A178530 Numbers n with the property that there exist nonnegative integers a and b such that n = concat(a,b) = a^2+b^2.

0, 1, 100, 101, 1233, 8833, 10100, 990100, 5882353, 94122353, 1765038125, 2584043776, 7416043776, 8235038125, 116788321168, 123288328768, 876712328768, 883212321168, 7681802663025, 8896802846976, 13793103448276, 15348303604525, 84651703604525, 86206903448276, 91103202846976, 92318202663025, 106058810243728

Offset

1,3

Comments

The sum of two numbers a1 and a2 that share a common b has the form of 10^n. Example: 12 + 88 = 100

The ordered pair of the final digit of a and b is always one of (0,0), (0,1), (0,5), (0,6), (2,3), (8,3), (2,8), or (8,8).

If b has k decimal digits, then (2a - 10^k)^2 + (2b - 1)^2 = 10^(2k) + 1 giving a way for efficient computation of many terms. - Max Alekseyev, Aug 17 2013

Links

Max Alekseyev, Table of n, a(n) for n = 1..200

Example

0 = 0^2+0^2 [this seems a bit far-fetched. - N. J. A. Sloane, Dec 23, 2010]
1=0^2+1^2 [ditto]
100=10^2+0^2.
101=10^2+1^2.
1233=12^2+33^2.

Mathematica

Sort[Reap[Do[n=a^2+b^2; If[n==FromDigits[Join[IntegerDigits[a], IntegerDigits[b]]], Sow[n]], {a, 0, 1000}, {b, 0, 1000}]][[2, 1]]]

Crossrefs

See A055616, A064942, A101311 for closely related sequences.

Sequence in context: A091254 A323823 A085251 * A063010 A215022 A094027

Adjacent sequences: A178527 A178528 A178529 * A178531 A178532 A178533

Keyword

nonn,base

Author

Karsten Meyer, Dec 23 2010

Extensions

Edited by N. J. A. Sloane, Dec 23 2010

a(11)-a(14) from Nathaniel Johnston, Jan 03 2011

Terms a(15) onward from Max Alekseyev, Aug 17 2013

Status

approved