A193829 -id:A193829 - cp's OEIS frontend

A337498 a(0) = 0; for n>0, a(n) = the smallest positive integer m not yet in the sequence with property that none of the numbers in row m of A193829 have occurred previously in the sequence.

0, 2, 5, 6, 11, 13, 17, 18, 23, 25, 29, 31, 37, 41, 42, 47, 53, 54, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 145, 149, 151, 155, 157, 162, 167, 169, 173, 179, 181, 185, 191, 193, 197, 199, 205, 209, 211, 215, 221, 223, 227, 229

Offset: 0

Scott R. Shannon, Sep 26 2020

nonn

The vast majority of terms are odd - in the first one million terms only 74 even numbers appear. Interestingly most of those even number appear in A014741, although that sequence contains many even numbers that do not appear here, e.g. 126, while this sequence contains even numbers not appearing in that sequence, e.g. 326.

			a(1) = 2 as the consecutive divisors of 2 are 1,2 the difference of which is 1, which has not occurred previously in the sequence.
a(2) = 5. The consecutive divisors of 3 are 1,3, the difference of which is 2, but a(1) = 2 so 3 cannot appear. The consecutive divisors of 4 are 1,2,4, the differences of which are 1,2, but a(1) = 2 so 4 cannot appear. The consecutive divisors of 5 are 1,5, the difference of which is 4 and as 4 has not occurred previously in the sequence a(2) = 5.
a(3) = 6 as the consecutive divisors of 6 are 1,2,3,6, the differences of which are 1,1,3, and as neither 1 or 3 has occurred previously a(3) = 6.
a(4) = 11. The divisors of 7 are 1,7 with a difference of 6, the divisors of 8 are 1,2,4,8 with differences 1,2,4, the divisors of 9 are 1,3,9 with differences 2,6, and the divisors of 10 are 1,2,5,10, with differences 1,3,5. All of 6,2,2,5 have all occurred previously in the sequence. The divisors of 11 are 1,11 with a difference of 10, which has not occurred previously so a(4) = 11.

Cf. A027750, A193829, A014741.

A060680 Smallest difference between consecutive divisors of n.

Original entry on oeis.org

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, 2, 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, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 6, 1, 2, 1, 4

Offset: 2

Labos Elemer, Apr 19 2001

nonn

a(n) = 1 if n is even and a(n) is even if n is odd.

a(n) = least m>0 such that n!+1+m and n-m are not relatively prime. - Clark Kimberling, Jul 21 2012

			For n = 35, divisors = {1,5,7,35}; differences = {4,2,28}; a(35) = smallest difference = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Antal Balog, Paul Erdős and Gérald Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.

Cf. A060681 (largest difference), A060682, A060683, A060684.

Cf. A193829, A027750.

a060680 = minimum . a193829_row  -- Reinhard Zumkeller, Jun 25 2015

read("transforms") :
A060680 := proc(n)
    sort(convert(numtheory[divisors](n),list)) ;
    DIFF(%) ;
    min(op(%)) ;
end proc:
seq(A060680(n),n=2..60) ; # R. J. Mathar, May 23 2018

a[n_] := Min@@(Drop[d=Divisors[n], 1]-Drop[d, -1]);
(* Second program: *)
a[n_] := Min[Differences[Divisors[n]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Oct 16 2024 *)

a(n) = {my(m = n, d1); fordiv(n, d, if(d > 1 && d - d1 < m, m = d - d1); d1 = d); m;} \\ Amiram Eldar, Mar 17 2025

a(2n+1) = A060684(n).

Corrected by David W. Wilson, May 04 2001

Edited by Dean Hickerson, Jan 22 2002

A287170 a(n) = number of runs of consecutive prime numbers among the prime divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Rémy Sigrist, Jun 04 2017

nonn

a(n) = 0 iff n = 1.
a(n) = 1 iff n belongs to A073491.
a(p) = 1 for any prime p.
a(A002110(n)) = 1 for any n > 0.
a(n!) = 1 for any n > 1.
a(A066205(n)) = n for any n > 0.
a(n) = a(A007947(n)) for any n > 0.
a(n) = a(A003961(n)) for any n > 0.
a(n*m) <= a(n) + a(m) for any n > 0 and m > 0.
Each number n can be uniquely represented as a product of a(n) distinct terms from A073491; this representation is minimal relative to the number of terms.

			See illustration of the first terms in the Links section.
The prime indices of 18564 are {1,1,2,4,6,7}, which separate into maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3; this corresponds to the ordered factorization 18564 = 12 * 7 * 221. - _Gus Wiseman_, Sep 03 2022

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, Scatterplot of the ordinal transform of the first 1000000 terms

Cf. A002110, A003961, A007947, A069010, A087207.
Positions of first appearances are A066205.
These are the row-lengths of A356226 and A356234. Other statistics are:
- length: A287170 (this sequence)
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356603 or A356232 (sorted)
A001222 counts prime factors, distinct A001221.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A000005, A053251, A055932, A061395, A073493, A107428, A132747, A137921, A193829, A286470.

Table[Length[Select[First/@If[n==1,{},FactorInteger[n]],!Divisible[n,NextPrime[#]]&]],{n,30}] (* Gus Wiseman, Sep 03 2022 *)

a(n) = my (f=factor(n)); if (#f~==0, return (0), return (#f~ - sum(i=1, #f~-1, if (primepi(f[i,1])+1 == primepi(f[i+1,1]), 1, 0))))

from sympy import factorint, primepi
def a087207(n):
    f=factorint(n)
    return sum([2**primepi(i - 1) for i in f])
def a069010(n): return sum(1 for d in bin(n)[2:].split('0') if len(d)) # this function from Chai Wah Wu
def a(n): return a069010(a087207(n)) # Indranil Ghosh, Jun 06 2017

a(n) = A069010(A087207(n))

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}]

A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 10 2022

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.

			Triangle  begins: {}, {1}, {1}, {2}, {1}, {2}, {1}, {3}, {2}, {1,1}, {1}, {3}, {1}, {1,1}, {2}, {4}, {1}, {3}, {1}, {2,1}, ... For example, the prime indices of 20 are {1,1,3}, which separates into maximal gapless submultisets {{1,1},{3}}, so row 20 is (2,1).
The prime indices of 18564 are {1,1,2,4,6,7}, which separates into {1,1,2}, {4}, {6,7}, so row 18564 is (3,1,2). This corresponds to the factorization 18564 = 12 * 7 * 221.

Row sums are A001222.
Singleton row positions are A073491, complement A073492.
Length-2,3,4 row positions are A073493-A073495.
Row lengths are A287170, firsts A066205.
Row minima are A356227.
Row maxima are A356228.
Bisected run-lengths are A356229.
Standard composition numbers of rows are A356230.
Heinz numbers of rows are A356231.
Positions of first appearances are A356232.
A001221 counts distinct prime factors, with sum A001414.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A055874, A060680-A060683, A137921, A193829, A286470, A328166.

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

A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

Original entry on oeis.org

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, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 2, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 2, 1, 78, 1

Offset: 2

Ivan Neretin, May 29 2015

nonn

a(n) = 1 for even n; a(p) = p-1 for prime p.
a(n) is even for odd n (since all divisors of n are odd).
It appears that a(n) = A052409(A005179(n)), i.e., it is the largest integer power of the smallest number with exactly n divisors. - Michel Marcus, Nov 10 2015
Conjecture: GCD of all (p-1) for prime p|n. - Thomas Ordowski, Sep 14 2016
Conjecture is true, because the set of numbers == 1 (mod g) is closed under multiplication. - Robert Israel, Sep 14 2016
Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n. See A328163 and A328164. - Gus Wiseman, Oct 16 2019

			65 has divisors 1, 5, 13, and 65, hence a(65) = gcd(1-1,5-1,13-1,65-1) = gcd(0,4,12,64) = 4.

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A049559, A057237, A060680, A063994, A187730, A027750.
Cf. A084190 (similar but with LCM).
Looking at prime indices instead of divisors gives A328167.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000005, A060681, A060683, A193829, A289508.

a258409 n = foldl1 gcd $ map (subtract 1) $ tail $ a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015

f:= n -> igcd(op(map(`-`,numtheory:-factorset(n),-1))):
map(f, [$2..100]); # Robert Israel, Sep 14 2016

Table[GCD @@ (Divisors[n] - 1), {n, 2, 100}]

a(n) = my(g=0); fordiv(n, d, g = gcd(g, d-1)); g; \\ Michel Marcus, May 29 2015

a(n) = gcd(apply(x->x-1, divisors(n))); \\ Michel Marcus, Nov 10 2015

a(n)=if(n%2==0, return(1)); if(n%3==0, return(2)); if(n%5==0 && n%4 != 1, return(2)); gcd(apply(p->p-1, factor(n)[,1])) \\ Charles R Greathouse IV, Sep 19 2016

A356232 Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 80, 100, 110, 128, 160, 200, 220, 250, 256, 320, 400, 440, 500, 512, 550, 640, 800, 880, 1000, 1024, 1100, 1210, 1250, 1280, 1600, 1760, 1870, 2000, 2048, 2200, 2420, 2500, 2560, 2750, 3200, 3520, 3740, 4000, 4096, 4400

Offset: 1

Gus Wiseman, Aug 20 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.

Also positions of first appearances of rows in A356226.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     10: {1,3}
     16: {1,1,1,1}
     20: {1,1,3}
     32: {1,1,1,1,1}
     40: {1,1,1,3}
     50: {1,3,3}
     64: {1,1,1,1,1,1}
     80: {1,1,1,1,3}
    100: {1,1,3,3}
    110: {1,3,5}
    128: {1,1,1,1,1,1,1}
    160: {1,1,1,1,1,3}
    200: {1,1,1,3,3}
    220: {1,1,3,5}
    250: {1,3,3,3}
    256: {1,1,1,1,1,1,1,1}
    320: {1,1,1,1,1,1,3}
    400: {1,1,1,1,3,3}

The partitions with these Heinz numbers are counted by A053251.
This is the odd restriction of A055932.
A subset of A066208 (numbers with all odd prime indices).
This is the sorted version of A356603.
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
- positions of first appearances: A356232 (this sequence)
A001221 counts distinct prime factors, with sum A001414.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A000005, A001222, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[1000],normQ[(primeMS[#]+1)/2]&]

A060682 Number of distinct differences between consecutive divisors of n (ordered by size).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 3, 1, 4, 3, 3, 1, 4, 2, 3, 3, 5, 1, 5, 1, 5, 3, 3, 3, 5, 1, 3, 3, 5, 1, 4, 1, 5, 4, 3, 1, 5, 2, 5, 3, 5, 1, 4, 3, 6, 3, 3, 1, 7, 1, 3, 4, 6, 3, 5, 1, 5, 3, 6, 1, 6, 1, 3, 3, 5, 3, 5, 1, 7, 4, 3, 1, 6, 3, 3, 3, 7, 1, 7, 2, 5, 3, 3, 3, 6, 1, 5, 4, 6, 1, 5, 1, 7, 5, 3

Offset: 2

Labos Elemer, Apr 19 2001

nonn

Number of all differences for n is d(n)-1 = A000005(n)-1. Increments are not necessarily different, so a(n)<=d(n)-1.

			For n=70, divisors={1,2,5,7,10,14,35,70}; differences={1,3,2,3,4,21,35}; a(70) = number of distinct differences = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
A. Balog, P. Erdős and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.
Jason Earls, Smarandache iterations of the first kind on functions involving divisors and prime factors, in Smarandache Notions Journal (2004), Vol. 14.1, page 264.

Cf. A000005, A060680, A060681, A060683.

Cf. A193829.

import Data.List (nub, genericLength)
a060682 = genericLength . nub . a193829_row
-- Reinhard Zumkeller, Jun 25 2015

a[n_ ] := Length[Union[Drop[d=Divisors[n], 1]-Drop[d, -1]]]

a(n) = my(d=divisors(n)); #vecsort(vector(#d-1, k, d[k+1] - d[k]),,8); \\ Michel Marcus, Jul 04 2017

Edited by Dean Hickerson, Jan 22 2002

A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 5, 3, 4, 2, 5, 2, 4, 3, 7, 2, 5, 2, 6, 4, 4, 2, 7, 3, 4, 5, 6, 2, 5, 2, 11, 4, 4, 3, 7, 2, 4, 4, 10, 2, 6, 2, 6, 5, 4, 2, 11, 3, 6, 4, 6, 2, 7, 4, 10, 4, 4, 2, 7, 2, 4, 6, 13, 4, 6, 2, 6, 4, 6, 2, 11, 2, 4, 5, 6, 3, 6, 2, 14, 7, 4, 2, 10

Offset: 1

Gus Wiseman, Aug 18 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
A multiset 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}. These have lengths (3,1,2), with Heinz number 30, so a(18564) = 30.

Positions of prime terms are A073491, complement A073492.
Positions of terms with bigomega 2-4 are A073493-A073495.
Applying bigomega gives A287170, firsts A066205, even bisection A356229.
These are the Heinz numbers of the rows of A356226.
Minimal/maximal prime indices are A356227/A356228.
A version for standard compositions is A356230, firsts A356232/A356603.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A055932, A060680-A060683, A193829, A286470, A328166, A356233-A356237.

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

A001222(a(n)) = A287170(n).
A055396(a(n)) = A356227(n).
A061395(a(n)) = A356228(n).

A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 2, 5, 11, 12, 13, 2, 7, 15, 16, 17, 18, 19, 4, 5, 3, 7, 2, 11, 23, 24, 25, 2, 13, 27, 4, 7, 29, 30, 31, 32, 3, 11, 2, 17, 35, 36, 37, 2, 19, 3, 13, 8, 5, 41, 6, 7, 43, 4, 11, 45, 2, 23, 47, 48, 49, 2, 25, 3, 17, 4, 13, 53, 54, 5, 11, 8

Offset: 1

Gus Wiseman, Aug 28 2022

Row-products are the positive integers 1, 2, 3, ...

			The first 16 rows:
   1 =
   2 = 2
   3 = 3
   4 = 4
   5 = 5
   6 = 6
   7 = 7
   8 = 8
   9 = 9
  10 = 2 * 5
  11 = 11
  12 = 12
  13 = 13
  14 = 2 * 7
  15 = 15
  16 = 16
The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.

Row-lengths are A287170, firsts A066205, even bisection A356229.
Applying bigomega to all parts gives A356226, statistics A356227-A356232.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.

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

cp's OEIS Frontend

A337498 a(0) = 0; for n>0, a(n) = the smallest positive integer m not yet in the sequence with property that none of the numbers in row m of A193829 have occurred previously in the sequence.

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A060680 Smallest difference between consecutive divisors of n.

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A287170 a(n) = number of runs of consecutive prime numbers among the prime divisors of n.

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

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A356226 Irregular triangle giving the lengths of maximal gapless submultisets of the prime indices of n.

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A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

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A356232 Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.

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A060682 Number of distinct differences between consecutive divisors of n (ordered by size).

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