A005361 -id:A005361 - cp's OEIS frontend

A051903 Maximum exponent in the prime factorization of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Labos Elemer, Dec 16 1999

Smallest number of factors of all factorizations of n into squarefree numbers, see also A128651, A001055. - Reinhard Zumkeller, Mar 30 2007
Maximum number of invariant factors among abelian groups of order n. - Álvar Ibeas, Nov 01 2014
a(n) is the highest of the frequencies of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1..r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24) = 3; indeed, the partition having Heinz number 24 = 2*2*2*3 is [1,1,1,2], where the distinct parts 1 and 2 have frequencies 3 and 1, respectively. - Emeric Deutsch, Jun 04 2015
From Thomas Ordowski, Dec 02 2019: (Start)
a(n) is the smallest k such that b^(phi(n)+k) == b^k (mod n) for all b.
The Euler phi function can be replaced by the Carmichael lambda function.
Problems:
(*) Are there composite numbers n > 4 such that n == a(n) (mod phi(n))? By Lehmer's totient conjecture, there are no such squarefree numbers.
(**) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod lambda(n))? These are odd numbers n such that a(n) > 1 and b^n == b^a(n) (mod n) for all b.
(***) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod ord_{n}(2))? These are odd numbers n such that a(n) > 1 and 2^n == 2^a(n) (mod n).
Note: if (***) do not exist, then (**) do not exist. (End)
Niven (1969) proved that the asymptotic mean of this sequence is 1 + Sum_{j>=2} 1 - (1/zeta(j)) (A033150). - Amiram Eldar, Jul 10 2020

			For n = 72 = 2^3*3^2, a(72) = max(exponents) = max(3,2) = 3.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Benjamin Merlin Bumpus and Zoltan A. Kocsis, Spined categories: generalizing tree-width beyond graphs, arXiv:2104.01841 [math.CO], 2021.
Cao Hui-Zhong, The Asymptotic Formulas Related to Exponents in Factoring Integers, Math. Balkanica, Vol. 5 (1991), Fasc. 2.
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant.
Index entries for sequences computed from exponents in factorization of n

Average value is A033150 = 1.7052....

Cf. A002322, A005361, A008479, A028234, A051904, A052409, A067029, A091050, A129132, A327295, A328310, A329885.

a051903 1 = 0
a051903 n = maximum $ a124010_row n -- Reinhard Zumkeller, May 27 2012

A051903 := proc(n)
        a := 0 ;
        for f in ifactors(n)[2] do
                a := max(a,op(2,f)) ;
        end do:
        a ;
end proc: # R. J. Mathar, Apr 03 2012
# second Maple program:
a:= n-> max(0, seq(i[2], i=ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, May 09 2020

Table[If[n == 1, 0, Max @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=if(n>1,vecmax(factor(n)[,2]),0) \\ Charles R Greathouse IV, Oct 30 2012

from sympy import factorint
def A051903(n):
    return max(factorint(n).values()) if n > 1 else 0
# Chai Wah Wu, Jan 03 2015

;; With memoization-macro definec.
(definec (A051903 n) (if (= 1 n) 0 (max (A067029 n) (A051903 (A028234 n))))) ;; Antti Karttunen, Aug 08 2016

a(n) = max_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(1) = 0; for n > 1, a(n) = max(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 08 2016
Conjecture: a(n) = a(A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - Velin Yanev, Sep 02 2017
Comment from David J. Seal, Sep 18 2017: (Start)
This conjecture seems very easily provable to me: if the factorization of n is p1^k1 * p2^k2 * ... * pm^km, then the factorization of the largest squarefree divisor of n is p1 * p2 * ... * pm. So the factorization of A003557(n) is p1^(k1-1) * p2^(k2-1) * ... * pm^(km-1) if exponents of zero are allowed, or with the product terms that have an exponent of zero removed if they're not (if that results in an empty product, consider it to be 1 as usual).
The formula then follows from the fact that provided all ki >= 1, Max(k1, k2, ..., km) = Max(k1-1, k2-1, ..., km-1) + 1, and Max(k1-1, k2-1, ..., km-1) is not altered by removing the ki-1 values that are 0, provided we treat the empty Max() as being 0. That proves the formula and the provisos about empty products and Max() correspond to a(1) = 0.
Also, for any n, applying the formula Max(k1, k2, ..., km) times to n = p1^k1 * p2^k2 * ... * pm^km reduces all the exponents to zero, i.e., to the case a(1) = 0, so that case and the formula generate the sequence. (End)
Sum_{k=1..n} (-1)^k * a(k) ~ c * n, where c = Sum_{k>=2} 1/((2^k-1)*zeta(k)) = 0.44541445377638761933... . - Amiram Eldar, Jul 28 2024
a(n) <= log(n)/log(2). - Hal M. Switkay, Jul 03 2025

A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

If n = Product p_i^a_i, d = Product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.
Also the number of squarefree divisors of n. - Labos Elemer
Also number of divisors of the squarefree kernel of n: a(n) = A000005(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
Also shadow transform of pronic numbers A002378.
For n >= 1 define an n X n (0,1) matrix A by A[i,j] = 1 if lcm(i,j) = n, A[i,j] = 0 if lcm(i,j) <> n for 1 <= i,j <= n. a(n) is the rank of A. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 11 2003
a(n) is also the number of solutions to x^2 - x == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
a(n) is the number of squarefree divisors of n, but in general the set of unitary divisors of n is not the set of squarefree divisors (compare the rows of A077610 and A206778). - Jaroslav Krizek, May 04 2009
Row lengths of the triangles in A077610 and in A206778. - Reinhard Zumkeller, Feb 12 2012
a(n) is also the number of distinct residues of k^phi(n) (mod n), k=0..n-1. - Michel Lagneau, Nov 15 2012
a(n) is the number of irreducible fractions y/x that satisfy x*y=n (and gcd(x,y)=1), x and y positive integers. - Luc Rousseau, Jul 09 2017
a(n) is the number of (x,y) lattice points satisfying both x*y=n and (x,y) is visible from (0,0), x and y positive integers. - Luc Rousseau, Jul 10 2017
Conjecture: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the unitary divisors of n is divisible by the sum of the k-th powers of the odd unitary divisors of n (note that this sequence lists the sum of the 0th powers of the unitary divisors of n). - Ivan N. Ianakiev, Feb 18 2018
a(n) is the number of one-digit numbers, k, when written in base n such that k and k^2 end in the same digit. - Matthew Scroggs, Jun 01 2018
Dirichlet convolution of A271102 and A000005. - Vaclav Kotesovec, Apr 08 2019
Conjecture: Let b(i; n), n > 0, be multiplicative sequences for some fixed integer i >= 0 with b(i; p^e) = (Sum_{k=1..i+1} A164652(i, k) * e^(k-1)) * (i+2) / (i!) for prime p and e > 0. Then we have Dirichlet generating functions: Sum_{n > 0} b(i; n) / n^s = (zeta(s))^(i+2) / zeta((i+2) * s). Examples for i=0 this sequence, for i=1 A226602, and for i=2 A286779. - Werner Schulte, Feb 17 2022
The smallest integer with 2^m unitary divisors, or equivalently, the smallest integer with 2^m squarefree divisors, is A002110(m). - Bernard Schott, Oct 04 2022

			a(12) = 4 because the four unitary divisors of 12 are 1, 3, 4, 12, and also because the four squarefree divisors of 12 are 1, 2, 3, 6.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.

T. D. Noe, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 49-50.
Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
Jon Maiga, Upper bound of Fibonacci entry points, 2019.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.
Index entries for sequences computed from exponents in factorization of n

Cf. A077610, A048105, A000173, A013928, A000079, A001221, A002110, A034448, A047994, A061142, A277561.
Sum of the k-th powers of the squarefree divisors of n for k=0..10: this sequence (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: this sequence (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), this sequence (k=10).
Cf. A020821 (Dgf at s=2), A177057 (Dgf at s=4).

a034444 = length . a077610_row  -- Reinhard Zumkeller, Feb 12 2012

[#[d:d in Divisors(n)|Gcd(d,n div d) eq 1]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

[&+[Abs(MoebiusMu(d)):d in Divisors(n)]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

with(numtheory): for n from 1 to 200 do printf(`%d,`,2^nops(ifactors(n)[2])) od:
with(numtheory);
# returns the number of unitary divisors of n and a list of them
f:=proc(n)
local ct,i,t1,ans;
ct:=0; ans:=[];
t1:=divisors(n);
for i from 1 to nops(t1) do
d:=t1[i];
if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi;
od:
RETURN([ct,ans]);
end;
# N. J. A. Sloane, May 01 2013
# alternative Maple program:
a:= n-> 2^nops(ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Jan 23 2024
a := n -> 2^NumberTheory:-NumberOfPrimeFactors(n, distinct):  # Peter Luschny, May 13 2025

a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* Jean-François Alcover, Apr 05 2011 *)
Table[2^PrimeNu[n],{n,110}] (* Harvey P. Dale, Jul 14 2011 *)

a(n)=1<Charles R Greathouse IV, Feb 11 2011

for(n=1, 100, print1(direuler(p=2, n, (1+X)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

from sympy import divisors, gcd
def a(n):
    return sum(1 for d in divisors(n) if gcd(d, n//d)==1)
# Indranil Ghosh, Apr 16 2017

from sympy import primefactors
def a(n): return 2**len(primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 16 2017

(define (A034444 n) (if (= 1 n) n (* 2 (A034444 (A028234 n))))) ;; Antti Karttunen, May 29 2017

a(n) = Sum_{d|n} abs(mu(n)) = 2^(number of different primes dividing n) = 2^A001221(n), with mu(n) = A008683(n). [Added Möbius formula. - Wolfdieter Lang, Jan 11 2020]
a(n) = Product_{ primes p|n } (1 + Legendre(1, p)).
Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley, Oct 25 2001
a(n) = Sum_{d|n} tau(d^2)*mu(n/d), Dirichlet convolution of A048691 and A008683. - Benoit Cloitre, Oct 03 2002
Dirichlet generating function: zeta(s)^2/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
Inverse Mobius transform of A008966. - Franklin T. Adams-Watters, Sep 11 2005
Asymptotically [Finch] the cumulative sum of a(n) = Sum_{n=1..N} a(n) ~ (6/(Pi^2))*N*log(N) + (6/(Pi^2))*(2*gamma - 1 - (12/(Pi^2))*zeta'(2))*N + O(sqrt(N)). - Jonathan Vos Post, May 08 2005 [typo corrected by Vaclav Kotesovec, Sep 13 2018]
a(n) = Sum_{d|n} floor(rad(d)/d), where rad is A007947 and floor(rad(n)/n) = A008966(n). - Enrique Pérez Herrero, Nov 13 2009
a(n) = A000005(n) - A048105(n); number of nonzero terms in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013
G.f.: Sum_{n>0} A008966(n)*x^n/(1-x^n). - Mircea Merca, Feb 25 2014
a(n) = Sum_{d|n} lambda(d)*mu(d), where lambda is A008836. - Enrique Pérez Herrero, Apr 27 2014
a(n) = A277561(A156552(n)). - Antti Karttunen, May 29 2017
a(n) = A005361(n^2)/A005361(n). - Velin Yanev, Jul 26 2017
L.g.f.: -log(Product_{k>=1} (1 - mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A001615(d) * A023900(n/d). - Torlach Rush, Jan 20 2020
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = 1. - Amiram Eldar, May 29 2020
a(n) = lim_{k->oo} A000005(n^(2*k))/A000005(n^k). - Velin Yanev and Amiram Eldar, Jan 10 2025

More terms from James Sellers, Jun 20 2000

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

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520

Offset: 1

Paul Boddington, Jul 21 2005

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)
Index to divisibility sequences.
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to primorial numbers.

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

Original entry on oeis.org

0, 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, 5, 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, 6, 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

Eric W. Weisstein

nonn

Greatest common divisor of all prime-exponents in canonical factorization of n for n>1: a(n)>1 iff n is a perfect power; a(A001597(k))=A025479(k). - Reinhard Zumkeller, Oct 13 2002
a(1) set to 0 since there is no largest finite integer power m for which a representation of the form 1 = 1^m exists (infinite largest m). - Daniel Forgues, Mar 06 2009
A052410(n)^a(n) = n. - Reinhard Zumkeller, Apr 06 2014
Positions of 1's are A007916. Smallest base is given by A052410. - Gus Wiseman, Jun 09 2020

			n = 72 = 2*2*2*3*3: GCD[exponents] = GCD[3,2] = 1. This is the least n for which a(n) <> A051904(n), the minimum of exponents.
For n = 10800 = 2^4 * 3^3 * 5^2, GCD[4,3,2] = 1, thus a(10800) = 1.

Daniel Forgues, Table of n, a(n) for n = 1..100000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
Eric Weisstein's World of Mathematics, Power
Eric Weisstein's World of Mathematics, Perfect Power
Index entries for sequences computed from exponents in factorization of n

Cf. A052410, A005361, A051903, A072411-A072414, A124010, A075802, A072776, A270492.
Apart from the initial term essentially the same as A253641.
Differs from A051904 for the first time at n=72, where a(72) = 1, while A051904(72) = 2.
Differs from A158378 for the first time at n=10800, where a(10800) = 1, while A158378(10800) = 2.
Cf. A000005, A000961, A001597, A052410, A303386, A327501.

a052409 1 = 0
a052409 n = foldr1 gcd $ a124010_row n
-- Reinhard Zumkeller, May 26 2012

# See link.
#
a:= n-> igcd(map(i-> i[2], ifactors(n)[2])[]):
seq(a(n), n=1..120);  # Alois P. Heinz, Oct 20 2019

Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=my(k=ispower(n)); if(k, k, n>1) \\ Charles R Greathouse IV, Oct 30 2012

from math import gcd
from sympy import factorint
def A052409(n): return gcd(*factorint(n).values()) # Chai Wah Wu, Aug 31 2022

(define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; Antti Karttunen, Aug 07 2017

a(1) = 0; for n > 1, a(n) = gcd(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 07 2017

More terms from Labos Elemer, Jun 17 2002

A051904 Minimal exponent in prime factorization of n.

Original entry on oeis.org

0, 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, 5, 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, 6, 1, 1, 1, 1, 1, 1, 1, 2, 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

Labos Elemer, Dec 16 1999

The asymptotic mean of this sequence is 1 (Niven, 1969). - Amiram Eldar, Jul 10 2020

Let k = A007947(n), then for n > 1 k^a(n) is the greatest power of k which divides n; see example. - David James Sycamore, Sep 07 2023

			For n = 72 = 2^3*3^2, a(72) = min(exponents) = min(3,2) = 2.
For n = 72, using alternative definition: rad(72) = 6; and 6^2 = 36 divides 72 but no higher power of 6 divides 72, so a(72) = 2.
For n = 432, rad(432) = 6 and 6^3 = 216 divides 432 but no higher power of 6 divides 432, therefore a(432) = 3. - _David James Sycamore_, Sep 08 2023

Daniel Forgues, Table of n, a(n) for n = 1..100000
Cao Hui-Zhong, The Asymptotic Formulas Related to Exponents in Factoring Integers, Math. Balkanica, Vol. 5 (1991), Fasc. 2.
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A001221, A005361, A008479, A051903, A052409, A076558, A124010, A362974.

Cf. A007947.

a051904 1 = 0
a051904 n = minimum $ a124010_row n  -- Reinhard Zumkeller, Jul 15 2012

a := proc (n) if n = 1 then 0 else min(seq(op(2, op(j, op(2, ifactors(n)))), j = 1 .. nops(op(2, ifactors(n))))) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, May 20 2015

Table[If[n == 1, 0, Min @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=vecmin(factor(n)[,2]) \\ Charles R Greathouse IV, Nov 19 2012

from sympy import factorint
def a(n):
    f = factorint(n)
    l = [f[p] for p in f]
    return 0 if n == 1 else min(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 13 2017

(define (A051904 n) (cond ((= 1 n) 0) ((= 1 (A001221 n)) (A001222 n)) (else (min (A067029 n) (A051904 (A028234 n)))))) ;; Antti Karttunen, Jul 12 2017

a(n) = min_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(1) = 0, for n > 1, if A001221(n) = 1 (when n is in A000961), a(n) = A001222(n), otherwise a(n) = min(A067029(n), a(A028234(n))). - Antti Karttunen, Jul 12 2017
Sum_{k=1..n} a(k) ~ n + zeta(3/2)*n^(1/2)/zeta(3) + (zeta(2/3)/zeta(2) + c0)*n^(1/3), where c0 = A362974 = Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)) [Cao Hui-Zhong, 1991]. - Vaclav Kotesovec, Mar 24 2025

A260443 Prime factorization representation of Stern polynomials: a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = a(n)*a(n+1).

Original entry on oeis.org

1, 2, 3, 6, 5, 18, 15, 30, 7, 90, 75, 270, 35, 450, 105, 210, 11, 630, 525, 6750, 245, 20250, 2625, 9450, 77, 15750, 3675, 47250, 385, 22050, 1155, 2310, 13, 6930, 5775, 330750, 2695, 3543750, 128625, 1653750, 847, 4961250, 643125, 53156250, 18865, 24806250, 202125, 727650, 143, 1212750, 282975, 57881250, 29645, 173643750, 1414875, 18191250, 1001

Offset: 0

Antti Karttunen, Jul 28 2015

The exponents in the prime factorization of term a(n) give the coefficients of the n-th Stern polynomial. See A125184 and the examples.
None of the terms have prime gaps in their factorization, i.e., all can be found in A073491.
Contains neither perfect squares nor prime powers with exponent > 1. A277701 gives the positions of the terms that are 2*square. - Antti Karttunen, Oct 27 2016
Many of the derived sequences (like A002487) have similar "Fir forest" or "Gaudian cathedrals" style scatter plot. - Antti Karttunen, Mar 21 2017

			n    a(n)   prime factorization    Stern polynomial
------------------------------------------------------------
0       1   (empty)                B_0(x) = 0
1       2   p_1                    B_1(x) = 1
2       3   p_2                    B_2(x) = x
3       6   p_2 * p_1              B_3(x) = x + 1
4       5   p_3                    B_4(x) = x^2
5      18   p_2^2 * p_1            B_5(x) = 2x + 1
6      15   p_3 * p_2              B_6(x) = x^2 + x
7      30   p_3 * p_2 * p_1        B_7(x) = x^2 + x + 1
8       7   p_4                    B_8(x) = x^3
9      90   p_3 * p_2^2 * p_1      B_9(x) = x^2 + 2x + 1

Alois P. Heinz, Table of n, a(n) for n = 0..4096 (first 1025 terms from Antti Karttunen)

Same sequence sorted into ascending order: A260442.
Cf. A000040, A000079, A000225, A001222, A002487, A003415, A003961, A005811, A007949, A046523, A056239, A073491, A090880, A097249, A101979, A125184, A178590, A186891, A206284, A277314, A277315, A277325, A277326, A277329, A277330, A277701, A277705, A277899, A278243, A278530, A278544, A284010, A284011.
Cf. also A048675, A277333 (left inverses).
Cf. A277323, A277324 (bisections), A277200 (even terms sorted), A277197 (first differences), A277198.
Cf. A277316 (values at primes), A277318.
Cf. A023758 (positions of squarefree terms), A101082 (of terms not squarefree), A277702 (positions of records), A277703 (their values).
Cf. A283992, A283993 (number of irreducible, reducible polynomials in range 1 .. n).
Cf. also A206296 (Fibonacci polynomials similarly represented).

b:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
a:= proc(n) option remember; `if`(n<2, n+1,
      `if`(irem(n, 2, 'h')=0, b(a(h)), a(h)*a(n-h)))
    end:
seq(a(n), n=0..56);  # Alois P. Heinz, Jul 04 2024

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a@ n, {n, 0, 56}] (* Michael De Vlieger, Apr 05 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ After Charles R Greathouse IV's code for "ps" in A186891.
\\ Antti Karttunen, Oct 11 2016

from sympy import factorint, prime, primepi
from functools import reduce
from operator import mul
def a003961(n):
    F = factorint(n)
    return 1 if n==1 else reduce(mul, (prime(primepi(i) + 1)**F[i] for i in F))
def a(n): return n + 1 if n<2 else a003961(a(n//2)) if n%2==0 else a((n - 1)//2)*a((n + 1)//2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

;; Uses memoization-macro definec:
(definec (A260443 n) (cond ((<= n 1) (+ 1 n)) ((even? n) (A003961 (A260443 (/ n 2)))) (else (* (A260443 (/ (- n 1) 2)) (A260443 (/ (+ n 1) 2))))))
;; A more standalone version added Oct 10 2016, requiring only an implementation of A000040 and the memoization-macro definec:
(define (A260443 n) (product_primes_to_kth_powers (A260443as_coeff_list n)))
(define (product_primes_to_kth_powers nums) (let loop ((p 1) (nums nums) (i 1)) (cond ((null? nums) p) (else (loop (* p (expt (A000040 i) (car nums))) (cdr nums) (+ 1 i))))))
(definec (A260443as_coeff_list n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_coeff_list (/ n 2)))) (else (add_two_lists (A260443as_coeff_list (/ (- n 1) 2)) (A260443as_coeff_list (/ (+ n 1) 2))))))
(define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))

a(0) = 1, a(1) = 2, a(2n) = A003961(a(n)), a(2n+1) = a(n)*a(n+1).
Other identities. For all n >= 0:
A001221(a(n)) = A277314(n). [#nonzero coefficients in each polynomial.]
A001222(a(n)) = A002487(n). [When each polynomial is evaluated at x=1.]
A048675(a(n)) = n.          [at x=2.]
A090880(a(n)) = A178590(n). [at x=3.]
A248663(a(n)) = A264977(n). [at x=2 over the field GF(2).]
A276075(a(n)) = A276081(n). ["at factorials".]
A156552(a(n)) = A277020(n). [Converted to "unary-binary" encoding.]
A051903(a(n)) = A277315(n). [Maximal coefficient.]
A277322(a(n)) = A277013(n). [Number of irreducible polynomial factors.]
A005361(a(n)) = A277325(n). [Product of nonzero coefficients.]
A072411(a(n)) = A277326(n). [And their LCM.]
A007913(a(n)) = A277330(n). [The squarefree part.]
A000005(a(n)) = A277705(n). [Number of divisors.]
A046523(a(n)) = A278243(n). [Filter-sequence.]
A284010(a(n)) = A284011(n). [True for n > 1. Another filter-sequence.]
A003415(a(n)) = A278544(n). [Arithmetic derivative.]
A056239(a(n)) = A278530(n). [Weighted sum of coefficients.]
A097249(a(n)) = A277899(n).
a(A000079(n)) = A000040(n+1).
a(A000225(n)) = A002110(n).
a(A000051(n)) = 3*A002110(n).
For n >= 1, a(A000918(n)) = A070826(n).
A007949(a(n)) is the interleaving of A000035 and A005811, probably A101979.
A061395(a(n)) = A277329(n).
Also, for all n >= 1:
A055396(a(n)) = A001511(n).
A252735(a(n)) = A061395(a(n)) - 1 = A057526(n).
a(A000040(n)) = A277316(n).
a(A186891(1+n)) = A277318(n). [Subsequence for irreducible polynomials].

More linking formulas added by Antti Karttunen, Mar 21 2017

A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

Original entry on oeis.org

1, 9, 4, 3, 5, 9, 6, 4, 3, 6, 8, 2, 0, 7, 5, 9, 2, 0, 5, 0, 5, 7, 0, 7, 0, 3, 6, 2, 5, 7, 4, 7, 6, 3, 4, 3, 7, 1, 8, 7, 8, 5, 8, 5, 0, 1, 7, 6, 7, 8, 0, 5, 7, 1, 6, 0, 2, 6, 6, 3, 5, 6, 8, 8, 9, 0, 0, 5, 3, 4, 9, 5, 0, 6, 9, 3, 5, 5, 4, 0, 5, 3, 9, 4, 8, 1, 7, 9, 1, 0, 0, 8, 2, 1, 1, 1, 1, 3, 0, 1, 0, 6, 9, 0, 5

Offset: 1

Benoit Cloitre, Apr 12 2003

Equals the Dirichlet zeta-function Sum_{n>=1} A001615(n)/n^s at s=3. - R. J. Mathar, Apr 02 2011
Dressler shows that this is the average value of A014197, that is, the number of values m such that phi(m) <= n is asymptotically n times this constant. Erdős had shown earlier that this limit exists. - Charles R Greathouse IV, Nov 26 2013
From Stanislav Sykora, Nov 14 2014: (Start)
Equals lim_{n->infinity} (Sum_{k=1..n} k/phi(k))/n, i.e., the limit mean value of k/phi(k), where phi(k) is Euler's totient function.
Also equals lim_{n->infinity} (Sum_{k=1..n} 1/phi(k))/log(n).
Proofs are trivial using the formulas for Sum_{k=1..n} k/phi(k) and Sum_{k=1..n} 1/phi(k) listed in the Wikipedia link.
For the limit mean value of phi(k)/k, see A059956. (End)
The asymptotic mean of A005361. - Amiram Eldar, Apr 13 2020

			1.94359643682075920505707036257476343718785850176780571602663568890 ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.7, p. 116.
Joe Roberts, Lure of the Integers, Mathematical Association of America, 1992. See p. 74.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Paul T. Bateman, The distribution of values of the Euler function, Acta Arithmetica 21:1 (1972), pp. 329-345.
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
Robert E. Dressler, A density which counts multiplicity, Pacific J. Math. 34 (1970), pp. 371-378.
Paul Erdős, Some remarks on Euler's ϕ function and some related problems, Bull. Amer. Math. Soc. 51 (1945), pp. 540-544.
J. von zur Gathen et al., Average order in cyclic groups, J. Theor. Nombres Bordeaux, 16 (2004), 107-123. Lists several other papers where this constant arises.
S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
D. Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013.
Eric Weisstein's World of Mathematics, Totient Summatory Function.
Eric Weisstein's World of Mathematics, Powerful Number.
Wikipedia, Euler's totient function.

Cf. A000010, A001615, A001694, A002117, A005117, A005361, A013661, A013664, A014197, A059956, A068468, A070243, A082696 (continued fraction), A157292.

First@RealDigits[ Zeta[2]*Zeta[3]/Zeta[6], 10, 100]
RealDigits[ 315 Zeta[3]/(2 Pi^4), 10, 111][[1]] (* Robert G. Wilson v, Aug 11 2014 *)

PARI
```
zeta(3)*315/2/Pi^4
```

Decimal expansion of Product_{p prime} (1+1/p/(p-1)) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707...
The sum of the reciprocals of the powerful numbers, A001694. - T. D. Noe, May 03 2006
Equals A013661 * A002117 / A013664 = 1 / A068468 = zeta(3) * 315/(2*Pi^4) = zeta(3) * A157292.
Equals Sum_{k>=1} mu(k)^2/(k*phi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Euler totient function values, A000010). - Amiram Eldar, Aug 18 2020

New definition from Eric W. Weisstein, May 04 2006

A057723 Sum of positive divisors of n that are divisible by every prime that divides n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 14, 12, 10, 11, 18, 13, 14, 15, 30, 17, 24, 19, 30, 21, 22, 23, 42, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 72, 37, 38, 39, 70, 41, 42, 43, 66, 60, 46, 47, 90, 56, 60, 51, 78, 53, 78, 55, 98, 57, 58, 59, 90, 61, 62, 84, 126, 65, 66, 67, 102, 69, 70

Offset: 1

Leroy Quet, Oct 27 2000

			The divisors of 12 that are divisible by both 2 and 3 are 6 and 12. So a(12) = 6 + 12 = 18.

Ivan Neretin, Table of n, a(n) for n = 1..10000
József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma- function.

Row sums of triangle A284318.
Cf. A000203 (sigma), A007947 (rad), A005361 (number of these divisors).
Cf. A049060 and A060640 (other sigma-like functions).
Cf. A065487, A330596.

[&*PrimeDivisors(n)*SumOfDivisors(n div &*PrimeDivisors(n)): n in [1..70]]; // Vincenzo Librandi, May 14 2015

seq(mul(f[1]*(f[1]^f[2]-1)/(f[1]-1), f = ifactors(n)[2]), n = 1 .. 100); # Robert Israel, May 13 2015

Table[(b = Times @@ FactorInteger[n][[All, 1]])*DivisorSigma[1, n/b], {n, 70}] (* Ivan Neretin, May 13 2015 *)
f[p_, e_] := (p^(e+1)-1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, f[i,2]=1); my(pp = factorback(f)); sumdiv(n, d, if (! (d % pp), d, 0));} \\ Michel Marcus, May 14 2015

If n = Product p_i^e_i then a(n) = Product (p_i + p_i^2 + ... + p_i^e_i).
a(n) = rad(n)*sigma(n/rad(n)) = A007947(n)*A000203(A003557(n)). - Ivan Neretin, May 13 2015
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product(p prime, 1 - p^(-s) + p^(1-2*s)). - Robert Israel, May 13 2015
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A330596 = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.7485352596823635646442150486379106016416403430053244045... - Vaclav Kotesovec, Dec 18 2019
a(n) = Sum_{d|n, rad(d)=rad(n)} d. - R. J. Mathar, Jun 02 2020
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Jun 10 2020
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * sigma(d). - Ilya Gutkovskiy, Apr 15 2021

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.
This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:
                                      1
                                      |
                   ...................2....................
                  6                                       12
       30......../ \........60                 180......../ \......360
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
  210       420      1260       2520     6300       12600   37800       75600
etc.

Cf. A129912 (same terms, but sorted into ascending order).
Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005, A324287, A324289, A324341, A324342, A324343.
Cf. A005940, A052330, A322827, A323505 for other similar trees.
Cf. also A260443.

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)

A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019

from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a108951(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022

(define (A283477 n) (A108951 (A019565 n)))
;; Recursive "binary tree" implementation, using memoization-macro definec:
(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
a(n) = A108951(A019565(n)).
A097248(a(n)) = A283475(n).
A007814(a(n)) = A051903(a(n)) = A000120(n).
A001221(a(n)) = A070939(n).
A001222(a(n)) = A029931(n).
A048675(a(n)) = A005187(n).
A248663(a(n)) = A006068(n).
A090880(a(n)) = A283483(n).
A276075(a(n)) = A283984(n).
A276085(a(n)) = A283985(n).
A046660(a(n)) = A124757(n).
A056169(a(n)) = A065120(n). [seems to be]
A005361(a(n)) = A284001(n).
A072411(a(n)) = A284002(n).
A007913(a(n)) = A284003(n).
A000005(a(n)) = A284005(n).
A324286(a(n)) = A324287(n).
A276086(a(n)) = A324289(n).
A267263(a(n)) = A324341(n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017

Four more linking formulas added by Antti Karttunen, Feb 25 2019

cp's OEIS Frontend

A051903 Maximum exponent in the prime factorization of n.

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A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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

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A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

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A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

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