A055874 -id:A055874 - cp's OEIS frontend

A328510 Smallest number whose divisors have n non-singleton runs.

1, 2, 20, 90, 630, 1260, 3780, 21420, 41580, 128520, 270270, 554400, 706860, 1413720, 2042040, 4324320, 4084080, 9189180, 6126120, 43825320, 12252240, 18378360, 82162080, 36756720, 85765680, 73513440, 183783600, 306306000, 257297040, 563603040, 514594080

Offset: 0

Gus Wiseman, Oct 18 2019

nonn

			The sequence of terms together with their non-singleton runs of divisors begins:
    1: {}
    2: {{1,2}}
   20: {{1,2},{4,5}}
   90: {{1,2,3},{5,6},{9,10}}
  630: {{1,2,3},{5,6,7},{9,10},{14,15}}

Giovanni Resta, Table of n, a(n) for n = 0..54

Equal {1} followed by the positions of first appearances in A328511 (times 2).
The longest run of divisors of n has length A055874.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The number of singleton runs of divisors is A132881.
Cf. A000005, A027750, A033676, A060775, A119313, A132747, A181063, A199970, A328166, A328457.

dv=Table[Length[DeleteCases[Length/@Split[Divisors[n],#2==#1+1&],1]],{n,1000}];
Table[Position[dv,i][[1,1]],{i,Union[dv]}]

Offset changed to 0 and a(10)-a(30) added by Giovanni Resta, Oct 25 2019

A328511 Number of non-singleton runs of divisors of 2n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 18 2019

nonn

			The divisors of 90 have runs: {{1, 2, 3}, {5, 6}, {9, 10}, {15}, {18}, {30}, {45}, {90}}, so a(45) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Run (cards)

Positions of first appearances are A328510.
The longest run of divisors of n has length A055874.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The number of singleton runs of divisors is A132881.
Cf. A000005, A033676, A060775, A119313, A132747, A181063, A199970, A328166, A328457.

f:= proc(n) local D,B,R;
  D:= sort(convert(numtheory:-divisors(2*n),list));
  B:= D[2..-1]-D[1..-2];
  R:= select(j -> (j=1 or B[j-1]>1) and B[j]=1, [$1..nops(B)]);
  nops(R);
end proc:
map(f, [$1..100]); # Robert Israel, Oct 25 2019

Table[Length[DeleteCases[Length/@Split[Divisors[2*n],#2==#1+1&],1]],{n,100}]

A368777 a(n) is the largest divisor of n that is a term of the sequence A003418, the least common multiple of the first k natural numbers.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 60, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12

Offset: 1

Hal M. Switkay, Jan 11 2024

The graph of this sequence gives it the appearance of a ruler-like function. If n is odd, a(n) = 1. If n is even and not a multiple of 6, a(n) = 2. If n is a multiple of 6 but not of 12, a(n) = 6, and so on.

			a(18) = 6 as 18 is divisible by lcm(1, 2, 3) = 6 but not by lcm(1, 2, 3, 4) = 12. so 6 is the largest divisor of 18 that is a term of A003418. - _David A. Corneth_, Jan 28 2024

Hal M. Switkay, Table of n, a(n) for n = 1..5040

Cf. A003418, A051451, A027750, A055874.

seq[max_] := Module[{lcms = Table[LCM @@ Range[k], {k, max}]}, Table[Max[Select[Divisors[k], MemberQ[lcms, #] &]], {k, 1, max}]]; seq[100] (* Amiram Eldar, Jan 12 2024 *)

a(n) = for(i = 2, n, if(n%i != 0, return(lcm([1..i-1])))); n \\ David A. Corneth, Jan 27 2024

a(n) = A003418(A055874(n))

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A328510 Smallest number whose divisors have n non-singleton runs.

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A328511 Number of non-singleton runs of divisors of 2n.

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A368777 a(n) is the largest divisor of n that is a term of the sequence A003418, the least common multiple of the first k natural numbers.

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