A131847 Positive integers n which each have at least one "isolated isolated divisor". An isolated isolated divisor is an isolated divisor that is not adjacent to any other isolated divisors in the list of all positive divisors of n. A divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n.
1, 4, 6, 56, 72, 110, 112, 132, 144, 156, 182, 210, 216, 224, 288, 380, 392, 432, 440, 448, 506, 550, 576, 616, 648, 720, 728, 780, 784, 792, 864, 880, 896, 924, 936, 952, 1050, 1064, 1152, 1200, 1210, 1224, 1232, 1260, 1274, 1288, 1296, 1320, 1368, 1404
Offset: 1
Keywords
Examples
The positive divisors of 56 are: 1,2,4,7,8,14,28,56. Of these, 1 and 2 differ by 1 and 7 and 8 differ by 1. Therefore the isolated divisors of 56 are 4,14,28,56. But 4 is not next to any isolated divisors in the list of all positive divisors of 56. (4 is next to 2 and 7, neither of which is isolated.) So 4 is an isolated isolated divisor of 56 and 56, therefore has at least one isolated isolated divisor.
Crossrefs
Cf. A133779.
Programs
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Maple
isIso := proc(k,divs) if not k-1 in divs and not k+1 in divs then true ; else false ; fi ; end: isA131847 := proc(n) local divs,i,isos ; divs := convert(numtheory[divisors](n),list) ; isos := [] ; for i from 1 to nops(divs) do isos := [op(isos),isIso(op(i,divs),divs)] ; od: if nops(isos) = 1 then RETURN(true) ; fi ; if op(1,isos) = true and op(2,isos) = false then RETURN(true) ; fi ; for i from 2 to nops(isos)-1 do if op(i,isos) = true and op(i-1,isos)=false and op(i+1,isos) = false then RETURN(true) ; fi ; od: if op(-1,isos) = true and op(-2,isos) = false then RETURN(true) ; fi ; RETURN(false) ; end: for n from 1 to 1500 do if isA131847(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Oct 24 2007
Extensions
More terms from R. J. Mathar, Oct 24 2007