Displaying 1-10 of 12 results found.
Number of times n occurs in A197863.
+20
1
1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2
FORMULA
Multiplicative with a(p^e) = 0 if e = 1, 2 if e = 2, and 1 otherwise.
MATHEMATICA
f[p_, e_] := Switch[e, 1, 0, 2, 2, _, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 28 2023 *)
PROG
(PARI) a(n)=my(es=factor(n)[, 2]~); prod(k=1, #es, if(es[k]==1, 0, if(es[k]==2, 2, 1)))
Powerfree part of n: product of primes that divide n only once.
+10
57
1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 1, 73, 74, 3, 19, 77, 78, 79, 5
COMMENTS
The previous name was: Write n = K^2*F where F is squarefree and F = g*f where g = gcd(K,F) and f = F/g; then a(n) = f(n) = F(n)/g(n). Thus gcd(K^2,f) = 1.
Differs from A007913; they coincide if and only if g(n) = 1. a(n) is the powerfree part of n; i.e., if n=Product(pi^ei) over all i (prime factorization) then a(n)=Product(pi^ei) over those i with ei=1; if n=b*c^2*d^3 then a(n) is minimum possible value of b. - Henry Bottomley, Sep 01 2000 Largest unitary squarefree number dividing n (the unitary squarefree kernel of n). - Steven Finch, Mar 01 2004 a(n) = 1 iff n is a squareful number ( A001694). 1 < a(n) < n iff n is a nonsquarefree number that is not squareful ( A332785). a(n) = n iff n is a squarefree number ( A005117). (End)
FORMULA
Multiplicative with a(p) = p and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Nov 01 2001 Dirichlet g.f.: zeta(s)*Product_{primes p} (1 + p^(1-s) - p^(-s) - p^(1-2s) + p^(-2s)). - R. J. Mathar, Dec 21 2011 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 - p^(1-3*s) + p^(2-3*s) - p^(2-2*s) + p^(-2*s) - p^(-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.394913518073109872954607634745304266741971541072... (End)
MAPLE
a := 1 ;
if n > 1 then
for f in ifactors(n)[2] do
if op(2, f) = 1 then
a := a*op(1, f) ;
end if;
end do:
end if;
a ;
MATHEMATICA
f[p_, e_] := If[e==1, p, 1]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)
PROG
local(a=1);
f=factor(n) ;
for(i=1, matsize(f)[1],
if( f[i, 2] ==1, a *= f[i, 1]
)
) ;
a ;
(PARI) a(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ Michel Marcus, Aug 27 2017 (Python)
from math import prod
from sympy import factorint
def A055231(n): return prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022
CROSSREFS
Cf. A008833, A007913, A007947, A000188, A057521, A055773 (computed for n!), A056169 (number of prime divisors), A056671 (number of divisors), A092261 (sum of divisors of the n-th term), A197863, A332785.
EXTENSIONS
Incorrect comments and example deleted by Peter Munn, Nov 30 2022
Largest cubefree number dividing n.
+10
22
1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 36, 73
FORMULA
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... ( A065465). - Amiram Eldar, Oct 13 2022
MATHEMATICA
Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> p^Min[e, 2]], {n, 73}] (* Michael De Vlieger, Jul 18 2017 *)
PROG
(Haskell)
a007948 = last . filter ((== 1) . a212793) . a027750_row
(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = min(f[i, 2], 2)); factorback(f); \\ Michel Marcus, Jun 09 2014
a(n) is the smallest exponentially odd number that is divisible by n.
+10
13
1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65
FORMULA
Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911..., f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)
MATHEMATICA
f[p_, e_] := If[OddQ[e], p^e, p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]%2, f[i, 1]^f[i, 2], f[i, 1]^(f[i, 2]+1)))};
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023
a(n) is the smallest cubefull exponentially odd number ( A335988) that is divisible by n.
+10
12
1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336
FORMULA
Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - Amiram Eldar, Nov 13 2022
MATHEMATICA
f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
PROG
(PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]%2, f[i, 1]^max(f[i, 2], 3), f[i, 1]^(f[i, 2]+1)))};
Smallest perfect power (at least a square) that is a multiple of n.
+10
11
1, 4, 9, 4, 25, 36, 49, 8, 9, 100, 121, 36, 169, 196, 225, 16, 289, 36, 361, 100, 441, 484, 529, 144, 25, 676, 27, 196, 841, 900, 961, 32, 1089, 1156, 1225, 36, 1369, 1444, 1521, 400, 1681, 1764, 1849, 484, 225, 2116, 2209, 144, 49, 100, 2601, 676, 2809, 216, 3025
COMMENTS
The prime signature of a(n) is not determined by the prime signature of n. For example, a(2^3*3^5) = 2^3*3^6, but a(2^3*5^5) = 2^5*5^5. - David Wasserman, May 03 2005 Likewise, this sequence is not multiplicative. The smallest exception is a(24) = 144 > 72 = a(3)*a(8). - Franklin T. Adams-Watters, Oct 18 2011
FORMULA
a(n)/n <= A007947(n) (the squarefree kernel of n). a(p^k) = p^k, a(k) = A007947(k)^2 for cubefree k. Furthermore, the upper bound on a(n)/n can be tightened to A007913(n). - Charlie Neder, Dec 26 2018
EXAMPLE
a(54) = 216 = 6^3. 54 is the least n such that a(n)/n does not divide A007947(n).
MATHEMATICA
a[n_] := n * SelectFirst[Range[n], GCD @@ FactorInteger[n*#][[;; , 2]] > 1 &]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)
a(n) is the smallest cubefull number ( A036966) that is a multiple of n.
+10
9
1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 16, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336
FORMULA
Multiplicative with a(p^e) = p^max(e,3).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p-2)/(p^3*(p-1))) = 1.76434793373691907811... . - Amiram Eldar, Jul 29 2022 Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(3)/4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.1559111567... . - Amiram Eldar, Nov 13 2022
MATHEMATICA
f[p_, e_] := p^Max[e, 3]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
PROG
(PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i, 1]^max(f[i, 2], 3))};
a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.
+10
9
1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 256, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 256, 65, 66, 67
FORMULA
Multiplicative with a(p^e) = p^(2^ceiling(log_2(e))).
MATHEMATICA
f[p_, e_] := p^(2^Ceiling[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) s(n) = {my(e=logint(n, 2)); if(n == 2^e, n, 2^(e+1))};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i, 1]^s(f[i, 2]))};
The number of divisors of the smallest powerful number that is a multiple of n.
+10
4
1, 3, 3, 3, 3, 9, 3, 4, 3, 9, 3, 9, 3, 9, 9, 5, 3, 9, 3, 9, 9, 9, 3, 12, 3, 9, 4, 9, 3, 27, 3, 6, 9, 9, 9, 9, 3, 9, 9, 12, 3, 27, 3, 9, 9, 9, 3, 15, 3, 9, 9, 9, 3, 12, 9, 12, 9, 9, 3, 27, 3, 9, 9, 7, 9, 27, 3, 9, 9, 27, 3, 12, 3, 9, 9, 9, 9, 27, 3, 15, 5, 9, 3
FORMULA
Multiplicative with a(p) = 3 and a(p^e) = e+1 for e >= 2.
a(n) >= A000005(n), with equality if and only if n is powerful ( A001694). Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 1/p^s - 2/p^(2*s) + 1/p^(3*s)).
Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 3/p^(2*s) + 3/p^(3*s) - 1/p^(4*s)).
Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 3/p^(3*s) - 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2), where
f(1) = Product_{primes p} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,
f'(1) = f(1) * Sum_{primes p} (6*p^2 - 9*p + 4) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 2.35603132119230949914708478515883136510141335620960622673206366...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-p*(12*p^5 - 27*p^4 + 16*p^3 + 9*p^2 - 12*p + 3) * log(p)^2 / (p^4 - 3*p^2 + 3*p - 1)^2) = f'(1)^2/f(1) + f(1) * (-7.3049026768735124341194605967271037971153161932236518820258070165876...),
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)
MATHEMATICA
f[p_, e_] := If[e == 1, 3, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> if(x == 1, 3, x+1), factor(n)[, 2]));
The sum of divisors of the smallest powerful number that is a multiple of n.
+10
4
1, 7, 13, 7, 31, 91, 57, 15, 13, 217, 133, 91, 183, 399, 403, 31, 307, 91, 381, 217, 741, 931, 553, 195, 31, 1281, 40, 399, 871, 2821, 993, 63, 1729, 2149, 1767, 91, 1407, 2667, 2379, 465, 1723, 5187, 1893, 931, 403, 3871, 2257, 403, 57, 217, 3991, 1281, 2863
FORMULA
Multiplicative with a(p) = p^2 + p + 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 2.
a(n) >= A000203(n), with equality if and only if n is powerful ( A001694). Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 1.01304866467771286896... .
MATHEMATICA
f[p_, e_] := If[e == 1, p^2 + p + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 1, f[i, 2] = 2)); sigma(f); }
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