A084760 Squarefree numbers in ascending order such that the difference of successive terms is unique. a(m) - a(m-1) = a(k) - a(k-1) iff m = k.

2, 3, 5, 10, 13, 17, 23, 30, 38, 47, 57, 69, 82, 93, 107, 122, 138, 155, 173, 193, 214, 233, 255, 278, 302, 327, 353, 381, 410, 437, 467, 498, 530, 563, 597, 633, 670, 705, 743, 782, 822, 863, 905, 949, 994, 1037, 1085, 1131, 1178, 1227, 1277, 1329, 1382, 1433

Offset

1,1

Comments

The sequence of first differences is 1, 2, 5, 3, 4, 6, 7, 8, 9, 10, 12, 13, 11, 14, 15, 16, 17, 18, 20, 21, 19, ... Conjecture: (1) every number is a term of this sequence. For every number r there exists some k such that a(k) - a(k-1) = r. Question: What is the longest string of consecutive integers in this sequence (of successive differences)?

Answer: 5, as exemplified by the 6 values 17 to 57. Any longer series with differences consecutive integers must include a multiple of 4, as can be seen by enumerating all possibilities modulo 4. - Franklin T. Adams-Watters, Jul 14 2006

Links

Table of n, a(n) for n=1..54.

Example

After 5 the next term is 10 and not 6 or 7, as 6-5 = 3-2 =1 and 7-5 = 5-3 = 2.

Crossrefs

Cf. A005117, A084758, A084759.

Sequence in context: A022426 A005677 A162404 * A186082 A103746 A071848

Adjacent sequences: A084757 A084758 A084759 * A084761 A084762 A084763

Keyword

nonn

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 17 2003

Extensions

More terms from Franklin T. Adams-Watters, Jul 14 2006

Status

approved