A089723 -id:A089723 - cp's OEIS frontend

A316789 Number of same-tree-factorizations of n.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

A constant factorization of n is a finite nonempty constant multiset of positive integers greater than 1 with product n. Constant factorizations correspond to perfect divisors (A089723). A same-tree-factorization of n is either (case 1) the number n itself or (case 2) a finite sequence of two or more same-tree-factorizations, one of each factor in a constant factorization of n.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(64) = 14 same-tree-factorizations:
  64
  (8*8)
  (4*4*4)
  (8*(2*2*2))
  ((2*2*2)*8)
  (4*4*(2*2))
  (4*(2*2)*4)
  ((2*2)*4*4)
  (2*2*2*2*2*2)
  (4*(2*2)*(2*2))
  ((2*2)*4*(2*2))
  ((2*2)*(2*2)*4)
  ((2*2*2)*(2*2*2))
  ((2*2)*(2*2)*(2*2))

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A001678, A003238, A007916, A052409, A052410, A067824, A089723, A281118, A281145, A294336, A316790.

a[n_]:=1+Sum[a[n^(1/d)]^d,{d,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}]
Array[a,100]

a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(d>1, a(z^(e/d))^d)))} \\ Andrew Howroyd, Nov 18 2018

a(n) = 1 + Sum_{n = x^y, y > 1} a(x)^y.

a(2^n) = A281145(n).

A316790 Number of orderless same-tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

A constant factorization of n is a finite nonempty constant multiset of positive integers greater than 1 with product n. Constant factorizations correspond to perfect divisors (A089723). An orderless same-tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless same-tree-factorizations, one of each factor in a constant factorization of n.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(64) = 9 orderless same-tree-factorizations:
  64
  (8*8)
  (4*4*4)
  (4*4*(2*2))
  (8*(2*2*2))
  (2*2*2*2*2*2)
  (4*(2*2)*(2*2))
  ((2*2*2)*(2*2*2))
  ((2*2)*(2*2)*(2*2))

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A003238, A052409, A052410, A067824, A089723, A281118, A281145, A289078, A292504, A316789.

a[n_]:=1+Sum[Binomial[a[n^(1/d)]+d-1,d],{d,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}]
Array[a,100]

a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(d>1, binomial(a(z^(e/d)) + d - 1, d))))} \\ Andrew Howroyd, Nov 18 2018

a(n) = 1 + Sum_{n = x^y, y > 1} binomial(a(x) + y - 1, y).

a(2^n) = A289078(n).

A327695 Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 22 2019

nonn

			The factorizations of 6, 12, 30, 48, 60, 180, and 210:
  (2*3)  (3*4)    (5*6)    (3*16)       (3*20)     (4*45)       (3*70)
         (2*2*3)  (2*15)   (3*4*4)      (4*15)     (5*36)       (5*42)
                  (3*10)   (2*2*2*2*3)  (5*12)     (9*20)       (6*35)
                  (2*3*5)               (3*4*5)    (4*5*9)      (7*30)
                                        (2*2*15)   (5*6*6)      (10*21)
                                        (2*2*3*5)  (2*2*45)     (14*15)
                                                   (3*3*20)     (2*105)
                                                   (2*2*5*9)    (5*6*7)
                                                   (3*3*4*5)    (2*3*35)
                                                   (2*2*3*3*5)  (2*5*21)
                                                                (2*7*15)
                                                                (3*5*14)
                                                                (3*7*10)
                                                                (2*3*5*7)

Factorizations that are constant or whose distinct parts are pairwise coprime are counted by A327399.
Numbers with pairwise coprime distinct prime indices are A304711.
Cf. A001055, A089723, A281116, A318721, A302569, A319269, A327407, A327517.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],CoprimeQ@@Union[#]&]],{n,100}]

a(n) = A327399(n) - A089723(n).

A346403 a(1)=1; for n>1, a(n) gives the sum of the exponents in the different ways to write n as n = x^y, 2 <= x, 1 <= y.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 15 2021

nonn

Denoted by tau(n) in Mycielski (1951), Fehér et al. (2006), and Awel and Küçükaslan (2020).

This function depends only on the prime signature of n (see the Formula section).

			4 = 2^2, gcd(2) = 2, sigma(2) = 3, so a(4) = 3. The representations are 4^1 and 2^2, and 1 + 2 = 3.
144 = 2^4 * 3^2, gcd(4,2) = 2, sigma(2) = 3, so a(144) = 3. The representations are 144^1 and 12^2, and 1 + 2 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Abdu Awel and M. Küçükaslan, A Note on Statistical Limit and Cluster Points of the Arithmetical Functions a_p(n), gamma(n), and tau(n) in the Sense of Density, Journal of the Indonesian Mathematical Society, Vol. 26, No. 2 (2020), pp. 224-233.
Zoltán Fehér, Béla László, Martin Mačaj and Tibor Šalát, Remarks on arithmetical functions a_p(n), gamma(n), tau(n), Annales Mathematicae et Informaticae, Vol. 33 (2006), pp. 35-43.
Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260.

Cf. A000203, A013661, A089723, A052409, A253641.

A253641:=proc(n) if n in {0,1} then 1 else igcd(map(i->i[2], ifactors(n)[2])[]); fi; end: seq(numtheory[sigma](A253641(n)), n=1..120); # Ridouane Oudra, Jun 04 2025

a[n_] := DivisorSigma[1, GCD @@ FactorInteger[n][[;; , 2]]]; Array[a, 100]

a(n) = if (n==1, 1, sigma(gcd(factor(n)[,2]))); \\ Michel Marcus, Jul 16 2021

If n = Product_{i} p_i^e_i, then a(n) = sigma(gcd()).
Sum_{n>=1} (a(n)-1)/n = Pi^2/6 + 1 (= A013661 + 1) (Mycielski, 1951).
a(n) = sigma(A052409(n)), for n>1. - Ridouane Oudra, Nov 23 2024
a(n) = sigma(A253641(n)). - Ridouane Oudra, Jun 04 2025

A363265 Number of integer factorizations of n with a unique mode.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 27 2023

nonn

An integer factorization of n is a multiset of positive integers > 1 with product n.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
Conjecture: 9 is missing from this sequence.

			The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:
  (2)  (4)    (16)       (24)       (48)         (72)
       (2*2)  (4*4)      (2*2*6)    (3*4*4)      (2*6*6)
              (2*2*4)    (2*2*2*3)  (2*2*12)     (3*3*8)
              (2*2*2*2)             (2*2*2*6)    (2*2*18)
                                    (2*2*3*4)    (2*2*2*9)
                                    (2*2*2*2*3)  (2*2*3*6)
                                                 (2*3*3*4)
                                                 (2*2*2*3*3)

The complement for partitions is A362607, ranks A362605.
The version for partitions is A362608, ranks A356862.
A001055 counts factorizations, strict A045778, ordered A074206.
A089723 counts constant factorizations.
A316439 counts factorizations by length, A008284 partitions.
A339846 counts even-length factorizations, A339890 odd-length.
Cf. A240219, A326622, A333487, A335434, A347438, A362610, A362611, A362612, A362614, A363723.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n],Length[modes[#]]==1&]],{n,100}]

A381872 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks having a common sum.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 14 2025

nonn

First differs from A321455 at a(144) = 4, A321455(144) = 3.

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, sum A056239.

			The prime indices of 144 are {1,1,1,1,2,2}, with the following 4 multiset partitions having common block sum:
  {{1,1,1,1,2,2}}
  {{2,2},{1,1,1,1}}
  {{1,1,2},{1,1,2}}
  {{2},{2},{1,1},{1,1}}
with sums: 8, 4, 4, 2, of which 3 are distinct, so a(144) = 3.
The prime indices of 1296 are {1,1,1,1,2,2,2,2}, with the following 7 multiset partitions having common block sum:
  {{1,1,1,1,2,2,2,2}}
  {{2,2,2},{1,1,1,1,2}}
  {{1,1,2,2},{1,1,2,2}}
  {{2,2},{2,2},{1,1,1,1}}
  {{2,2},{1,1,2},{1,1,2}}
  {{1,2},{1,2},{1,2},{1,2}}
  {{2},{2},{2},{2},{1,1},{1,1}}
with sums: 12, 6, 6, 4, 4, 3, 2, of which 5 are distinct, so a(1296) = 5.

With equal blocks instead of sums we have A089723.
Without equal sums we have A317141, before sums A001055, lower A300383.
Positions of terms > 1 are A321454.
Before taking sums we had A321455.
With distinct instead of equal sums we have A381637, before sums A321469.
A000041 counts integer partitions, strict A000009, constant A000005.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For sets of constant multisets (A050361) see A381715.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
Cf. A000720, A000961, A001222, A279787, A279789, A305551, A306017, A321451, A321452, A321453.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],SameQ@@Total/@#&]]],{n,100}]

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A316789 Number of same-tree-factorizations of n.

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A316790 Number of orderless same-tree-factorizations of n.

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A327695 Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

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A346403 a(1)=1; for n>1, a(n) gives the sum of the exponents in the different ways to write n as n = x^y, 2 <= x, 1 <= y.

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A363265 Number of integer factorizations of n with a unique mode.

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A381872 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks having a common sum.

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