A193829 -id:A193829 - cp's OEIS frontend

A356228 Greatest size of a gapless submultiset of the prime indices of n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 3, 1, 2, 1, 2, 3, 1, 1, 5, 2, 2, 1, 2, 1, 4, 1, 3, 1, 1, 1, 4, 1, 1, 2, 6, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 3, 2, 2, 2, 1, 4, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless intervals: {2,3}, {5,5,6}, {9}.

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.

			The prime indices of 700 are {1,1,3,3,4}, with maximal gapless submultisets {1,1}, {3,3,4}, so a(700) = 3.
The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-maxima of A356226, firsts A356232.
The smallest instead of greatest size is A356227.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356069 counts gapless divisors.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A328162, A328457, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Max@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333766(A356230(n)).

a(n) = A061395(A356231(n)).

A356603 Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).

Original entry on oeis.org

1, 2, 4, 10, 8, 20, 50, 110, 16, 40, 100, 220, 250, 550, 1210, 1870, 32, 80, 200, 440, 500, 1100, 2420, 3740, 1250, 2750, 6050, 9350, 13310, 20570, 31790, 43010, 64, 160, 400, 880, 1000, 2200, 4840, 7480, 2500, 5500, 12100, 18700, 26620, 41140, 63580, 86020

Offset: 0

Gus Wiseman, Aug 30 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The image consists of all numbers whose prime indices are odd and cover an initial interval of odd positive integers.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     10: {1,3}
      8: {1,1,1}
     20: {1,1,3}
     50: {1,3,3}
    110: {1,3,5}
     16: {1,1,1,1}
     40: {1,1,1,3}
    100: {1,1,3,3}
    220: {1,1,3,5}
    250: {1,3,3,3}
    550: {1,3,3,5}
   1210: {1,3,5,5}
   1870: {1,3,5,7}

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The partitions with these Heinz numbers are counted by A053251.
A subset of A066208 (numbers with all odd prime indices).
Up to permutation, these are the positions of first appearances of rows in A356226. Other statistics are:
- length: A287170, firsts A066205
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
The sorted version is A356232.
An ordered version is counted by A356604.
A001221 counts distinct prime factors, sum A001414.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A000005, A001222, A055932, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=1/2 Total[2^Accumulate[Reverse[q]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
sq=stcinv/@Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,1000}];
Table[Position[sq,k][[1,1]],{k,0,mnrm[Rest[sq]]}]

A328026 Number of divisible pairs of consecutive divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 2, 2, 2, 3, 4, 1, 2, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 2, 2, 3, 2, 4, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 2, 6, 2, 4, 1, 4, 2, 2, 1, 2, 1, 2, 3, 4, 2, 4, 1, 4, 4, 2, 1, 2, 2, 2, 2, 6, 1, 2, 2, 4, 2, 2, 2, 2, 1, 3, 4, 6, 1, 4, 1, 6, 2

Offset: 1

Gus Wiseman, Oct 03 2019

nonn

The number m = 2^n, n >= 0, is the smallest for which a(m) = n. - Marius A. Burtea, Nov 20 2019

			The divisors of 500 are {1,2,4,5,10,20,25,50,100,125,250,500}, with consecutive divisible pairs {1,2}, {2,4}, {5,10}, {10,20}, {25,50}, {50,100}, {125,250}, {250,500}, so a(500) = 8.

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

Positions of 1's are A000040.
Positions of 0's and 2's are A328028.
Positions of terms > 2 are A328189.
Successive pairs of consecutive divisors are counted by A129308.
Cf. A000005, A027750, A033676, A060680, A060681, A060682, A060683, A193829, A328023, A328024, A328194.

f:=func;  g:=func; [g(n):n in [1..100]]; // Marius A. Burtea, Nov 20 2019

Table[Length[Split[Divisors[n],!Divisible[#2,#1]&]]-1,{n,100}]

a(n) = {my(d=divisors(n), nb=0); for (i=2, #d, if ((d[i] % d[i-1]) == 0, nb++)); nb;} \\ Michel Marcus, Oct 05 2019

a(p^k) = k for any prime number p and k >= 0. - Rémy Sigrist, Oct 05 2019

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

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 1, 1, 1, 1, 4, 2, 1, 3, 1, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.

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.

			The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 1.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-minima of A356226, firsts A356232.
The greatest instead of smallest size is A356228.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Min@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333768(A356230(n)).

a(n) = A055396(A356231(n)).

A057237 Maximum k <= n such that 1, 2, ..., k are all relatively prime to n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 4, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 4, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 6, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 6, 1, 2, 1

Offset: 1

Leroy Quet, Sep 20 2000

nonn

In reduced residue system for n [=RRS(n)] the [initial] segment of consecutive integers, i.e. of which no number is missing is {1,2,....,a[n]}. The first missing term from RRS(n) is 1+a(n), the least prime divisor.. E.g. n=121 : RRS[121] = {1,2,3,4,5,6,7,8,9,10,lag,12,..}, i.e. no 11 is in RRS; a[n] is the length of longest lag-free number segment consisting of consecutive integers, since A020639[n] divides n. - Labos Elemer, May 14 2003

a(n) is also the difference between the smallest two divisors of n, (the column 1 of A193829), if n >= 2. - Omar E. Pol, Aug 31 2011

			a(25) = 4 because 1, 2, 3 and 4 are relatively prime to 25.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A020639, A083218.

Cf. A066169.

Join[{1},Table[Length[Split[Boole[CoprimeQ[n,Range[n-1]]]][[1]]],{n,2,100}]] (* Harvey P. Dale, Dec 28 2021 *)

a(n) = if (n==1, 1, factor(n)[1,1] - 1); \\ Michel Marcus, May 29 2015

For n >= 2, a(n) = (smallest prime dividing n) - 1 = A020639(n) - 1.

For n >= 2, a(n) = (n-1) mod (smallest prime dividing n); cf. A083218. - Reinhard Zumkeller, Apr 22 2003

A060765 Numbers n such that every difference between consecutive divisors (ordered by increasing magnitude) of n is also a divisor of n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 42, 48, 54, 60, 64, 72, 96, 100, 108, 120, 128, 144, 156, 162, 168, 180, 192, 216, 240, 256, 272, 288, 294, 300, 324, 342, 360, 384, 432, 480, 486, 500, 504, 512, 576, 600, 648, 720, 768, 840, 900, 960, 972, 1008, 1024

Offset: 1

Labos Elemer, Apr 24 2001

nonn

Equivalently, A060763(n)=0.
Powers of 2 and factorials up to 7! are here.
For each k=1..A000005(a(n))-1 exists k' < A000005(a(n)) such that A193829(a(n),k) = A027750(a(n),k'). - Reinhard Zumkeller, Jun 25 2015
From Robert Israel, Jul 03 2017: (Start)
Also includes 3*2^k and 2*3^k for all k>= 1.
All terms except 1 are even. (End)
Conjecture: a(n) has the property that for each prime divisor p, p-1|a(n)/p. If this conjecture is true then terms can be searched by distinct prime divisors. - David A. Corneth, Jul 06 2017
The divisors of a(n) form a Brauer chain. See A079301 for the definition of a Brauer chain. - Zizheng Fang, Jan 30 2020

			For n = 12, divisors={1, 2, 3, 4, 6, 12}; differences={1, 1, 1, 2, 6}; every difference is a divisor, so 12 is in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..750

Cf. A060683, A060763, A193829, A027750.

import Data.List (sort, nub); import Data.List.Ordered (subset)
a060765 n = a060765_list !! (n-1)
a060765_list = filter
(\x -> sort (nub $ a193829_row x) `subset` a027750_row' x) [1..]
-- Reinhard Zumkeller, Jun 25 2015

[k:k in [1..1025]| forall{i:i in [2..#Divisors(k)]|k mod (d[i]-d[i-1]) eq 0 where d is Divisors(k)}]; // Marius A. Burtea, Jan 30 2020

f:= proc(n) local D,L;
  D:= numtheory:-divisors(n);
  L:= sort(convert(D,list));
  nops(convert(L[2..-1]-L[1..-2],set) minus D);
end proc:
select(f=0, [$1..1000]); # Robert Israel, Jul 03 2017

test[n_ ] := Length[Complement[Drop[d=Divisors[n], 1]-Drop[d, -1], d]]==0; Select[Range[1, 1024], test]
(* Second program: *)
Select[Range[2^10], Function[n, AllTrue[Differences@ Divisors@ n, Divisible[n, #] &]]] (* Michael De Vlieger, Jul 12 2017 *)

isok(n)=my(d=divisors(n), v=vecsort(vector(#d-1, k, d[k+1]-d[k]),,8)); #select(x->setsearch(d, x), v) == #v; \\ Michel Marcus, Jul 06 2017

is(n)=my(t); fordiv(n,d, if(n%(d-t), return(0)); t=d); 1 \\ Charles R Greathouse IV, Jul 12 2017

Edited by Dean Hickerson, Jan 22 2002

A060766 Least common multiple of differences between consecutive divisors of n (ordered by size).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 6, 15, 10, 6, 12, 35, 10, 8, 16, 9, 18, 10, 28, 99, 22, 12, 20, 143, 18, 42, 28, 60, 30, 16, 88, 255, 28, 18, 36, 323, 130, 60, 40, 21, 42, 154, 60, 483, 46, 24, 42, 75, 238, 234, 52, 27, 132, 84, 304, 783, 58, 60, 60, 899, 84, 32, 104, 165, 66, 442

Offset: 2

Labos Elemer, Apr 24 2001

nonn

			For n=98, divisors={1,2,7,14,49,98}; differences={1,5,7,35,49}; a(98) = LCM of differences = 245.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

The GCD version appears to be A258409.
The LCM of the prime indices of n is A290103(n).
The differences between consecutive divisors of n are row n of A193829.
Cf. A000005, A027750, A060680, A060681, A060683, A328023, A328026, A328219.

a[n_ ] := LCM@@(Drop[d=Divisors[n], 1]-Drop[d, -1])
Table[LCM@@Differences[Divisors[n]],{n,2,70}] (* Harvey P. Dale, Oct 08 2012 *)

a(n) = A290103(A328023(n)). - Gus Wiseman, Oct 16 2019

Edited by Dean Hickerson, Jan 22 2002

A328023 Heinz number of the multiset of differences between consecutive divisors of n.

Original entry on oeis.org

1, 2, 3, 6, 7, 20, 13, 42, 39, 110, 29, 312, 37, 374, 261, 798, 53, 2300, 61, 3828, 903, 1426, 79, 18648, 497, 2542, 2379, 21930, 107, 86856, 113, 42294, 4503, 5546, 2247, 475800, 151, 7906, 8787, 370620, 173, 843880, 181, 249798, 92547, 12118, 199, 5965848

Offset: 1

Gus Wiseman, Oct 02 2019

nonn

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
            1: ()
            2: (1)
            3: (2)
            6: (2,1)
            7: (4)
           20: (3,1,1)
           13: (6)
           42: (4,2,1)
           39: (6,2)
          110: (5,3,1)
           29: (10)
          312: (6,2,1,1,1)
           37: (12)
          374: (7,5,1)
          261: (10,2,2)
          798: (8,4,2,1)
           53: (16)
         2300: (9,3,3,1,1)
           61: (18)
         3828: (10,5,2,1,1)
For example, the divisors of 6 are {1,2,3,6}, with differences {1,1,3}, with Heinz number 20, so a(6) = 20.

The sorted version is A328024.
a(n) is the Heinz number of row n of A193829, A328025, or A328027.
Cf. A000005, A027750, A056239, A060682, A060683, A112798, A129308, A193829.

Table[Times@@Prime/@Differences[Divisors[n]],{n,100}]

A056239(a(n)) = n - 1. In words, the integer partition with Heinz number a(n) is an integer partition of n - 1.
A055396(a(n)) = A060680(n).
A061395(a(n)) = A060681(n).
A001221(a(n)) = A060682(n).
A001222(a(n)) = A000005(n).

A060684 Smallest difference between consecutive divisors (ordered by size) of 2n+1.

Original entry on oeis.org

2, 4, 6, 2, 10, 12, 2, 16, 18, 2, 22, 4, 2, 28, 30, 2, 2, 36, 2, 40, 42, 2, 46, 6, 2, 52, 4, 2, 58, 60, 2, 4, 66, 2, 70, 72, 2, 4, 78, 2, 82, 4, 2, 88, 6, 2, 4, 96, 2, 100, 102, 2, 106, 108, 2, 112, 4, 2, 6, 10, 2, 4, 126, 2, 130, 6, 2, 136, 138, 2, 2, 4, 2, 148, 150, 2, 4, 156, 2, 6

Offset: 1

Labos Elemer, Apr 19 2001

nonn

Successively greater values of a(n) occur when 2n+1 is prime.

			For n=38, 2n+1=77; divisors={1,7,11,77}; differences={6,4,66}; a(38) = smallest difference = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A060680.
Different from A071294.
Cf. A193829, A027750.

a060684 = minimum . a193829_row . (+ 1) . (* 2)
-- Reinhard Zumkeller, Jun 25 2015

a[n_ ] := Min@@(Drop[d=Divisors[2n+1], 1]-Drop[d, -1])
Array[Min[Differences[Divisors[2*#+1]]]&,80] (* Harvey P. Dale, Dec 08 2013 *)

A060680(2n+1)

Edited by Dean Hickerson, Jan 22 2002

A328024 Heinz numbers of multisets representing the differences between some positive integer's consecutive divisors.

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 20, 29, 37, 39, 42, 53, 61, 79, 107, 110, 113, 151, 173, 181, 199, 239, 261, 271, 281, 312, 317, 349, 359, 374, 397, 421, 457, 497, 503, 541, 557, 577, 593, 613, 701, 733, 769, 787, 798, 857, 863, 903, 911, 953, 983, 1021, 1061, 1069, 1151

Offset: 1

Gus Wiseman, Oct 02 2019

nonn

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

There is exactly one entry with any given sum of prime indices A056239.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     3: {2}
     6: {1,2}
     7: {4}
    13: {6}
    20: {1,1,3}
    29: {10}
    37: {12}
    39: {2,6}
    42: {1,2,4}
    53: {16}
    61: {18}
    79: {22}
   107: {28}
   110: {1,3,5}
   113: {30}
   151: {36}
   173: {40}
   181: {42}
   199: {46}
   239: {52}
   261: {2,2,10}
   271: {58}
   281: {60}
   312: {1,1,1,2,6}
For example, the divisors of 8 are {1,2,4,8}, with differences {1,2,4}, with Heinz number 42, so 42 belongs to the sequence.

A permutation of A328023.
Also the set of possible Heinz numbers of rows of A193829, A328025, or A328027.
Cf. A001222, A000005, A027750, A056239, A060680, A060681, A060682, A112798.

nn=1000;
Select[Union[Table[Times@@Prime/@Differences[Divisors[n]],{n,nn}]],#<=nn&]

cp's OEIS Frontend

A356228 Greatest size of a gapless submultiset of the prime indices of n.

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A356603 Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).

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A328026 Number of divisible pairs of consecutive divisors of n.

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A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

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A057237 Maximum k <= n such that 1, 2, ..., k are all relatively prime to n.

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A060765 Numbers n such that every difference between consecutive divisors (ordered by increasing magnitude) of n is also a divisor of n.

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A060766 Least common multiple of differences between consecutive divisors of n (ordered by size).

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