A324924 -id:A324924 - cp's OEIS frontend

A196050 Number of edges in the rooted tree with Matula-Goebel number n.

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 6, 5, 5, 6, 5, 6, 6, 5, 5, 6, 6, 5, 6, 5, 6, 7, 6, 6, 6, 6, 7, 6, 6, 5, 7, 7, 6, 6, 6, 5, 7, 6, 6, 7, 6, 7, 7, 5, 6, 7, 7, 6, 7, 6, 6, 8, 6, 7, 7, 6, 7, 8, 6, 6, 7, 7, 6, 7, 7, 6, 8, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 6, 7, 7, 7, 8, 6, 6, 8, 6, 8

Offset: 1

Emeric Deutsch, Sep 27 2011

nonn

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
a(n) is, for n >= 2, the number of prime function prime(.) = A000040(.) operations in the complete reduction of n. See the W. Lang link with a list of the reductions for n = 2..100, where a curly bracket notation {.} is used for prime(.). - Wolfdieter Lang, Apr 03 2018
From Gus Wiseman, Mar 23 2019: (Start)
Every positive integer has a unique factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
In this factorization, a(n) is the number of factors counted with multiplicity. For example, a(11) = 4, a(50) = 7, a(360) = 10.
(End)
From Antti Karttunen, Oct 23 2023: (Start)
Totally additive with a(prime(n)) = 1 + a(n).
Number of iterations of A366385 (or equally, of A366387) needed to reach 1.
(End)

			a(7) = 3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Göbel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
Wolfdieter Lang, Complete prime function reduction for n = 2..100.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

One less than A061775.
Cf. A000040, A000081, A000720, A003963, A007097, A020639, A049084, A109082, A109129, A317713, A318995, A358729.
Cf. A324850, A324922, A324923, A324924, A324925, A324931, A324935, A366385, A366387, A366388.

import Data.List (genericIndex)
a196050 n = genericIndex a196050_list (n - 1)
a196050_list = 0 : g 2 where
   g x = y : g (x + 1) where
     y = if t > 0 then a196050 t + 1 else a196050 r + a196050 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 then 1+a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

a[1] = 0; a[n_?PrimeQ] := a[n] = 1 + a[PrimePi[n]]; a[n_] := Total[#[[2]] * a[#[[1]] ]& /@ FactorInteger[n]];
Array[a, 110] (* Jean-François Alcover, Nov 16 2017 *)
difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length[difac[n]],{n,100}] (* Gus Wiseman, Mar 23 2019 *)

a(n) = my(f=factor(n)); [self()(primepi(p))+1 |p<-f[,1]]*f[,2]; \\ Kevin Ryde, May 28 2021

from functools import lru_cache
from sympy import isprime, primepi, factorint
@lru_cache(maxsize=None)
def A196050(n):
    if n == 1 : return 0
    if isprime(n): return 1+A196050(primepi(n))
    return sum(e*A196050(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

a(1)=0; if n = prime(t) (the t-th prime), then a(n)=1 + a(t); if n = r*s (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.
a(n) = A061775(n) - 1.
a(n) = A109129(n) + A366388(n) = A109082(n) + A358729(n). - Antti Karttunen, Oct 23 2023

A109129 Width (i.e., number of non-root vertices having degree 1) of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Keith Briggs, Aug 17 2005

nonn

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m >= 2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
A non-root vertex having degree 1 is called a leaf.
Every positive integer has a unique factorization (see A324924) into factors q(i) = prime(i)/i for i > 0. The number of ones in this factorization is a(n). For example, 30 = q(1)^3 q(2)^2 q(3), so a(30) = 3. - Gus Wiseman, Mar 23 2019

			a(7)=2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A061775, A091233.
Cf. A049084, A020639.
Cf. A000081, A000720, A001222, A007097, A109082, A196050, A317713.
Cf. A324850, A324922, A324923, A324924, A324931.

import Data.List (genericIndex)
a109129 n = genericIndex a109129_list (n - 1)
a109129_list = 0 : 1 : g 3 where
   g x = y : g (x + 1) where
     y = if t > 0 then a109129 t else a109129 r + a109129 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

Nest[Function[{a, n}, Append[a, If[PrimeQ@ n, a[[PrimePi@ n]], Total@ Map[#2 a[[#1]] & @@ # &, FactorInteger[n]] ]]] @@ {#, Length@ # + 1} &, {0, 1}, 105] (* Michael De Vlieger, Mar 24 2019 *)

ML(n) = if( n==1, 1, my(f=factor(n)); sum(k=1,matsize(f)[1],ML(primepi(f[k,1]))*f[k,2])) ;
A109129(n) = if( n==1, 0, ML(n) ); \\ François Marques, Mar 16 2021

from functools import lru_cache
from sympy import primepi, isprime, factorint
@lru_cache(maxsize=None)
def A109129(n):
    if n <= 2: return n-1
    if isprime(n): return A109129(primepi(n))
    return sum(e*A109129(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

a(1)=0; a(2)=1; if n = p(t) (= the t-th prime) and t >= 2, then a(n) = a(t); if n = rs (r, s >= 2), then a(n) = a(r) + a(s). The Maple program is based on this recursive formula.

The Gutman et al. references contain a different recursive formula.

Typo in formula fixed by Reinhard Zumkeller, Sep 03 2013

A324850 Numbers divisible by the product of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 36, 48, 56, 60, 64, 72, 96, 112, 120, 128, 144, 152, 156, 168, 180, 192, 216, 224, 240, 256, 288, 304, 312, 330, 336, 360, 384, 432, 448, 476, 480, 512, 576, 608, 624, 660, 672, 720, 768, 784, 828, 840, 848, 864, 888, 896

Offset: 1

Gus Wiseman, Mar 18 2019

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, with product A003963(n). For example, the prime indices of 30 are {1,2,3}, with product 6, which divides 30, so 30 is in the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  48: {1,1,1,1,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
  72: {1,1,1,2,2}
  96: {1,1,1,1,1,2}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003963, A036844, A120383, A324847, A324756, A324758, A324847, A324848, A324849, A324852, A324853, A324856, A324923, A324924, A324925, A324931.

Select[Range[100],Divisible[#,Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]]&]

isok(n) = my(f=factor(n)); !(n % prod(k=1, #f~, primepi(f[k,1])^f[k,2])); \\ Michel Marcus, Mar 22 2019

n/A003963(n) = A324933(n)/A324934(n).

A324923 Number of distinct factors in the factorization of n into factors q(i) = prime(i)/i, i > 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 20 2019

nonn

Also the number of distinct proper terminal subtrees of the rooted tree with Matula-Goebel number n. See illustrations in A061773.

			The factorization 22 = q(1)^2 q(2) q(3) q(5) has four distinct factors, so a(22) = 4.

One less than A317713.
Cf. A000081, A003963, A061773, A061775, A109082, A109129, A120383, A196050.
Cf. A324850, A324922, A324924, A324925, A324931, A324934, A324935, A324936, A366385, A366386.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length[Union[difac[n]]],{n,100}]

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A324923(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #Set(lista); }; \\ Antti Karttunen, Oct 23 2023

a(n) = A317713(n) - 1.

a(n) = A196050(n) - A366386(n). - Antti Karttunen, Oct 23 2023

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

A330950 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 22, 15, 30, 42, 77, 42, 101, 56, 176, 176, 231, 135, 490, 490, 490, 792, 1002, 490, 1575, 627, 3010, 2436, 2436, 3718, 5604, 1958, 4565, 6842, 12310, 3718, 14883, 4565, 21637, 26015, 17977, 8349, 53174, 44583, 63261

Offset: 1

Gus Wiseman, Jan 15 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(10) = 11 partitions:
  1  11  21  211   32   321    43    5111      522      631
             1111  311  2211   421   32111     3222     3331
                        21111  4111  41111     4221     4321
                                     221111    22221    5311
                                     311111    32211    32221
                                     2111111   222111   33211
                                     11111111  2211111  43111
                                                        322111
                                                        331111
                                                        3211111
                                                        31111111
For example, the Heinz number of (3,2) is 15, which is divisible by 5, so (3,2) is counted under a(5).

The Heinz numbers of these partitions are given by A324851.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by all parts are A330952.
Partitions whose Heinz number is divisible by their product are A324925.
Cf. A056239, A112798, A196050, A324850, A324924, A330953, A330954, A331379, A331381, A331383, A331384.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@Prime/@#,n]&]],{n,20}]

A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 28, 8, 36, 60, 330, 24, 156, 56, 180, 16, 476, 72, 152, 120, 168, 660, 828, 48, 900, 312, 216, 112, 1740, 360, 10230, 32, 1980, 952, 840, 144, 888, 304, 936, 240, 6396, 336, 2408, 1320, 1080, 1656, 8460, 96, 784, 1800, 2856, 624, 848, 432

Offset: 1

Gus Wiseman, Mar 20 2019

Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0 given by the rows of A324924. Then a(n) is the number obtained by encoding this factorization as a standard factorization into prime numbers (A112798).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Sorting the sequence gives A324850.
Cf. A000081, A003963, A061775, A109129, A112798, A120383, A196050, A317713.
Cf. A324848, A324849, A324923, A324924, A324925, A324931, A324934.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
difac[n_]:=If[n==1,{},With[{m=Product[Prime[i]/i,{i,primeMS[n]}]},Sort[Join[primeMS[n],difac[n/m]]]]];
Table[Times@@Prime/@difac[n],{n,30}]

a(n) = my (f=factor(n)); prod (i=1, #f~, (f[i,1] * a(primepi(f[i,1])))^f[i,2]) \\ Rémy Sigrist, Jul 18 2019

a(n) = Product_t mg(t) where the product is over all (not necessarily distinct) terminal subtrees of the rooted tree with Matula-Goebel number n, and mg(t) is the Matula-Goebel number of t.

Completely multiplicative with a(prime(n)) = prime(n) * a(n). - Rémy Sigrist, Jul 18 2019

Keyword mult added by Rémy Sigrist, Jul 18 2019

A324931 Integers in the list of quotients of positive integers by their product of prime indices.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 12, 7, 5, 32, 9, 24, 14, 10, 64, 18, 48, 28, 20, 128, 36, 19, 13, 21, 15, 96, 27, 56, 40, 256, 72, 38, 26, 11, 42, 30, 192, 54, 112, 17, 80, 512, 144, 76, 52, 22, 84, 60, 384, 49, 23, 35, 53, 108, 37, 224, 25, 57, 39, 34, 160, 63, 1024

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

These quotients are given by A324932(n)/A324933(n).

This is a permutation of the positive integers, with inverse A324934.

			The sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324934.

Select[Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}],IntegerQ]

a(n) = A324850(n)/A003963(A324850(n)).

A324925 Number of integer partitions y of n such that Product_{i in y} prime(i)/i is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 5, 8, 9, 11, 17, 19, 21, 28, 32, 40, 51, 57, 67, 83, 96, 118, 142, 160, 189, 224, 260, 307, 363, 412, 479, 561, 649, 749, 874, 997, 1141, 1321, 1518, 1734, 1994, 2274, 2582, 2960, 3374, 3837, 4370, 4950, 5604, 6371, 7208, 8157, 9231, 10392

Offset: 0

Gus Wiseman, Mar 20 2019

nonn

The Heinz numbers of these integer partitions are given by A324850.

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (11)  (21)   (211)   (2111)   (321)     (3211)     (32111)
             (111)  (1111)  (11111)  (411)     (4111)     (41111)
                                     (2211)    (22111)    (221111)
                                     (21111)   (211111)   (2111111)
                                     (111111)  (1111111)  (11111111)
For example, (3,2,1,1) is such a partition because (2/1) * (2/1) * (3/2) * (5/3) = 10 is an integer.

Cf. A003963, A109129, A324850, A324922, A324923, A324924, A324931, A324934.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Product[Prime[i]/i,{i,#}]]&]],{n,0,30}]

A324935 Matula-Goebel numbers of rooted trees whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 24, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 42, 43, 44, 48, 51, 52, 53, 56, 57, 58, 59, 62, 64, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 86, 88, 89, 91, 95, 96, 101, 102, 104

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0. This sequence consists of all numbers where this factorization has all distinct factors, except possibly for any multiplicity of q(1). For example, 22 = q(1)^2 q(2) q(3) q(5) is in the sequence, while 50 = q(1)^3 q(2)^2 q(3)^2 is not.

The enumeration of these trees by number of vertices is A324936.

			The sequence of trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  31: (((((o)))))

Gus Wiseman, The first 36 rooted trees whose non-leaf terminal subtrees are all different, together with their Matula-Goebel numbers.

Cf. A000081, A004111, A007097, A061775, A196050, A276625, A290822, A317713.

Cf. A324850, A324922, A324923, A324924, A324931, A324934, A324936.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Select[Range[100],UnsameQ@@DeleteCases[difac[#],1]&]

A324934 Inverse permutation to A324931.

Original entry on oeis.org

1, 2, 4, 3, 10, 6, 9, 5, 12, 15, 35, 8, 24, 14, 26, 7, 41, 17, 23, 20, 25, 47, 52, 13, 58, 34, 28, 19, 79, 37, 184, 11, 87, 61, 53, 22, 56, 33, 60, 30, 145, 36, 92, 70, 65, 75, 164, 18, 51, 82, 98, 46, 54, 39, 178, 29, 59, 106, 293, 49, 122, 245, 63, 16, 125

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Cf. A003963, A120383, A196050, A317713, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324931.

cp's OEIS Frontend

A196050 Number of edges in the rooted tree with Matula-Goebel number n.

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A109129 Width (i.e., number of non-root vertices having degree 1) of the rooted tree with Matula-Goebel number n.

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A324850 Numbers divisible by the product of their prime indices.

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A324923 Number of distinct factors in the factorization of n into factors q(i) = prime(i)/i, i > 0.

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A330950 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

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A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.

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A324931 Integers in the list of quotients of positive integers by their product of prime indices.

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