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

Author

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

Keywords

• nonn

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

Examples

			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.

Extensions

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