A328458 -id:A328458 - cp's OEIS frontend

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

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

A356233 Number of integer factorizations of n into gapless numbers (A066311).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 1, 1, 2, 5, 1, 4, 1, 2, 1, 1, 1, 7, 2, 1, 3, 2, 1, 4, 1, 7, 1, 1, 2, 9, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 12, 2, 2, 1, 2, 1, 7, 1, 3, 1, 1, 1, 8, 1, 1, 2, 11, 1, 2, 1, 2, 1, 2, 1, 16, 1, 1, 4, 2, 2, 2, 1, 5, 5, 1, 1, 4, 1, 1

Offset: 1

Gus Wiseman, Aug 28 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. We define a number to be gapless (listed by A066311) iff its prime indices cover an interval of positive integers.

			The counted factorizations of n = 2, 4, 8, 12, 24, 36, 48:
  (2)  (4)    (8)      (12)     (24)       (36)       (48)
       (2*2)  (2*4)    (2*6)    (3*8)      (4*9)      (6*8)
              (2*2*2)  (3*4)    (4*6)      (6*6)      (2*24)
                       (2*2*3)  (2*12)     (2*18)     (3*16)
                                (2*2*6)    (3*12)     (4*12)
                                (2*3*4)    (2*2*9)    (2*3*8)
                                (2*2*2*3)  (2*3*6)    (2*4*6)
                                           (3*3*4)    (3*4*4)
                                           (2*2*3*3)  (2*2*12)
                                                      (2*2*2*6)
                                                      (2*2*3*4)
                                                      (2*2*2*2*3)

The shortest of these factorizations is listed at A356234, length A287170.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356226 lists the lengths of maximal gapless submultisets of prime indices:
- length: A287170
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356232
Cf. A001222, A060680-A060683, A073491-A073495, A193829, A328195, A328335-A328458.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sqq[n_]:=Max@@Differences[primeMS[n]]<=1;
Table[Length[Select[facs[n],And@@sqq/@#&]],{n,100}]

A328457 Length of the longest run of divisors > 1 of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

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

Records occur at A328448.
Positions of 0's and 1's are A088725.
The version that looks at all divisors (including 1) 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).
The longest run of nontrivial divisors of n is A328458(n).
Cf. A000005, A027750, A129308, A181063, A199970, A328162, A328195, A328449, A360128.

Table[If[n==1,0,Max@@Length/@Split[Rest[Divisors[n]],#2==#1+1&]],{n,100}]

A328457(n) = { my(rl=0,pd=0,m=0); fordiv(n, d, if(d>1, if(d>(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

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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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A356233 Number of integer factorizations of n into gapless numbers (A066311).

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A328457 Length of the longest run of divisors > 1 of n.

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