A053806 Numbers where a gap begins in the sequence of squarefree numbers (A005117).
4, 8, 12, 16, 18, 20, 24, 27, 32, 36, 40, 44, 48, 52, 54, 56, 60, 63, 68, 72, 75, 80, 84, 88, 90, 92, 96, 98, 104, 108, 112, 116, 120, 124, 128, 132, 135, 140, 144, 147, 150, 152, 156, 160, 162, 164, 168, 171, 175, 180, 184, 188, 192, 196, 198, 200, 204, 207, 212
Offset: 1
Examples
The first gap is at 4 and has length 1; the next starts at 8 and has length 2 (since neither 8 nor 9 are squarefree).
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- M. Filaseta and O. Trifonov, On Gaps between Squarefree Numbers. In Analytic Number Theory, Vol 85, 1990, Birkhäuser, Basel, pp. 235-253.
- E. Fogels, On the average values of arithmetic functions, Proc. Cambridge Philos. Soc. 1941, 37: 358-372.
- L. Marmet, First occurrences of squarefree gaps...
- L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012.
- K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 1951 (2) 26:263-268.
Programs
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PARI
is(n)=!issquarefree(n) && issquarefree(n-1) \\ Charles R Greathouse IV, Nov 05 2017
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PARI
list(lim)=my(v=List(),t); forfactored(n=4,lim\1, if(vecmax(n[2][,2])>1, if(!t, listput(v,n[1])); t=1, t=0)); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017