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A077214
a(1) = 1. a(n+1) is the smallest number which is greater than a(n), coprime to it and has a different prime signature.
1
1, 2, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 37, 38, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 88, 89, 90
OFFSET
1,2
COMMENTS
Subsidiary sequence: First occurrence of a difference n between successive terms. Conjecture : the maximum value of a(k+1) - a(k) = 7 that too occurs only once for k = 2.( i.e. the subsediary sequence has only 7 terms.)
I conjecture that a(n+1)-a(n) is unbounded. The simplest way to get such a gap is to find n divisible by lcm(1,2,...,k) where n+1 has the same prime signature as n. I expect that such numbers exist, although they are likely to be quite large. (Since n-1 lacks small divisors, it is very likely to be in the sequence and so n is also very likely to be in the sequence.) Note that the sequence includes every prime except 3, 5 and 7. - Franklin T. Adams-Watters, Jul 18 2006
LINKS
MATHEMATICA
sig[n_] := Sort@FactorInteger[n][[;; , 2]]; f[1] = 2; f[n_] := Module[{sign = sig[n], k = n + 1}, While[! CoprimeQ[n, k] || sig[k] == sign, k++]; k]; NestList[f, 1, 100] (* Amiram Eldar, Dec 29 2020 *)
CROSSREFS
Sequence in context: A131463 A065644 A043065 * A167451 A135782 A281899
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Nov 02 2002
EXTENSIONS
More terms from Franklin T. Adams-Watters, Jul 18 2006
STATUS
approved