A046101 -id:A046101 - cp's OEIS frontend

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller, Mar 15 1996

Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]
a(n) = A128651(A129132(n-1) + 2) for n > 1. - Reinhard Zumkeller, Mar 30 2007
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length.  For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017

			G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandache-type multiplicative sequences
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
Neville Holmes, Integer Sequences [Broken link]
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
Wikipedia, Radical of an integer.

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
Range of values is A005117.
Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
Bisections: A099984, A099985.
Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
A003961, A059896 are used to express relationship between terms of this sequence.
Cf. A001615, A008683, A076479, A078615.

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
seq(A007947(i),i=1..69); # Peter Luschny, Mar 22 2011
A:= n -> convert(numtheory:-factorset(n),`*`):
seq(A(n),n=1..100); # Robert Israel, Aug 10 2014
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
From Antti Karttunen, Jun 18 2017: (Start)
a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
a(n) = A019565(A087207(n)). (End)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
From Peter Munn, Jan 01 2020: (Start)
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A003961(n)) = A003961(a(n)).
a(n^2) = a(n).
a(A225546(n)) = A019565(A267116(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1

Offset: 1

N. J. A. Sloane

Equivalently, number of Abelian groups with n conjugacy classes. - Michael Somos, Aug 10 2010
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
Also number of rings with n elements that are the direct product of fields; these are the commutative rings with n elements having no nilpotents; likewise the commutative rings where for every element x there is a k > 0 such that x^(k+1) = x. - Franklin T. Adams-Watters, Oct 20 2006
Range is A033637.
a(n) = 1 if and only if n is from A005117 (squarefree numbers). See the Ahmed Fares comment there, and the formula for n>=2 below. - Wolfdieter Lang, Sep 09 2012
Also, from a theorem of Molnár (see [Molnár]), the number of (non-isomorphic) abelian groups of order 2*n + 1 is equal to the number of non-congruent lattice Z-tilings of R^n by crosses, where a "cross" is a unit cube in R^n for which at each facet is attached another unit cube (Z, R are the integers and reals, respectively). (Cf. [Horak].) - L. Edson Jeffery, Nov 29 2012
Zeta(k*s) is the Dirichlet generating function of the characteristic function of numbers which are k-th powers (k=1 in A000012, k=2 in A010052, k=3 in A010057, see arXiv:1106.4038 Section 3.1). The infinite product over k (here) is the number of representations n=product_i (b_i)^(e_i) where all exponents e_i are distinct and >=1. Examples: a(n=4)=2: 4^1 = 2^2. a(n=8)=3: 8^1 = 2^1*2^2 = 2^3. a(n=9)=2: 9^1 = 3^2. a(n=12)=2: 12^1 = 3*2^2. a(n=16)=5: 16^1 = 2*2^3 = 4^2 = 2^2*4^1 = 2^4. If the e_i are the set {1,2} we get A046951, the number of representations as a product of a number and a square. - R. J. Mathar, Nov 05 2016
See A060689 for the number of non-abelian groups of order n. - M. F. Hasler, Oct 24 2017
Kendall & Rankin prove that the density of {n: a(n) = m} exists for each m. - Charles R Greathouse IV, Jul 14 2024

			a(1) = 1 since the trivial group {e} is the only group of order 1, and it is Abelian; alternatively, since the only factorization of 1 into prime powers is the empty product.
a(p) = 1 for any prime p, since the only factorization into prime powers is p = p^1, and (in view of Lagrange's theorem) there is only one group of prime order p; it is isomorphic to (Z/pZ,+) and thus Abelian.
From _Wolfdieter Lang_, Jul 22 2011: (Start)
a(8) = 3 because 8 = 2^3, hence a(8) = pa(3) = A000041(3) = 3 from the partitions (3), (2, 1) and (1, 1, 1), leading to the 3 factorizations of 8: 8, 4*2 and 2*2*2.
a(36) = 4 because 36 = 2^2*3^2, hence a(36) = pa(2)*pa(2) = 4 from the partitions (2) and (1, 1), leading to the 4 factorizations of 36: 2^2*3^2, 2^2*3^1*3^1, 2^1*2^1*3^2 and 2^1*2^1*3^1*3^1.
(End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Speiser, Die Theorie der Gruppen von endlicher Ordnung, 4. Auflage, Birkhäuser, 1956.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tak-Shing T. Chan, and Y.-H. Yang, Polar n-Complex and n-Bicomplex Singular Value Decomposition and Principal Component Pursuit, IEEE Transactions on Signal Processing ( Volume: 64, Issue: 24, Dec.15, 15 2016 ); DOI: 10.1109/TSP.2016.2612171.
I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34.
I. G. Connell, Letter to N. J. A. Sloane, no date
P. Erdős and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
Steven R. Finch, Abelian Group Enumeration Constants [Broken link]
Steven R. Finch, Abelian Group Enumeration Constants [broken link?] [From the Wayback machine]
P. Horak, Error-correcting codes and Minkowski's conjecture, Tatra Mt. Math. Publ., 45 (2010), p. 40.
B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.
D. G. Kendall, R. A. Rankin, On the number of Abelian groups of a given order, Q. J. Math. 18 (1947) 197-208.
Nobushige Kurokawa and Masato Wakayama, Zeta extensions. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 126--130. MR1930216 (2003h:11112).
E. Molnár, Sui mosaici dello spazio di dimensione n, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 51 (1971), 177-185.
H.-E. Richert, Über die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr. 56 (1952) 21-32.
Marko Riedel, Counting Abelian Groups, Mathematics Stack Exchange, October 2014.
Laszlo Toth, A note on the number of abelian groups of a given order, arXiv:1203.6473 [math.NT], (2012).
Eric Weisstein's World of Mathematics, Abelian Group
Eric Weisstein's World of Mathematics, Finite Group
Eric Weisstein's World of Mathematics, Kronecker Decomposition Theorem
Index entries for sequences related to groups
Index entries for "core" sequences

Cf. A000001, A021002, A060689, A000041, A000961, A001055, A005361, A034382, A046054, A046055, A046056, A046101, A050360, A055653, A057521, A101872 (bisection), A101876 (quadrisection), A124010, A050361, A051532, A129667 (Dirichlet inverse), A188585.

Cf. A080729 (Dgf at s=2), A369634 (Dgf at s=3).

a000688 = product . map a000041 . a124010_row
-- Reinhard Zumkeller, Aug 28 2014

with(combinat): readlib(ifactors): for n from 1 to 120 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,ans): od: # James Sellers, Dec 07 2000

f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* Robert G. Wilson v, Sep 22 2006 *)
Table[FiniteAbelianGroupCount[n], {n, 200}] (* Requires version 7.0 or later. - Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)

A000688(n)=local(f);f=factor(n);prod(i=1,matsize(f)[1],numbpart(f[i,2])) \\ Michael B. Porter, Feb 08 2010

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,numbpart(f[i])) \\ Charles R Greathouse IV, Apr 16 2015

from sympy import factorint, npartitions
from math import prod
def A000688(n): return prod(map(npartitions,factorint(n).values())) # Chai Wah Wu, Jan 14 2022

def a(n):
    F=factor(n)
    return prod([number_of_partitions(F[i][1]) for i in range(len(F))])
# Ralf Stephan, Jun 21 2014

Multiplicative with a(p^k) = number of partitions of k = A000041(k); a(mn) = a(m)a(n) if (m, n) = 1.
a(2n) = A101872(n).
a(n) = Product_{j = 1..N(n)} A000041(e(j)), n >= 2, if
  n  = Product_{j = 1..N(n)} prime(j)^e(j), N(n) = A001221(n). See the Richert reference, quoting A. Speiser's book on finite groups (in German, p. 51 in words). - Wolfdieter Lang, Jul 23 2011
In terms of the cycle index of the symmetric group: Product_{q=1..m} [z^{v_q}] Z(S_v) 1/(1-z) where v is the maximum exponent of any prime in the prime factorization of n, v_q are the exponents of the prime factors, and Z(S_v) is the cycle index of the symmetric group on v elements. - Marko Riedel, Oct 03 2014
Dirichlet g.f.: Sum_{n >= 1} a(n)/n^s = Product_{k >= 1} zeta(ks) [Kendall]. - Álvar Ibeas, Nov 05 2014
a(n)=2 for all n in A054753 and for all n in A085987.  a(n)=3 for all n in A030078 and for all n in A065036.  a(n)=4 for all n in A085986.  a(n)=5 for all n in A030514 and for all n in A178739.  a(n)=6 for all n in A143610. - R. J. Mathar, Nov 05 2016
A050360(n) = a(A025487(n)). a(n) = A050360(A101296(n)). - R. J. Mathar, May 26 2017
a(n) = A000001(n) - A060689(n). - M. F. Hasler, Oct 24 2017
From Amiram Eldar, Nov 01 2020: (Start)
a(n) = a(A057521(n)).
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A021002. (End)
a(n) = A005361(n) except when n is a term of A046101, since A000041(x) = x for x <= 3. - Miles Englezou, Feb 17 2024
Inverse Moebius transform of A188585: a(n) = Sum_{d|n} A188585(d). - Amiram Eldar, Jun 10 2025

A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336

Offset: 1

Eric W. Weisstein

nonn

Also called cubeful numbers, but this term is ambiguous and is best avoided.
Numbers n such that A007427(n) = sum(d|n,mu(d)*mu(n/d)) == 0. - Benoit Cloitre, Apr 17 2002
The convention in the OEIS is that squareful, cubeful, biquadrateful (A046101), ... 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. Added by N. J. A. Sloane, Apr 25 2023:  This suggestion has not been a success. It is hopeless to try to make a distinction between "cubeful" and "cubefull". To avoid ambiguity, do not use either term, but instead say exactly what you mean.
Also solutions to equation tau_{-2}(n)=0, where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Amiram Eldar, Jul 09 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cubefree.

Complement of A004709.
Subsequences: A000578 and A030078.
Cf. A001694, A036966, A046101, A088453.

a046099 n = a046099_list !! (n-1)
a046099_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012

isA046099 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) >= 3 then
            return true;
        end if;
    end do:
    false ;
end proc:
for n from 1 do
    if isA046099(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015

lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>2, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 15 2008 *)

is(n)=n>7 && vecmax(factor(n)[,2])>2 \\ Charles R Greathouse IV, Sep 17 2015

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) != n
print(list(filter(ok, range(1, 337)))) # Michael S. Branicky, Aug 16 2021

from sympy import mobius, integer_nthroot
def A046099(n):
    def f(x): return n+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)
    return m # Chai Wah Wu, Aug 05 2024

A212793(a(n)) = 0. - Reinhard Zumkeller, May 27 2012

Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(3*s)-1))/zeta(3*s). - Amiram Eldar, Dec 27 2022

More terms from Vladimir Joseph Stephan Orlovsky, Aug 15 2008

Edited by N. J. A. Sloane, Jul 27 2009

A046100 Biquadratefree numbers: numbers that are not divisible by any 4th power greater than 1.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

nonn

Differs from A023809 at entries 0, 81, 162, 225, 226, etc. - R. J. Mathar, Oct 18 2008
Density is 1/zeta(4) = A215267 = 0.923938.... - Charles R Greathouse IV, Sep 02 2015
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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.
Eric Weisstein's World of Mathematics, Biquadratefree.

Cf. A046101, A005117 (2-free), A004709 (3-free).
Cf. A023809, A051903, A215267.
Subsequence of A209061.

a046100 n = a046100_list !! (n-1)
a046100_list = filter ((< 4) . a051903) [1..]
-- Reinhard Zumkeller, Sep 03 2015

A046100 := proc(n)
    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;
end proc: # R. J. Mathar, Aug 08 2012

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 *)

is(n)=n==1 || vecmax(factor(n)[,2])<4 \\ Charles R Greathouse IV, Jun 16 2012

from sympy import mobius, integer_nthroot
def A046100(n):
    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)
    return m # Chai Wah Wu, Aug 05 2024

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)]
A046100_list(76) # Peter Luschny, Aug 08 2012

A051903(a(n)) < 4. - Reinhard Zumkeller, Sep 03 2015

Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(4*s), for s > 1. - Amiram Eldar, Dec 27 2022

Name edited by Amiram Eldar, Jul 29 2024

A036967 4-full numbers: if a prime p divides k then so does p^4.

Original entry on oeis.org

1, 16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 1296, 2048, 2187, 2401, 2592, 3125, 3888, 4096, 5184, 6561, 7776, 8192, 10000, 10368, 11664, 14641, 15552, 15625, 16384, 16807, 19683, 20000, 20736, 23328, 28561, 31104, 32768, 34992

Offset: 1

N. J. A. Sloane

a(m) mod prime(n) > 0 for m < A258601(n); a(A258601(n)) = A030514(n) = prime(n)^4. - Reinhard Zumkeller, Jun 06 2015

E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 300 terms from T. D. Noe)

A030514 is a subsequence.

Cf. A001694, A036966, A046101, A258601.

import Data.Set (singleton, deleteFindMin, fromList, union)
a036967 n = a036967_list !! (n-1)
a036967_list = 1 : f (singleton z) [1, z] zs where
   f s q4s p4s'@(p4:p4s)
     | m < p4 = m : f (union (fromList $ map (* m) ps) s') q4s p4s'
     | otherwise = f (union (fromList $ map (* p4) q4s) s) (p4:q4s) p4s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030514_list
-- Reinhard Zumkeller, Jun 03 2015

Join[{1},Select[Range[35000],Min[Transpose[FactorInteger[#]][[2]]]>3&]] (* Harvey P. Dale, Jun 05 2012 *)

is(n)=n==1 || vecmin(factor(n)[,2])>3 \\ Charles R Greathouse IV, Sep 17 2015

M(v,u,lim)=vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))
Gen(lim,k)={my(v=[1]); forprime(p=2, sqrtnint(lim, k), v=M(v, concat([1], vector(logint(lim,p)-k+1,e,p^(e+k-1))), lim));v}
Gen(35000,4) \\ Andrew Howroyd, Sep 10 2024

from sympy import factorint
A036967_list = [n for n in range(1,10**5) if min(factorint(n).values(),default=4) >= 4] # Chai Wah Wu, Aug 18 2021

from math import gcd
from sympy import integer_nthroot, factorint
def A036967(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for u in range(1,integer_nthroot(x,7)[0]+1):
            if all(d<=1 for d in factorint(u).values()):
                for w in range(1,integer_nthroot(a:=x//u**7,6)[0]+1):
                    if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
                        for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
                            if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
                                c -= integer_nthroot(z//y**5,4)[0]
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.1488462139214317030108176090790939019972506733993367867997411290952527... - Amiram Eldar, Jul 09 2020

More terms from Erich Friedman

Corrected by Vladeta Jovovic, Aug 17 2002

A007428 Moebius transform applied thrice to sequence 1,0,0,0,....

Original entry on oeis.org

1, -3, -3, 3, -3, 9, -3, -1, 3, 9, -3, -9, -3, 9, 9, 0, -3, -9, -3, -9, 9, 9, -3, 3, 3, 9, -1, -9, -3, -27, -3, 0, 9, 9, 9, 9, -3, 9, 9, 3, -3, -27, -3, -9, -9, 9, -3, 0, 3, -9, 9, -9, -3, 3, 9, 3, 9, 9, -3, 27, -3, 9, -9, 0, 9, -27, -3, -9, 9, -27, -3, -3, -3, 9, -9, -9, 9, -27

Offset: 1

N. J. A. Sloane

Dirichlet inverse of A007425. - R. J. Mathar, Jul 15 2010

abs(a(n)) is the number of ways to write n=xyz where x,y,z are squarefree numbers. - Benoit Cloitre, Jan 02 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Consecutive nested Dirichlet convolution: A063524, A008683 or A007427. - Enrique Pérez Herrero, Jul 12 2010

Cf. A124010.

a007428 n = product
   [a007318' 3 e * cycle [1,-1] !! fromIntegral e | e <- a124010_row n]
-- Reinhard Zumkeller, Oct 09 2013

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:
(möbius@@3)([1, seq(0, i=1..77)]); # Peter Luschny, Sep 08 2017

tau[1,n_Integer]:=1; SetAttributes[tau, Listable];
tau[k_Integer,n_Integer]:=Plus@@(tau[k-1,Divisors[n]])/; k > 1;
tau[k_Integer,n_Integer]:=Plus@@(tau[k+1,Divisors[n]]*MoebiusMu[n/Divisors[n]]); k<1;
A007428[n_]:=tau[ -3,n]; (* Enrique Pérez Herrero, Jul 12 2010 *)
a[n_] := Which[n==1, 1, PrimeQ[n], -3, True, Times @@ Map[Function[e, Binomial[3, e] (-1)^e], FactorInteger[n][[All, 2]]]];
Array[a, 100] (* Jean-François Alcover, Jun 20 2018 *)

a(n) = {my(f=factor(n)); for (k=1, #f~, e = f[k,2]; f[k,1] = binomial(3, e)*(-1)^e; f[k,2] = 1); factorback(f);} \\ Michel Marcus, Jan 03 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^3)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Multiplicative with a(p^e) = (3 choose e) (-1)^e.
Dirichlet g.f.: 1/zeta(s)^3.
From Enrique Pérez Herrero, Jul 12 2010: (Start)
a(n^3) = A008683(n).
a(s) = (-3)^A001221(s) provided s is a squarefree number (A005117). (End)
a(A046101(n)) = 0. - Enrique Pérez Herrero, Sep 07 2017
a(n) = Sum_{a*b*c=n} mu(a)*mu(b)*mu(c). - Benedict W. J. Irwin, Mar 02 2022

A130897 Numbers that are not exponentially squarefree.

Original entry on oeis.org

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

Offset: 1

Laszlo Toth, Mar 18 2011

nonn

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.
Product_{k = 1..A001221(n)} A008966(A124010(n,k)) = 0. - Reinhard Zumkeller, Mar 13 2012
The density of {a(n)} is 0.04407699... (see comment in A209061). - Peter J. C. Moses and Vladimir Shevelev, Sep 08 2015

			16=2^4, 48=2^4*3, 256=2^8 are non-e-squarefree, since 4 and 8 are nonsquarefree.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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.
Laszlo Toth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166.

Complement of A209061; subsequence of A013929, A046099, and A046101.

Cf. A005117, A166234, A049419, A051377, A008966, A124010.

a130897 n = a130897_list !! (n-1)
a130897_list = filter
   (any (== 0) . map (a008966 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Mar 13 2012

filter:=n ->  not andmap(t -> numtheory:-issqrfree(t[2]), ifactors(n)[2]);
select(filter, [$1..1000]); # Robert Israel, Sep 03 2015

Select[Range@ 1000, ! AllTrue[Last /@ FactorInteger@ #, SquareFreeQ] &] (* Michael De Vlieger, Sep 07 2015, Version 10 *)

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

A068782 Lesser of two consecutive numbers each divisible by a fourth power.

Original entry on oeis.org

80, 624, 1215, 1376, 2400, 2511, 2672, 3807, 3968, 4374, 5103, 5264, 6399, 6560, 7695, 7856, 8991, 9152, 9375, 10287, 10448, 10624, 11583, 11744, 12879, 13040, 14175, 14336, 14640, 15471, 15632, 16767, 16928, 18063, 18224, 19359, 19375

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

The asymptotic density of this sequence is 1 - 2/zeta(4) + Product_{p prime} (1 - 2/p^4) = 0.001856185541538432217... - Amiram Eldar, Feb 16 2021

Below 9508685764, it suffices to check for n such that either n or n+1 is divisible by p^4 for some p <= 19. - Charles R Greathouse IV, Jul 17 2024

			80 is a term as 80 and 81 both are divisible by a fourth power, 2^4 and 3^4 respectively.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A068140.

Cf. A068780, A068781, A068783, A068784, A068785, A046101, A215267.

Select[ Range[2, 25000], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 3 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 3 &]

has(n)=vecmax(factor(n)[,2])>3
is(n)=has(n+1)&&has(n) \\ Charles R Greathouse IV, Dec 19 2018

list(lim)=my(v=List(),x=1); forfactored(n=81,lim\1+1, if(vecmax(n[2][,2])>3, if(x,listput(v,n[1]-1),x=1),x=0)); Vec(v) \\ Charles R Greathouse IV, Dec 19 2018

a(0) = 0 removed by Charles R Greathouse IV, Dec 19 2018

A086709 Primes p such that p-1 and p+1 are both divisible by fourth powers.

Original entry on oeis.org

1249, 2753, 3727, 4049, 4801, 5023, 7937, 10529, 11503, 12799, 13121, 15391, 20897, 21871, 22193, 23167, 25759, 28351, 28751, 31249, 32561, 33857, 35153, 37423, 39041, 42929, 46817, 47791, 48751, 49409, 50383, 51679, 55889, 58481, 62047

Offset: 1

Amarnath Murthy, Jason Earls, Jul 28 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Vincenzo Librandi)

Cf. A046101, A075432, A086708, A089212.

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p-1]>=4&&f[p+1]>=4,AppendTo[lst,p]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
dfpQ[n_]:=Max[Transpose[FactorInteger[n]][[2]]]>3; Select[Prime[Range[ 6500]], dfpQ[#-1]&&dfpQ[#+1]&] (* Harvey P. Dale, May 11 2012 *)

A365171 The number of divisors d of n such that gcd(d, n/d) is a square.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Aug 25 2023

The sum of these divisors is A365172(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A000290, A000583, A004524, A005117, A008833, A034444, A046101, A046951, A182448, A365172, A365173.

f[p_, e_] := Floor[(e + 3)/4] + Floor[(e + 4)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+3)\4 + (x+4)\4, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X)^2 * (1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Multiplicative with a(p^e) = floor((e + 3)/4) + floor((e + 4)/4) = A004524(e+3).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034444(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) == 1 (mod 2) if and only if n is a fourth power (A000583).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s) + 1)).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s) + 1)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/(p^2 + 1)) = Pi^2/15 = A182448,
f'(1) = f(1) * Sum_{p prime} 2*log(p) / (p^2 + 1) = f(1) * 0.8852429263675811068149340172820329246145172848406469350087715037483367369...
and gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.

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A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.

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A046100 Biquadratefree numbers: numbers that are not divisible by any 4th power greater than 1.

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A036967 4-full numbers: if a prime p divides k then so does p^4.

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