A089723 -id:A089723 - cp's OEIS frontend

A295920 Number of twice-factorizations of n of type (P,R,R).

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 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, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 30 2017

nonn

a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) perfect divisors of d.

			The a(64) = 17 twice-factorizations are:
(2)*(2)*(2)*(2)*(2)*(2)  (2*2)*(2*2)*(2*2)  (2*2*2)*(2*2*2)  (2*2*2*2*2*2)
(2*2)*(2*2)*(4)          (2*2)*(4)*(2*2)    (4)*(2*2)*(2*2)
(2*2)*(4)*(4)            (4)*(2*2)*(4)      (4)*(4)*(2*2)
(2*2*2)*(8)              (8)*(2*2*2)
(4)*(4)*(4)              (4*4*4)
(8)*(8)                  (8*8)
(64)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001055, A052409, A052410, A089723, A279789, A281113, A295923, A295924, A295931, A295935.

Table[Sum[Length[Divisors[GCD@@FactorInteger[n^(1/d)][[All,2]]]]^d,{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A295920(n) = if(1==n,n,my(r); sumdiv(A052409(n), d, if(!ispower(n,d,&r),(1/0),numdiv(A052409(r))^d))); \\ Antti Karttunen, Dec 06 2018, after Mathematica-code

a(n) = Sum_{d|A052409(n)} A000005(A052409(n^(1/d)))^d. - Antti Karttunen, Dec 06 2018, after Mathematica-code

More terms from Antti Karttunen, Dec 06 2018

A296131 Number of twice-factorizations of n where the first factorization is strict and the latter factorizations are constant, i.e., type (P,Q,R).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 4, 4, 1, 5, 1, 9, 2, 2, 2, 9, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 13, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 11, 1, 2, 4, 16, 2, 5, 1, 4, 2, 5, 1, 18, 1, 2, 4, 4, 2, 5, 1, 13, 5, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

a(n) is the number of ways to choose a perfect divisor of each factor in a strict factorization of n.

			The a(24) = 8 twice-factorizations: (2)*(3)*(2*2), (2)*(3)*(4), (2)*(12), (3)*(2*2*2), (3)*(8), (2*2)*(6), (4)*(6), (24).

Cf. A001055, A005117, A045778, A063834, A089723, A279786, A281113, A296120, A296118.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Product[DivisorSigma[0,GCD@@FactorInteger[d][[All,2]]],{d,fac}],{fac,sfs[n]}],{n,100}]

Dirichlet g.f.: Product_{n > 1}(1 + A089723(n)/n^s).

A296134 Number of twice-factorizations of n of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 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, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

a(n) is the number of ways to choose a strict integer partition of a divisor of A052409(n).

			The a(16) = 4 twice-factorizations: (2)*(2*2*2), (2*2*2*2), (4*4), (16).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000009, A001055, A047966, A047968, A052409, A052410, A089723, A281113, A295923, A295924, A295931, A295935, A296133.

Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsQ],{n,100}]

A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A296134(n) = if(1==n,n,sumdiv(A052409(n),d,A000009(d))); \\ Antti Karttunen, Jul 29 2018

From Antti Karttunen, Jul 31 2018: (Start)
a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000009(d).
a(n) = A047966(A052409(n)). (End)

More terms from Antti Karttunen, Jul 29 2018

A363742 Number of integer factorizations of n with different mean, median, and mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 7

Offset: 1

Gus Wiseman, Jun 27 2023

nonn

An integer factorization of n is a multiset of positive integers > 1 with product n.
If there are multiple modes, then the mode is automatically considered different from the mean and median; otherwise, we take the unique mode.
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}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
Position of first appearance of n is: 1, 30, 48, 60, 72, 200, 160, 96, ...

			The a(n) factorizations for n = 30, 48, 60, 72, 96, 144:
  (2*3*5)  (2*3*8)    (2*5*6)    (2*4*9)    (2*6*8)    (2*8*9)
           (2*2*3*4)  (2*3*10)   (3*4*6)    (3*4*8)    (3*6*8)
                      (2*2*3*5)  (2*3*12)   (2*3*16)   (2*3*24)
                                 (2*2*3*6)  (2*4*12)   (2*4*18)
                                            (2*2*3*8)  (2*6*12)
                                            (2*2*4*6)  (3*4*12)
                                            (2*3*4*4)  (2*2*4*9)
                                                       (2*3*4*6)
                                                       (2*2*3*12)
                                                       (2*2*3*3*4)

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

Just (mean) != (median): A359911, complement A359909, partitions A359894.
The version for partitions is A363720, equal A363719, ranks A363730.
For equal instead of unequal we have A363741.
A001055 counts factorizations, strict A045778, ordered A074206.
A316439 counts factorizations by length, A008284 partitions.
A363265 counts factorizations with a unique mode.
Cf. A089723, A237984, A240219, A326622, A339846, A339890, A359910, A362608, A363725, A363727.

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],{Mean[#]}!={Median[#]}!=modes[#]&]],{n,100}]

median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
uniqmode(lista) = { my(freqs=Map(),v); for(i=1,#lista,if(!mapisdefined(freqs,lista[i],&v), v = 0); mapput(freqs,lista[i],1+v)); my(keys=Vec(freqs), fr, mc=0, mf=0, isuniq=0); for(i=1,#keys, fr = mapget(freqs,keys[i]); if(fr>=mf, isuniq = (fr>mf); mf = fr; mc = keys[i])); if(!isuniq, -1, mc); }; \\ Returns -1 if not unique mode.
all_different(facs) = { my(mean=(vecsum(facs)/#facs), med=median(facs), mode=uniqmode(facs)); ((mean!=med) &&  (mean!=mode) && (med!=mode)); };
A363742(n, m=n, facs=List([])) = if(1==n, (#facs>0 && all_different(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A363742(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

A175084 Possible values for product of perfect divisors of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Jaroslav Krizek, Jan 24 2010

nonn

Perfect divisor of n is divisor d such that d^k = n for some k >= 1. All terms of this sequence occur only once. See A089723 (number of perfect divisors of n) and A175068 (product of perfect divisors of n).

Complement of A175085. - Jaroslav Krizek, Jan 30 2010

Cf. A175087 (characteristic function).

With[{nn = 78}, TakeWhile[#, # <= nn &] &@ Union@ Table[Apply[Times, Select[Divisors@ n, Or[# == 1, #^IntegerExponent[n, #] == n] &]], {n, nn}] ] (* Michael De Vlieger, Nov 21 2017 *)

A255165 a(n) = Sum_{k=2..n} floor(log(n)/log(k)), n >= 1.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Richard R. Forberg, Feb 15 2015

The sum jumps up by 2 or more where n is a power of one or more k < n, otherwise it gains 1 with each increase in n.
First differences = A089723.
This calculation is analogous to that used for the sum of the number of divisors for all integers <= n in A006218.
a(n)+n gives the number of digits in the representations of n from base 2 to base n+1. - Christina Steffan, Dec 06 2015
Without floor, Sum_{k=2..n} log(n)/log(k) ~ n * (1 + 1/log(n) + 2/log(n)^2 + 6/log(n)^3 + 24/log(n)^4 + 120/log(n)^5 + ...). - Vaclav Kotesovec, Apr 06 2021

			The first jump is at n = 4 where, in the summation, log(4)/log(2), as it reaches a new floor.
Note: Possible complications exist calculating the floor function on ratios of logs that produce exact integers (e.g., in Mathematica). Adding an infinitesimal amount to n solves it.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum 2 (1951), 254-260. See T(n) pp. 258-259.

Cf. A006218, A089361, A089723, A043000.

[0] cat [&+[Floor(Log(n)/Log(k)):k in [2..n]]:n in [2..70]]; // Marius A. Burtea, Nov 13 2019

Table[Sum[Floor[Log[n]/Log[k]], {k, 2, n}], {n, 1, 100}]

a(n)=sum(k=2,n,log(n)\log(k)) \\ Anders Hellström, Dec 06 2015

a(n) = Sum_{k=2..n} floor(log(n)/log(k)), n >= 1.
It appears that a(n) = A089361(n) + n - 1. - Michel Marcus, Feb 17 2015
From Ridouane Oudra, Nov 13 2019: (Start)
a(n) = Sum_{i=2..n} floor(n^(1/i)).
a(n) = Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1).
a(n) = A043000(n) - n + 1. (End)
a(n) ~ n. - Vaclav Kotesovec, Apr 06 2021

A259362 a(1) = 1, for n > 1: a(n) is the number of ways to write n as a nontrivial perfect power.

Original entry on oeis.org

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

Offset: 1

Doug Bell, Jun 24 2015

nonn

a(n) = number of integer pairs (i,j) for distinct values of i where i > 0, j > 1 and n = i^j. Since 1 = 1^r for all real values of r, the requirement for a distinct i causes a(1) = 1 instead of a(1) = infinity.
Alternatively, the sequence can be defined as: a(1) = 1, for n > 1: a(n) = number of pairs (i,j) such that i > 0, j > 1 and n = i^j.
A007916 = n, where a(n) = 0.
A001597 = n, where a(n) > 0.
A175082 = n, where n = 1 or a(n) = 0.
A117453 = n, where n = 1 or a(n) > 1.
A175065 = n, where n > 1 and a(n) > 0 and this is the first occurrence in this sequence of a(n).
A072103 = n repeated a(n) times where n > 1.
A075802 = min(1, a(n)).
A175066 = a(n), where n = 1 or a(n) > 1. This sequence is an expansion of A175066.
A253642 = 0 followed by a(n), where n > 1 and a(n) > 0.
A175064 = a(1) followed by a(n) + 1, where n > 1 and a(n) > 0.
Where n > 1, A001597(x) = n (which implies a(n) > 0), i = A025478(x) and j = A253641(n), then a(n) = A000005(j) - 1, which is the number of factors of j greater than 1. The integer pair (i,j) comprises the smallest value i and the largest value j where i > 0, j > 1 and n = i^j. The a(n) pairs of (a,b) where a > 0, b > 1 and n = a^b are formed with b = each of the a(n) factors of j greater than 1. Examples for n = {8,4096}:
  a(8) = 1, A001597(3) = 8, A025478(3) = 2, A253641(8) = 3, 8 = 2^3 and A000005(3) - 1 = 1 because there is one factor of 3 greater than 1 [3]. The set of pairs (a,b) is {(2,3)}.
  a(4096) = 5, A001597(82) = 4096, A025478(82) = 2, A253641(4096) = 12, 4096 = 2^12 and A000005(12) - 1 = 5 because there are five factors of 12 greater than 1 [2,3,4,6,12]. The set of pairs (a,b) is {(64,2),(16,3),(8,4),(4,6),(2,12)}.
A023055 = the ordered list of x+1 with duplicates removed, where x is the number of consecutive zeros appearing in this sequence between any two nonzero terms.
A070428(x) = number of terms a(n) > 0 where n <= 10^x.
a(n) <= A188585(n).

			a(6) = 0 because there is no way to write 6 as a nontrivial perfect power.
a(9) = 1 because there is one way to write 9 as a nontrivial perfect power: 3^2.
a(16) = 2 because there are two ways to write 16 as a nontrivial perfect power: 2^4, 4^2.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:       1, 2, 3, 4, 5, 6, 7, 8, 9, ...
Squares: 1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:   1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
...
Sum:    oo, 0, 0, 1, 0, 0, 0, 1, 1, ...
a(1)=1:  1, 0, 0, 1, 0, 0, 0, 1, 1, ... (= this sequence). (End)

Doug Bell, Table of n, a(n) for n = 1..5000

Cf. A001597, A007916, A023055, A025478, A070428, A072103, A075090, A075109, A075802, A117453, A175064, A175066, A175082, A188585, A253641, A253642.

Cf. A010052, A010057, A374016, A089361, A089723, A382691 (alternating).

a[n_] := If[n == 1, 1, Sum[Boole[IntegerQ[n^(1/k)]], {k, 2, Floor[Log[2, n]]}]]; Array[a, 100] (* Friedjof Tellkamp, Jun 14 2025 *)
a[n_] := If[n == 1, 1, DivisorSigma[0, Apply[GCD, Transpose[FactorInteger[n]][[2]]]] - 1]; Array[a, 100] (* Michael Shamos, Jul 06 2025 *)

a(n) = if (n==1, 1, sum(i=2, logint(n, 2), ispower(n, i))); \\ Michel Marcus, Apr 11 2025

a(1) = 1, for n > 1: a(n) = A000005(A253641(n)) - 1.
If n not in A001597, then a(n) = 0, otherwise a(n) = A175064(x) - 1 where A001597(x) = n.
From Friedjof Tellkamp, Jun 14 2025: (Start)
a(n) = A089723(n) - 1, for n > 1.
a(n) = A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.
Sum_{k>=2..n} a(k) = A089361(n), for n > 1.
G.f.: x + Sum_{j>=2, k>=2} x^(j^k).
Dirichlet g.f.: 1 + Sum_{k>=2} zeta(k*s)-1. (End)

A294339 Number of ways to write 2^n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 12, 5, 6, 2, 19, 2, 6, 6, 32, 2, 19, 2, 19, 6, 6, 2, 56, 5, 6, 12, 19, 2, 26, 2, 79, 6, 6, 6, 71, 2, 6, 6, 56, 2, 26, 2, 19, 19, 6, 2, 169, 5, 19, 6, 19, 2, 56, 6, 56, 6, 6, 2, 101, 2, 6, 19, 203, 6, 26, 2, 19, 6, 26, 2, 237, 2, 6, 19, 19

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(6) = 6 ways are 64, 8^2, (2^3)^2, 4^3, (2^2)^3, 2^6.

Hans Havermann, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294338.

f:= proc(n) option remember; local F,t,s,g,a;
  F:= ifactors(n)[2];
  g:= igcd(op(map(t -> t[2],F)));
  t:= 1;
  for s in numtheory:-divisors(g) minus {1} do
    t:= t + procname(mul(a[1]^(a[2]/s),a=F))*procname(s)
  od;
  t
end proc:
seq(f(2^n),n=1..100); # Robert Israel, Dec 01 2017

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

a(n) = A294338(2^n). - R. J. Mathar, Nov 27 2017

A296133 Number of twice-factorizations of n of type (Q,R,Q).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 6, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(36) = 7 twice-factorizations are (2*3)*(6), (6)*(2*3), (2*3*6), (2*18), (3*12), (4*9), (36).

Cf. A000005, A001055, A005117, A045778, A052409, A052410, A089723, A279790, A279791, A284639, A292886, A296132.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Join@@Function[fac,Select[Join@@Permutations/@sps[fac],SameQ@@Times@@@#&]]/@sfs[n]],{n,100}]

A304481 Turn the power-tower for n upside-down.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 32, 26, 27, 28, 29, 30, 31, 25, 33, 34, 35, 64, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 128, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 36, 65, 66, 67

Offset: 1

Gus Wiseman, May 13 2018

This is an involution of the positive integers.

The power-tower for n is defined as follows. Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. Then a(n) = c(x_k)^...^c(x_3)^c(x_2)^c(x_1).

			The power tower of 81 is 3^2^2, which turned upside-down is 2^2^3 = 256, so a(81) = 256.

Robert Israel, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304491, A304492, A304495.

f:= proc(n,r) local F,a,y;
     if n = 1 then return 1 fi;
     F:= ifactors(n)[2];
     y:= igcd(seq(t[2],t=F));
     if y = 1 then return n^r fi;
     a:= mul(t[1]^(t[2]/y),t=F);
     procname(y,a^r)
end proc:
seq(f(n,1),n=1..100); # Robert Israel, May 13 2018

tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
Table[Power@@Reverse[tow[n]],{n,100}]

cp's OEIS Frontend

A295920 Number of twice-factorizations of n of type (P,R,R).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A296131 Number of twice-factorizations of n where the first factorization is strict and the latter factorizations are constant, i.e., type (P,Q,R).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A296134 Number of twice-factorizations of n of type (R,Q,R).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A363742 Number of integer factorizations of n with different mean, median, and mode.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A175084 Possible values for product of perfect divisors of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A255165 a(n) = Sum_{k=2..n} floor(log(n)/log(k)), n >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A259362 a(1) = 1, for n > 1: a(n) is the number of ways to write n as a nontrivial perfect power.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica