Displaying 1-10 of 23 results found.
a(n) is the smallest number k such that k*n is an exponentially squarefree number ( A209061).
+20
5
1, 1, 1, 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, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1
FORMULA
Multiplicative with a(p^e) = p^ A081221(e). a(n) >= 1, with equality if and only if n is an exponentially squarefree number ( A209061). Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^f(k-1))/p^k) = 1.06562841319..., where f(k) = A081221(k) and f(0) = 0.
MATHEMATICA
f[p_, e_] := Module[{k = e}, While[! SquareFreeQ[k], k++]; p^(k-e)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) s(e) = {my(k = e); while(!issquarefree(k), k++); k - e; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }
a(n) is the smallest multiple of n that is an exponentially squarefree number ( A209061).
+20
4
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 96, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68
FORMULA
Multiplicative with a(p^e) = p^ A067535(e). a(n) >= n, with equality if and only if n is an exponentially squarefree number ( A209061). Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.532814206... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = A067535(k) and f(0) = 0.
MATHEMATICA
f[p_, e_] := Module[{k = e}, While[! SquareFreeQ[k], k++]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) s(e) = {my(k = e); while(!issquarefree(k), k++); k; };
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }
Characteristic function of exponentially squarefree numbers ( A209061).
+20
1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1
FORMULA
Multiplicative with a(p^e) = mu(e)^2, where mu is the Möbius function ( A008683). a(n) = 1 if and only if n is in A209061. a(n) = 0 if and only if n is in A130897. Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A262276.
MATHEMATICA
f[p_, e_] := MoebiusMu[e]^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> moebius(x)^2, factor(n)[, 2]));
Squarefree numbers: numbers that are not divisible by a square greater than 1.
(Formerly M0617)
+10
1751
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113
COMMENTS
1 together with the numbers that are products of distinct primes.
Also smallest sequence with the property that a(m)*a(k) is never a square for k != m. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001
Numbers k such that there is only one Abelian group with k elements, the cyclic group of order k (the numbers such that A000688(k) = 1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001 a(n) is the smallest m with exactly n squarefree numbers <= m. - Amarnath Murthy, May 21 2002 k is squarefree <=> k divides prime(k)# where prime(k)# = product of first k prime numbers. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 30 2004
The LCM of any finite subset is in this sequence. - Lekraj Beedassy, Jul 11 2006 This sequence and the Beatty Pi^2/6 sequence ( A059535) are "incestuous": the first 20000 terms are bounded within (-9, 14). - Ed Pegg Jr, Jul 22 2008 Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1) + ... + alpha(r)), sigma_0(n) number of divisors of n ( A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1) + ... + alpha(r) is sequence ( A086436). Function D(n) splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers ( A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. For D(n)=1/2 we have A048109, for D(n)=3/4 we have A067295. - Ctibor O. Zizka, Sep 21 2008 Numbers k such that sqrt(k) cannot be simplified. - Sean Loughran, Sep 04 2011 Indices m where A057918(m)=0, i.e., positive integers m for which there are no integers k in {1,2,...,m-1} such that k*m is a square. - John W. Layman, Sep 08 2011 It appears that these are numbers j such that Product_{k=1..j} (prime(k) mod j) = 0 (see Maple code). - Gary Detlefs, Dec 07 2011. - This is the same claim as Mohammed Bouayoun's Mar 30 2004 comment above. To see why it holds: Primorial numbers, A002110, a subsequence of this sequence, are never divisible by any nonsquarefree number, A013929, and on the other hand, the index of the greatest prime dividing any n is less than n. Cf. A243291. - Antti Karttunen, Jun 03 2014 Conjecture: For each n=2,3,... there are infinitely many integers b > a(n) such that Sum_{k=1..n} a(k)*b^(k-1) is prime, and the smallest such an integer b does not exceed (n+3)*(n+4). - Zhi-Wei Sun, Mar 26 2013 The probability that a random natural number belongs to the sequence is 6/Pi^2, A059956 (see Cesàro reference). - Giorgio Balzarotti, Nov 21 2013 Booker, Hiary, & Keating give a subexponential algorithm for testing membership in this sequence without factoring. - Charles R Greathouse IV, Jan 29 2014 Because in the factorizations into prime numbers these a(n) (n >= 2) have exponents which are either 0 or 1 one could call the a(n) 'numbers with a fermionic prime number decomposition'. The levels are the prime numbers prime(j), j >= 1, and the occupation numbers (exponents) e(j) are 0 or 1 (like in Pauli's exclusion principle). A 'fermionic state' is then denoted by a sequence with entries 0 or 1, where, except for the zero sequence, trailing zeros are omitted. The zero sequence stands for a(1) = 1. For example a(5) = 6 = 2^1*3^1 is denoted by the 'fermionic state' [1, 1], a(7) = 10 by [1, 0, 1]. Compare with 'fermionic partitions' counted in A000009. - Wolfdieter Lang, May 14 2014 The following is an Eratosthenes-type sieve for squarefree numbers. For integers > 1:
1) Remove even numbers, except for 2; the minimal non-removed number is 3.
2) Replace multiples of 3 removed in step 1, and remove multiples of 3 except for 3 itself; the minimal non-removed number is 5.
3) Replace multiples of 5 removed as a result of steps 1 and 2, and remove multiples of 5 except for 5 itself; the minimal non-removed number is 6.
4) Replace multiples of 6 removed as a result of steps 1, 2 and 3 and remove multiples of 6 except for 6 itself; the minimal non-removed number is 7.
5) Repeat using the last minimal non-removed number to sieve from the recovered multiples of previous steps.
Proof. We use induction. Suppose that as a result of the algorithm, we have found all squarefree numbers less than n and no other numbers. If n is squarefree, then the number of its proper divisors d > 1 is even (it is 2^k - 2, where k is the number of its prime divisors), and, by the algorithm, it remains in the sequence. Otherwise, n is removed, since the number of its squarefree divisors > 1 is odd (it is 2^k-1).
(End)
The lexicographically least sequence of integers > 1 such that each entry has an even number of proper divisors occurring in the sequence (that's the sieve restated). - Glen Whitney, Aug 30 2015 0 is nonsquarefree because it is divisible by any square. - Jon Perry, Nov 22 2014, edited by M. F. Hasler, Aug 13 2015 The Heinz numbers of partitions with distinct parts. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} prime(j) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] the Heinz number is 2*2*3*7*29 = 2436. The number 30 (= 2*3*5) is in the sequence because it is the Heinz number of the partition [1,2,3]. - Emeric Deutsch, May 21 2015 It is possible for 2 consecutive terms to be even; for example a(258)=422 and a(259)=426. - Thomas Ordowski, Jul 21 2015. [These form a subsequence of A077395 since their product is divisible by 4. - M. F. Hasler, Aug 13 2015] There are never more than 3 consecutive terms. Runs of 3 terms start at 1, 5, 13, 21, 29, 33, ... ( A007675). - Ivan Neretin, Nov 07 2015 Numbers k such that b^(phi(k)+1) == b (mod k) for every integer b. - Thomas Ordowski, Oct 09 2016 Boreico shows that the set of square roots of the terms of this sequence is linearly independent over the rationals. - Jason Kimberley, Nov 25 2016 (reference found by Michael Coons). The prime zeta function P(s) "has singular points along the real axis for s=1/k where k runs through all positive integers without a square factor". See Wolfram link. - Maleval Francis, Jun 23 2018 The Schnirelmann density of the squarefree numbers is 53/88 (Rogers, 1964). - Amiram Eldar, Mar 12 2021 Numbers k such that all groups of order k have a trivial Frattini subgroup [Dummit and Foote].
Let the group G have order n. If n is squarefree and n > 1, then G is solvable, and thus by Hall's Theorem contains a subgroup H_p of index p for all p | n. Each H_p is maximal in G by order considerations, and the intersection of all the H_p's is trivial. Thus G's Frattini subgroup Phi(G), being the intersection of G's maximal subgroups, must be trivial. If n is not squarefree, the cyclic group of order n has a nontrivial Frattini subgroup. (End)
Numbers for which the squarefree divisors ( A206778) and the unitary divisors ( A077610) are the same; moreover they are also the set of divisors ( A027750). - Bernard Schott, Nov 04 2022 Numbers n such that mu(n) != 0, where mu(n) is the Möbius function ( A008683). Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = mu(n)*n, where sigma(n) is the sum of divisors function ( A000203). (End) a(n) is the smallest root of x = 1 + Sum_{k=1..n-1} floor(sqrt(x/a(k))) greater than a(n-1). - Yifan Xie, Jul 10 2024
REFERENCES
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 165, p. 53, Ellipses, Paris, 2008.
Dummit, David S., and Richard M. Foote. Abstract algebra. Vol. 1999. Englewood Cliffs, NJ: Prentice Hall, 1991.
Ivan M. Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 251.
Michael Pohst and Hans J. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Srinivasa Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106. Eric Weisstein's World of Mathematics, Squarefree.
FORMULA
|a(n) - n*Pi^2/6| < 0.058377*sqrt(n) for n >= 268293; this result can be derived from Cohen, Dress, & El Marraki, see links. - Charles R Greathouse IV, Jan 18 2018 Sum_{n>=1} (-1)^(a(n)+1)/a(n)^2 = 9/Pi^2.
Sum_{k=1..n} 1/a(k) ~ (6/Pi^2) * log(n).
Sum_{k=1..n} (-1)^(a(k)+1)/a(k) ~ (2/Pi^2) * log(n).
(all from Scott, 2006) (End)
MAPLE
with(numtheory); a := [ ]; for n from 1 to 200 do if issqrfree(n) then a := [ op(a), n ]; fi; od:
t:= n-> product(ithprime(k), k=1..n): for n from 1 to 113 do if(t(n) mod n = 0) then print(n) fi od; # Gary Detlefs, Dec 07 2011 A005117 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[issqrfree](a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Jan 09 2013
MATHEMATICA
Select[Range[150], Max[Last /@ FactorInteger[ # ]] < 2 &] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
NextSquareFree[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sf = n + sgn; While[c < Abs[k], While[ ! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[ sgn < 0, sf--, sf++]; c++]; sf + If[ sgn < 0, 1, -1]]; NestList[ NextSquareFree, 1, 70] (* Robert G. Wilson v, Apr 18 2014 *)
PROG
(Magma) [ n : n in [1..1000] | IsSquarefree(n) ];
(PARI) bnd = 1000; L = vector(bnd); j = 1; for (i=1, bnd, if(issquarefree(i), L[j]=i; j=j+1)); L
(PARI) {a(n)= local(m, c); if(n<=1, n==1, c=1; m=1; while( c<n, m++; if(issquarefree(m), c++)); m)} /* Michael Somos, Apr 29 2005 */ (PARI) list(n)=my(v=vectorsmall(n, i, 1), u, j); forprime(p=2, sqrtint(n), forstep(i=p^2, n, p^2, v[i]=0)); u=vector(sum(i=1, n, v[i])); for(i=1, n, if(v[i], u[j++]=i)); u \\ Charles R Greathouse IV, Jun 08 2012 (PARI)
S(n) = my(s); forsquarefree(k=1, sqrtint(n), s+=n\k[1]^2*moebius(k)); s;
a(n) = my(min=1, max=231, k=0, sc=0); if(n >= 144, min=floor(zeta(2)*n - 5*sqrt(n)); max=ceil(zeta(2)*n + 5*sqrt(n))); while(min <= max, k=(min+max)\2; sc=S(k); if(abs(sc-n) <= sqrtint(n), break); if(sc > n, max=k-1, if(sc < n, min=k+1, break))); while(!issquarefree(k), k-=1); while(sc != n, my(j=1); if(sc > n, j = -1); k += j; sc += j; while(!issquarefree(k), k += j)); k; \\ Daniel Suteu, Jul 07 2022 (PARI) first(n)=my(v=vector(n), i); forsquarefree(k=1, if(n<268293, (33*n+30)\20, (n*Pi^2/6+0.058377*sqrt(n))\1), if(i++>n, return(v)); v[i]=k[1]); v \\ Charles R Greathouse IV, Jan 10 2023 (Haskell)
a005117 n = a005117_list !! (n-1)
a005117_list = filter ((== 1) . a008966) [1..]
(Python)
from sympy.ntheory.factor_ import core
def ok(n): return core(n, 2) == n
(Python)
from itertools import count, islice
from sympy import factorint
def A005117_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda n:all(x == 1 for x in factorint(n).values()), count(max(startvalue, 1)))
(Python)
from math import isqrt
from sympy import mobius
def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
CROSSREFS
Cf. A076259 (first differences), A173143 (partial sums), A000688, A003277, A013928, A020753, A020754, A020755, A030059, A030229, A033197, A034444, A039956, A048672, A053797, A057918, A059956, A071403, A072284, A120992, A133466, A136742, A136743, A160764, A243289, A243347, A243348, A243351, A215366, A046660, A265668, A265675.
Exponentially odd numbers.
+10
106
1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97
COMMENTS
The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
The density of the sequence is the constant given by A065463. The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - Amiram Eldar, Jan 22 2024
FORMULA
Sum_{a(n)<=x} 1 = C*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... ( A065463). Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - Amiram Eldar, Sep 26 2023
MATHEMATICA
Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)
PROG
(PARI) isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k, 2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016 (Python)
from itertools import count, islice
from sympy import factorint
def A268335_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda n:all(e&1 for e in factorint(n).values()), count(max(startvalue, 1)))
CROSSREFS
Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.
Exponentially 2^n-numbers: 1 together with positive integers k such that all exponents in prime factorization of k are powers of 2.
+10
56
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 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, 48, 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, 80, 81
COMMENTS
Previous name: sequence consists of products of distinct relatively prime terms of A084400. - Vladimir Shevelev, Sep 24 2015 These numbers are also called "compact integers."
The density of this sequence exists and equals 0.872497...
There exist only seven compact factorials A000142(n) for n=1,2,3,6,7,10 and 11.
FORMULA
Identities arising from the calculation of the density h of the sequence (cf. [Shevelev] and comment for a generalization in A209061): h = Product_{prime p} Sum_{j in {0 and 2^k}}(p-1)/p^(j+1) = Product_{prime p} (1 + Sum_{j>=2} (u(j) - u(j-1))/p^j) = (1/zeta(2))* Product_{p}(1 + 1/(p+1))*Sum_{i>=1}p^(-(2^i-1)), where u(n) is the characteristic function of set {2^k, k>=0}. - Vladimir Shevelev, Sep 24 2015
EXAMPLE
60 = 2^(2^1)*3^(2^0)*5^(2^0).
MAPLE
isA000079 := proc(n)
if n = 1 then
true;
else
type(n, 'even') and nops(numtheory[factorset](n))=1 ;
simplify(%) ;
end if;
end proc:
isA138302 := proc(n)
local p;
if n = 1 then
return true;
end if;
for p in ifactors(n)[2] do
if not isA000079(op(2, p)) then
return false;
end if;
end do:
true ;
end proc:
for n from 1 to 100 do
if isA138302(n) then
printf("%d, ", n) ;
end if;
MATHEMATICA
selQ[n_] := AllTrue[FactorInteger[n][[All, 2]], IntegerQ[Log[2, #]]&];
PROG
(PARI) is(n)=if(n<8, n>0, vecmin(apply(n->n>>valuation(n, 2)==1, factor(n)[, 2]))) \\ Charles R Greathouse IV, Dec 07 2012
Biquadratefree numbers: numbers that are not divisible by any 4th power greater than 1.
+10
43
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76
COMMENTS
The Schnirelmann density of the biquadratefree numbers is 145/157 (Orr, 1969). - Amiram Eldar, Mar 12 2021 This sequence has arbitrarily large gaps and hence is not a Beatty sequence. - Charles R Greathouse IV, Jan 27 2022
FORMULA
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(4*s), for s > 1. - Amiram Eldar, Dec 27 2022
MAPLE
option remember;
local a, p, is4free;
if n = 1 then
return 1;
else
for a from procname(n-1)+1 do
is4free := true ;
for p in ifactors(a)[2] do
if op(2, p) >= 4 then
is4free := false;
break;
end if;
end do:
if is4free then
return a;
end if;
end do:
end if;
MATHEMATICA
lst={}; Do[a=0; Do[If[FactorInteger[m][[n, 2]]>4, a=1], {n, Length[FactorInteger[m]]}]; If[a!=1, AppendTo[lst, m]], {m, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *) Select[Range[100], Max[FactorInteger[#][[;; , 2]]]<4&] (* Harvey P. Dale, Jul 13 2023 *)
PROG
(Sage)
def is_biquadratefree(n):
return all(c[1] < 4 for c in n.factor())
def A046100_list(n): return [i for i in (1..n) if is_biquadratefree(i)] (Haskell)
a046100 n = a046100_list !! (n-1)
a046100_list = filter ((< 4) . a051903) [1..]
(Python)
from sympy import mobius, integer_nthroot
def f(x): return n+x-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)
Exponentially evil numbers.
+10
24
1, 8, 27, 32, 64, 125, 216, 243, 343, 512, 729, 864, 1000, 1024, 1331, 1728, 1944, 2197, 2744, 3125, 3375, 4000, 4096, 4913, 5832, 6859, 7776, 8000, 9261, 10648, 10976, 12167, 13824, 15552, 15625, 16807, 17576, 19683, 21952, 23328, 24389, 25000, 27000, 27648, 29791
COMMENTS
Or the numbers whose prime power factorization contains primes only in evil exponents ( A001969): 0, 3, 5, 6, 9, 10, 12, ... If n is in the sequence, then n^2 is also in the sequence.
FORMULA
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^ A001969(k)) = Product_{p prime} f(1/p) = 1.2413599378..., where f(x) = (1/(1-x) + Product_{k>=0} (1 - x^(2^k)))/2. - Amiram Eldar, May 18 2023, Dec 01 2023
EXAMPLE
864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.
MATHEMATICA
{1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *) expEvilQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], EvenQ[DigitCount[#, 2, 1]] &]; With[{max = 30000}, Select[Union[Flatten[Table[i^2*j^3, {j, Surd[max, 3]}, {i, Sqrt[max/j^3]}]]], expEvilQ]] (* Amiram Eldar, Dec 01 2023 *)
PROG
(PARI) isok(n) = {my(f = factor(n)); for (i=1, #f~, if (hammingweight(f[i, 2]) % 2, return (0)); ); return (1); } \\ Michel Marcus, Sep 27 2015 (Haskell)
a262675 n = a262675_list !! (n-1)
a262675_list = filter
(all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
(Perl) use ntheory ":all"; sub isok { my @f = factor_exp($_[0]); return scalar(grep { !(hammingweight($_->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015
CROSSREFS
Apart from 1, a subsequence of A270421. Sequence A270437 sorted into ascending order. Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.
Numbers whose exponents in their prime power factorization are squares.
+10
18
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101
COMMENTS
Numbers whose prime factorization has the form Product_i p_i^e_i where the e_i are all squares.
Let h_k be the density of the subsequence of A197680 of numbers whose prime power factorization (PPF) has the form Product_i p_i^e_i where the e_i all squares <= k^2. Then for every k>1 there exists eps_k>0 such that for any x from the interval (h_k-eps_k, h_k) there is no a sequence S of positive integers such that x is the density of numbers whose PPF has the form Product_i p_i^e_i where the e_i are all in S. - For a proof, see [Shevelev], second link. - Vladimir Shevelev, Nov 17 2015
FORMULA
Sum_{i<=x, i is in A197680} 1 = h*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c=4*sqrt(2.4/log 2)=7.44308... and h=Product_{prime p} (1+Sum_{i>=2} (u(i)-u(i-1))/p^i)=0.641115... where u(n) is the characteristic function of sequence A000290. The calculations of h in the formula were done independently by Juan Arias-de-Reyna and Peter J. C. Moses. For a proof of the formula, see the first Shevelev link. - Vladimir Shevelev, Nov 17 2015
MAPLE
a:= proc(n) option remember; local k; for k from 1+
`if`(n=1, 0, a(n-1)) while 0=mul(`if`(issqr(
i[2]), 1, 0), i=ifactors(k)[2]) do od; k
end:
MATHEMATICA
Select[Range[100], Union[IntegerQ /@ Sqrt[Transpose[FactorInteger[#]][[2]]]][[1]] &] (* T. D. Noe, Oct 18 2011 *)
PROG
(PARI) isok(n) = {my(f = factor(n)[, 2]); #select(x->issquare(x), f) == #f; } \\ Michel Marcus, Oct 23 2015
Numbers that are not exponentially squarefree.
+10
8
16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 256, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 768, 784, 810, 816, 848, 880, 891, 912, 944, 976
COMMENTS
A positive integer is called exponentially squarefree (e-squarefree) if in its prime power factorization all the exponents are squarefree.
a(n) is the sequence of positive integers in which prime power factorization there is at least one nonsquarefree exponent.
n is non-e-squarefree iff f(n)=0, where f(n) is the exponential Moebius function A166234.
LINKS
M. V. Subbarao, On some arithmetic convolutions, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics No. 251, 247-271, Springer, 1972, doi:10.1007/BFb0058796.
EXAMPLE
16=2^4, 48=2^4*3, 256=2^8 are non-e-squarefree, since 4 and 8 are nonsquarefree.
MAPLE
filter:=n -> not andmap(t -> numtheory:-issqrfree(t[2]), ifactors(n)[2]);
MATHEMATICA
Select[Range@ 1000, ! AllTrue[Last /@ FactorInteger@ #, SquareFreeQ] &] (* Michael De Vlieger, Sep 07 2015, Version 10 *)
PROG
(Haskell)
a130897 n = a130897_list !! (n-1)
a130897_list = filter
(any (== 0) . map (a008966 . fromIntegral) . a124010_row) [1..]
(PARI) is(n)=my(f=factor(n)[, 2]); for(i=1, #f, if(!issquarefree(f[i]), return(1))); 0 \\ Charles R Greathouse IV, Sep 03 2015
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