Displaying 1-10 of 82 results found.
Non-cubefree numbers n such that 2n+1 is also non-cubefree ( A046099).
+20
2
40, 256, 312, 472, 688, 904, 1120, 1200, 1312, 1336, 1552, 1768, 1984, 2187, 2200, 2312, 2416, 2456, 2632, 2848, 2875, 3064, 3280, 3312, 3429, 3496, 3712, 3928, 3944, 4144, 4312, 4360, 4576, 4792, 5008, 5224, 5312, 5440, 5562, 5656, 5872, 6088, 6250, 6304, 6312, 6345, 6520, 6655, 6688, 6736, 6952
COMMENTS
The probability that a random integer is cubefree is 1/zeta(3) (see A088453).
EXAMPLE
312 is there because 2^3 divides 312 and 5^3 divides 312*2+1.
8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091, 2571353, 2685619, 3307949
COMMENTS
Numbers with exactly three factorizations: A001055(a(n)) = 3 (e.g., a(4) = 1*343 = 7*49 = 7*7*7). - Reinhard Zumkeller, Dec 29 2001 Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (9/7) * (28/26) * (124/126) * (344/342) * (1332/1330) * ... = 48/35. - Dimitris Valianatos, Mar 06 2020 There exist 5 groups of order p^3, when p prime, so this is a subsequence of A054397. Three of them are abelian: C_p^3, C_p^2 X C_p and C_p X C_p X C_p = (C_p)^3. For 8 = 2^3, the 2 nonabelian groups are D_8 and Q_8; for odd prime p, the 2 nonabelian groups are (C_p x C_p) : C_p, and C_p^2 : C_p (remark, for p = 2, these two semi-direct products are isomorphic to D_8). Here C, D, Q mean Cyclic, Dihedral, Quaternion groups of the stated order; the symbols X and : mean direct and semidirect products respectively. - Bernard Schott, Dec 11 2021
REFERENCES
Edmund Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen über Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.
LINKS
Wikipedia, p-group, Classification.
FORMULA
Product_{n>=1} (1 + 1/a(n)) = zeta(3)/zeta(6) ( A157289). Product_{n>=1} (1 - 1/a(n)) = 1/zeta(3) ( A088453). (End)
EXAMPLE
a(3) = 125; since the 3rd prime is 5, a(3) = 5^3 = 125.
PROG
(Sage)
(Haskell)
a030078 = a000578 . a000040
(Python)
from sympy import prime, primerange
def aupton(terms): return [p**3 for p in primerange(1, prime(terms)+1)]
CROSSREFS
Other sequences that are k-th powers of primes are: A000040 (k=1), A001248 (k=2), this sequence (k=3), A030514 (k=4), A050997 (k=5), A030516 (k=6), A092759 (k=7), A179645 (k=8), A179665 (k=9), A030629 (k=10), A079395 (k=11), A030631 (k=12), A138031 (k=13), A030635 (k=16), A138032 (k=17), A030637 (k=18).
Cubefree numbers: numbers that are not divisible by any cube > 1.
+10
178
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85
COMMENTS
Numbers n such that no smaller number m satisfies: kronecker(n,k)=kronecker(m,k) for all k. - Michael Somos, Sep 22 2005 The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = A088453. - Gerard P. Michon, May 06 2009 The Schnirelmann density of the cubefree numbers is 157/189 (Orr, 1969). - Amiram Eldar, Mar 12 2021 Numbers whose sets of unitary divisors ( A077610) and bi-unitary divisors ( A222266) coincide. Number whose all divisors are (1+e)-divisors, or equivalently, numbers k such that A049599(k) = A000005(k). (End)
LINKS
Eric Weisstein's World of Mathematics, Cubefree.
FORMULA
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Dec 27 2022
MAPLE
isA004709 := proc(n)
local p;
for p in ifactors(n)[2] do
if op(2, p) > 2 then
return false;
end if;
end do:
true ;
end proc:
MATHEMATICA
Select[Range[6!], FreeQ[FactorInteger[#], {_, k_ /; k > 2}] &] (* Jan Mangaldan, May 07 2014 *)
PROG
(PARI) {a(n)= local(m, c); if(n<2, n==1, c=1; m=1; while( c<n, m++; if( 3>vecmax(factor(m)[, 2]), c++)); m)} /* Michael Somos, Sep 22 2005 */ (Haskell)
a004709 n = a004709_list !! (n-1)
a004709_list = filter ((== 1) . a212793) [1..]
(Python)
from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) == n
(Python)
from sympy import mobius, integer_nthroot
def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
Moebius transform applied twice to sequence 1,0,0,0,....
(Formerly M0198)
+10
66
1, -2, -2, 1, -2, 4, -2, 0, 1, 4, -2, -2, -2, 4, 4, 0, -2, -2, -2, -2, 4, 4, -2, 0, 1, 4, 0, -2, -2, -8, -2, 0, 4, 4, 4, 1, -2, 4, 4, 0, -2, -8, -2, -2, -2, 4, -2, 0, 1, -2, 4, -2, -2, 0, 4, 0, 4, 4, -2, 4, -2, 4, -2, 0, 4, -8, -2, -2, 4, -8, -2, 0, -2, 4, -2, -2, 4, -8, -2, 0, 0
COMMENTS
|a(n)| is the number of ways to write n as a product of 2 squarefree numbers (i.e., number of ways to write n = x*y with 1 <= x <= n, 1 <= y <= n, x and y squarefree). - Benoit Cloitre, Jan 01 2003
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
Dirichlet g.f.: 1/zeta(s)^2.
Multiplicative function with a(p^e) = binomial(2, e)*(-1)^e for p prime and e >= 0.
Dirichlet g.f. of |a(n)|: zeta(s)^2/zeta(2s)^2 (conjectured). - Ralf Stephan, Jul 05 2013. The conjecture is correct because 1+Sum_{e>=1} binomial(2,e)/p^(e*s) = (p^s+1)^2/p^2s, whose product over p is zeta(s)^2/zeta(2s)^2. - Michael ShamosSum_{k=1..n} abs(a(k)) ~ (n/zeta(2)^2) * (log(n) + 2*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant ( A001620). - Amiram Eldar, Dec 24 2023
EXAMPLE
G.f. = x - 2*x^2 - 2*x^3 + x^4 - 2*x^5 + 4*x^6 - 2*x^7 + x^9 + 4*x^10 + ...
We have a(3^1) = C(2, 1)*(-1)^1 = -2, a(3^2) = C(2, 2)*(-1)^2 = 1, and a(3^m) = C(2, m)*(-1)^m = 0 for m >= 3. - Petros Hadjicostas, Jun 07 2019
MAPLE
möbius := proc(a) local b, i, mo: b := NULL:
mo := (m, n) -> `if`(irem(m, n) = 0, numtheory:-mobius(m/n), 0);
for i to nops(a) do b := b, add(mo(i, j)*a[j], j=1..i) od: [b] end:
MATHEMATICA
f[n_] := Plus @@ Times @@@ (MoebiusMu[{#, n/#}] & /@ Divisors@n); Array[f, 105] (* Robert G. Wilson v *) a[n_] := DivisorSum[n, MoebiusMu[#]*MoebiusMu[n/#]&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, (1 - X)^2)[n])}; /* Michael Somos, Nov 15 2002 */ (PARI) {a(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d)))}; /* Michael Somos, Nov 15 2002 */ (PARI) a(n)=if(n>1, my(f=factor(n)[, 2], s=sum(i=1, #f, f[i]==1)); if(vecmax(f)>2, 0, (-1)^s<<s), 1) \\ Charles R Greathouse IV, Jun 28 2011 (Haskell)
a007427 n = sum $ zipWith (*) mds $ reverse mds where
mds = a225817_row n
(Python)
from math import prod, comb
from sympy import factorint
def A007427(n): return prod(-comb(2, e) if e&1 else comb(2, e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024
EXTENSIONS
Added a proof of Stephan's conjecture about the Dirichlet g.f. of |a(n)|.
Number of factorizations of n with integer alternating product.
+10
39
1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 6, 1, 1, 1, 2, 1, 1, 1, 7
COMMENTS
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
EXAMPLE
The factorizations for n = 4, 16, 36, 48, 54, 64, 108:
(4) (16) (36) (48) (54) (64) (108)
(2*2) (4*4) (6*6) (2*4*6) (2*3*9) (8*8) (2*6*9)
(2*2*4) (2*2*9) (3*4*4) (3*3*6) (2*4*8) (3*6*6)
(2*2*2*2) (2*3*6) (2*2*12) (4*4*4) (2*2*27)
(3*3*4) (2*2*2*2*3) (2*2*16) (2*3*18)
(2*2*3*3) (2*2*4*4) (3*3*12)
(2*2*2*2*4) (2*2*3*3*3)
(2*2*2*2*2*2)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[facs[n], IntegerQ@*altprod]], {n, 100}]
PROG
(PARI) A347437(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if((d>1)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023
CROSSREFS
Allowing any alternating product <= 1 gives A339846. Allowing any alternating product > 1 gives A339890. The restriction to powers of 2 is A344607. The even-length case is A347438, also the case of alternating product 1. Allowing any alternating product < 1 gives A347440. Allowing any alternating product >= 1 gives A347456. A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors of n (reverse: A071322).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A062312, A088218, A119620, A316523, A330972, A332269, A347445, A347447, A347451, A347454.
Number of unordered factorizations of n with alternating product 1.
+10
37
1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 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
COMMENTS
Also the number of unordered factorizations of n with alternating sum 0.
Also the number of unordered factorizations of n with all even multiplicities.
This is the even-length case of A347437, the odd-length case being A347441. An unordered factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
EXAMPLE
The a(n) factorizations for n = 16, 64, 144, 256, 576:
4*4 8*8 12*12 16*16 24*24
2*2*2*2 2*2*4*4 2*2*6*6 2*2*8*8 3*3*8*8
2*2*2*2*2*2 3*3*4*4 4*4*4*4 4*4*6*6
2*2*2*2*3*3 2*2*2*2*4*4 2*2*12*12
2*2*2*2*2*2*2*2 2*2*2*2*6*6
2*2*3*3*4*4
2*2*2*2*2*2*3*3
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[facs[n], altprod[#]==1&]], {n, 100}]
PROG
(PARI) A347438(n, m=n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A347438(n/d, d, 1-k, t*(d^((-1)^k))))); (s)); \\ Antti Karttunen, Oct 30 2021
CROSSREFS
Positions of nonzero terms are A000290. The restriction to powers of 2 is A035363. Positions of non-1's are A213367 \ {1}. Sorted first positions are 1, 2, and all terms of A330972 squared. Allowing any alternating product <= 1 gives A339846. Allowing any alternating product > 1 gives A339890. Allowing any integer alternating product gives A347437. Allowing any integer reciprocal alternating product gives A347439. Allowing any alternating product < 1 gives A347440. Allowing any alternating product >= 1 gives A347456. A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A000041, A005117, A025047, A038548, A062312, A088218, A316523, A332269, A344607, A347442, A347446, A347463.
Characteristic function of cubefree numbers, A004709.
+10
33
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1
LINKS
Eric Weisstein's World of Mathematics, Cubefree.
FORMULA
Multiplicative with a(p^e) = 1 if e<=2, =0 if e>=3. - R. J. Mathar, Dec 17 2012 Sum_{n>0} a(n)/n^s = Product_{p prime} (1+p^(-s)+p^(-2s)) = zeta(s) / zeta(3s). - Ralf Stephan, Jul 07 2013 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/zeta(3) ( A088453). - Amiram Eldar, Jul 23 2022 Dirichlet g.f.: zeta(s)/zeta(3*s). - Amiram Eldar, Dec 27 2022
MATHEMATICA
Table[Boole[Max[FactorInteger[n][[All, 2]]] < 3], {n, 1, 100}] (* Geoffrey Critzer, Feb 25 2015 *)
PROG
(Haskell)
a212793 = cubeFree a000040_list 0 0 where
cubeFree ps'@(p:ps) q e x
| e > 2 = 0
| x == 1 = 1
| r > 0 = cubeFree ps p 0 x
| otherwise = cubeFree ps' p (e + 1) x' where (x', r) = divMod x p
(PARI) a(n) = {f = factor(n); for (i=1, #f~, if ((f[i, 2]) >=3, return(0)); ); return (1); } \\ Michel Marcus, Feb 10 2015
CROSSREFS
Cf. A000005, A000007, A004709, A008966, A046099, A053864, A060431, A088453, A112526, A124010, A307423.
Number of factorizations of n with integer reciprocal alternating product.
+10
29
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 6, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 1, 4, 0, 0, 0, 1, 0, 0, 0, 5
COMMENTS
All of these factorizations have an even number of factors, so their reverse-alternating product is also an integer.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
The value of a(n) does not depend solely on the prime signature of n. See the example comparing a(144) and a(400). - Antti Karttunen, Jul 28 2024
EXAMPLE
The a(n) factorizations for
n = 16, 36, 64, 72, 128, 144:
a(n) = 3, 4, 6, 5, 7, 11
--------------------------------------------------------------------------------
2*8 6*6 8*8 2*36 2*64 2*72
4*4 2*18 2*32 3*24 4*32 3*48
2*2*2*2 3*12 4*16 6*12 8*16 4*36
2*2*3*3 2*2*2*8 2*2*3*6 2*2*4*8 6*24
2*2*4*4 2*3*3*4 2*4*4*4 12*12
2*2*2*2*2*2 2*2*2*16 2*2*6*6
2*2*2*2*2*4 2*3*3*8
3*3*4*4
2*2*2*18
2*2*3*12
2*2*2*2*3*3
For n=400, there are 12 such factorizations:
2*200
4*100
5*80
10*40
20*20
2*2*2*50
2*2*5*20
2*2*10*10
2*4*5*10
2*5*5*8
4*4*5*5
2*2*2*2*5*5.
Note that 400 = 2^4 * 5^2 has the same prime signature as 144 = 2^4 * 3^2. 400 = 2*4*5*10 is the factorization for which there is no analogous factorization of 144, as 2*3*4*6 doesn't satisfy the condition of having an integer reciprocal alternating product.
(End)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i, {i, Length[q]}];
Table[Length[Select[facs[n], IntegerQ[recaltprod[#]]&]], {n, 100}]
PROG
(PARI) A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024 (PARI) A347439(n, m=0, ap=1, e=1) = if(1==n, 1==denominator(ap), sumdiv(n, d, if(d>1 && d>=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024
CROSSREFS
Positions of 0's are A005117 \ {1}. Positions of non-0's are 1 and A013929. Positions of 1's are 1 and A082293. Allowing any alternating product <= 1 gives A339846. Allowing any alternating product > 1 gives A339890. The non-reciprocal version is A347437. Allowing any alternating product < 1 gives A347440. The non-reciprocal reverse version is A347442. Allowing any alternating product >= 1 gives A347456. The restriction to perfect squares is A347459, non-reciprocal A347458. A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A236913, A316523, A330972, A332269, A344606, A344607, A347445, A347446, A347454, A347457, A347463.
Number of ordered factorizations of n with integer alternating product.
+10
26
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 7, 1, 4, 1, 4, 1, 1, 1, 6, 2, 1, 3, 4, 1, 1, 1, 11, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 4, 4, 1, 1, 20, 2, 4, 1, 4, 1, 6, 1, 6, 1, 1, 1, 8, 1, 1, 4, 26, 1, 1, 1, 4, 1, 1, 1, 35, 1, 1, 4, 4, 1, 1, 1, 20, 7, 1, 1, 8, 1, 1, 1, 6, 1, 8, 1, 4, 1, 1, 1, 32, 1, 4, 4, 18
COMMENTS
An ordered factorization of n is a sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
EXAMPLE
The ordered factorizations for n = 4, 8, 12, 16, 24, 32, 36:
4 8 12 16 24 32 36
2*2 4*2 6*2 4*4 12*2 8*4 6*6
2*2*2 2*2*3 8*2 2*2*6 16*2 12*3
3*2*2 2*2*4 3*2*4 2*2*8 18*2
2*4*2 4*2*3 2*4*4 2*2*9
4*2*2 6*2*2 4*2*4 2*3*6
2*2*2*2 4*4*2 2*6*3
8*2*2 3*2*6
2*2*4*2 3*3*4
4*2*2*2 3*6*2
2*2*2*2*2 4*3*3
6*2*3
6*3*2
9*2*2
2*2*3*3
2*3*3*2
3*2*2*3
3*3*2*2
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[Join@@Permutations/@facs[n], IntegerQ[altprod[#]]&]], {n, 100}]
PROG
(PARI) A347463(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if(d>1, A347463(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024
CROSSREFS
The restriction to powers of 2 is A116406. A046099 counts factorizations with no alternating permutations.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347438 counts factorizations with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A138364, A347440, A347441, A347453, A347454, A347456, A347458, A347459, A347464, A347705, A347708.
16, 32, 48, 64, 80, 81, 96, 112, 128, 144, 160, 162, 176, 192, 208, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 384, 400, 405, 416, 432, 448, 464, 480, 486, 496, 512, 528, 544, 560, 567, 576, 592, 608, 624, 625, 640, 648, 656, 672, 688, 704
COMMENTS
The convention in the OEIS is that squareful, cubeful ( A046099), biquadrateful, ... mean the same as "not squarefree" etc., while 2- or square-full, 3- or cube-full ( A036966), 4-full ( A036967) are used for Golomb's notion of powerful numbers ( A001694, see references there), when each prime factor occurs to a power > 1. - M. F. Hasler, Feb 12 2008 The asymptotic density of this sequence is 1 - 1/zeta(4) = 1 - 90/Pi^4 = 0.076061... - Amiram Eldar, Jul 09 2020
MAPLE
with(NumberTheory):
isBiquadrateful := n -> is(denom(Radical(n) / LargestNthPower(n, 2)) <> 1):
select(isBiquadrateful, [`$`(1..704)]); # Peter Luschny, Jul 12 2022
MATHEMATICA
lst={}; Do[a=0; Do[If[FactorInteger[m][[n, 2]]>3, a=1], {n, Length[FactorInteger[m]]}]; If[a==1, AppendTo[lst, m]], {m, 10^3}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 15 2008 *) Select[Range[1000], Max[Transpose[FactorInteger[#]][[2]]]>3&] (* Harvey P. Dale, May 25 2014 *)
PROG
(Haskell)
a046101 n = a046101_list !! (n-1)
a046101_list = filter ((> 3) . a051903) [1..]
(Python)
from sympy import mobius, integer_nthroot
def f(x): return n+sum(mobius(k)*(x//k**4) for k in range(1, integer_nthroot(x, 4)[0]+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
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