A181063 -id:A181063 - cp's OEIS frontend

A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n.

1, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 5, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1

Offset: 1

Gus Wiseman, Oct 17 2019

nonn

By convention, a(1) = 1, and a(p) = 0 for p prime.

			The non-singleton runs of the nontrivial divisors of 1260 are: {2,3,4,5,6,7} {9,10} {14,15} {20,21} {35,36}, so a(1260) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of first appearances are A328459.
Positions of 0's and 1's are A088723.
The version that looks at all divisors is A055874.
The number of successive pairs of divisors > 1 of n is A088722(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A033676, A060681, A060775, A070824, A088725, A129308, A181063, A199970, A328165, A163870, A328194, A328448, A328449.

Table[Switch[n,1,1,?PrimeQ,0,,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,100}]

A328458(n) = if(1==n,n,my(rl=0,pd=0,m=0); fordiv(n, d, if(1(1+pd), m = max(m,rl); rl=0); pd=d; rl++)); max(m,rl)); \\ Antti Karttunen, Feb 23 2023

Data section extended up to a(105) by Antti Karttunen, Feb 23 2023

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

Original entry on oeis.org

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}]

A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

Original entry on oeis.org

1, 2, 6, 12, 60, 420, 504, 840, 2520, 27720, 360360, 720720, 4084080

Offset: 0

Gus Wiseman, Oct 17 2019

			The sequence of terms > 1 together with their nontrivial divisors begins:
    2: {}
    6: {2,3}
   12: {2,3,4,6}
   60: {2,3,4,5,6,10,12,15,20,30}
  420: {2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210}
  504: {2,3,4,6,7,8,9,12,14,18,21,24,28,36,42,56,63,72,84,126,168,252}

Positions of first appearances in A328458.
The version for all divisors is A051451.
Cf. A000005, A033676, A060775, A070824, A119313, A129308, A181063, A199970, A163870, A328448, A328449.

dav=Table[Switch[n,1,1,_,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,1000}];
Table[Position[dav,i][[1,1]],{i,Union[dav]}]//Sort

a(12) from Robert Israel, Mar 31 2023

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A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of 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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A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

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