61, 87, 62, 64, 65, 66, 67, 68, 69, 70, 71, 83

Offset

1,2

Comments

From Daniel Forgues, Jan 13 2020: (Start)

This sequence is the concatenated rows of a 5 column array of positive integers T(n, k) such that, for row n >= 1:

* For 1 <= k <= 4: T(n, k) are the smallest positive integers, in increasing order, not appearing in previous rows;

* T(n, 5) is the smallest integer greater than T(n, 4) not appearing in previous rows such that the row sum is a perfect square.

The perfect squares seem to be weakly increasing - but are they? (End)

No, the sequence of squares is not weakly increasing. For 310 terms, the resulting square is 1600, but for 315 terms it is 1521. - Michel Marcus, Jan 17 2020

Links

Table of n, a(n) for n=1..75.

Sean A. Irvine, Java program (github)

Example

1+2+3+4+6=16, 5+7+8+9+20=49, 10+11+12+13+18=64, 14+15+16+17+19=81.
From Daniel Forgues, Jan 11 2020: (Start) The array begins:
   1,  2,  3,  4,  6:  4^2
   5,  7,  8,  9, 20:  7^2
  10, 11, 12, 13, 18:  8^2
  14, 15, 16, 17, 19:  9^2
  21, 22, 23, 24, 31: 11^2
  25, 26, 27, 28, 38: 12^2
  29, 30, 32, 33, 45: 13^2
  34, 35, 36, 37, 54: 14^2
  39, 40, 41, 42, 63: 15^2
  43, 44, 46, 47, 76: 16^2
  48, 49, 50, 51, 58: 16^2
  52, 53, 55, 56, 73: 17^2
  57, 59, 60, 61, 87: 18^2
  62, 64, 65, 66, 67: 18^2
...
(End).

Mathematica

s={}; ra=Range[1000]; Do[su=ra[[1]]+ra[[2]]+ra[[3]]+ra[[4]]; c=5; While[ !IntegerQ[Sqrt[su+ra[[c]]]], c++ ]; rac=ra[[c]]; s=Join[s, {ra[[1]], ra[[2]], ra[[3]], ra[[4]], rac}]; ra=Complement[ra, {ra[[1]], ra[[2]], ra[[3]], ra[[4]], rac}], {50}]; s

Crossrefs

Cf. A130108, A130109, A130110.

Sequence in context: A057508 A057164 A085175 * A104182 A371424 A371425

Adjacent sequences: A130108 A130109 A130110 * A130112 A130113 A130114

Keyword

nonn,tabf

Author

Zak Seidov, May 08 2007

Extensions

a(73)-a(75) from Michel Marcus, Jan 17 2020

Status

approved