A028234 -id:A028234 - cp's OEIS frontend

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

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

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4, 64, 5, 128, 6, 10, 17, 256, 5, 8, 33, 6, 10, 512, 7, 1024, 5, 18, 65, 12, 6, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 16, 9, 66, 34, 32768, 7, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 6, 36, 19

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.
However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.
The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020
If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

			From _Gus Wiseman_, May 22 2024: (Start)
The A018819(7) = 6 cases of binary rank 7 are the following, together with their prime indices:
   30: {1,2,3}
   40: {1,1,1,3}
   54: {1,2,2,2}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

Row 2 of A104244.
Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.
Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.
A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.
Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.
Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A248692,A056239), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).
See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.
The formula section details how the sequence maps the terms of A329050, A329332.
A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.
The term k appears A018819(k) times.
The inverse transformation is A019565 (Heinz number of binary indices).
The version for distinct prime indices is A087207.
Numbers k such that a(k) is prime are A277319, counts A372688.
Grouping by image gives A277905.
A014499 lists binary indices of prime numbers.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000720, A035100, A071814, A096111, A225620, A243055, A304818, A355536, A358136, A372890.
Cf. A087207, A097248.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.
A048675 := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;
# simpler alternative
f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Oct 10 2016

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*2^primepi(f[k,1]))/2; \\ Michel Marcus, Oct 10 2016

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:
v048675sigs = readvec("a048675.txt");
A048675(n) = if(n<=2,n-1,my(prsig=v048675sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

from sympy import factorint, primepi
def a(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).
Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]
From Antti Karttunen, Jul 29 2015: (Start)
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]
a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).
Other identities. For all n >= 0:
a(A019565(n)) = n.
a(A260443(n)) = n.
a(A206296(n)) = A000129(n).
a(A005940(n+1)) = A087808(n).
a(A007913(n)) = A248663(n).
a(A007947(n)) = A087207(n).
a(A283477(n)) = A005187(n).
a(A284003(n)) = A006068(n).
a(A285101(n)) = A028362(1+n).
a(A285102(n)) = A068052(n).
Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)
a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016
From Peter Munn, Jan 31 2020: (Start)
a(n^2) = a(A003961(n)) = 2 * a(n).
a(A297845(n,k)) = a(n) * a(k).
a(n) = a(A225546(n)).
a(A329332(n,k)) = n * k.
a(A329050(n,k)) = 2^(n+k).
(End)
From Antti Karttunen, Feb 02-25 2020, Feb 01 2021: (Start)
a(n) = Sum_{d|n} A297108(d) = Sum_{d|A225546(n)} A297108(d).
a(n) = a(A097248(n)).
For n >= 2:
A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).
A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).
For n >= 1, A331750(n) = a(A000203(n)).
For n >= 1, the following chains hold:
A293447(n) >= a(n) >= A331740(n) >= A331591(n).
a(n) >= A087207(n) >= A248663(n).
(End)
a(n) = A087207(A097248(n)). - Flávio V. Fernandes, Jul 16 2025

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

A005361 Product of exponents of prime factorization of n.

Original entry on oeis.org

1, 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, 4, 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, 6, 1, 1, 2, 2, 1, 1

Offset: 1

Jeffrey Shallit and Olivier Gérard

a(n) depends only on prime signature of n (cf. A025487, A052306). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
There was a comment here that said "a(n) is the number of nilpotents elements in the ring Z/nZ", but this is false, see A003557.
a(n) is the number of square-full divisors of n. a(n) is also the number of divisors d of n such that d and n have the same prime factors, i.e., A007947(d) = A007947(n). - Laszlo Toth, May 22 2009
Number of divisors u of n such that u|(u^n/n). Row lengths in triangle of A284318. - Juri-Stepan Gerasimov, Apr 05 2017

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

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
P. Erdős and T. Motzkin, Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681. (Incorrect solution!)
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
J. Knopfmacher, A prime-divisor function, Proc. Amer. Math. Soc., 40 (1973), 373-377.
MathStackExchange, Dirichlet series for zeta(s)*zeta(2*s)*zeta(3*s)*zeta(6*s)^(-1).
H. N. Shapiro, Problem 5735, Amer. Math. Monthly, 97 (1990), 937.
D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full divisors of an integer, Proc. Amer. Math. Soc., Vol. 34, No. 1 (1972), pp. 79-80.
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A002110, A005117 (indices of ones), A028234, A052306, A067029, A072411, A082695, A284318.
Cf. A360969, A360970, A361132.
Cf. A340065 (Dgf at s=2).

a005361 = product . a124010_row -- Reinhard Zumkeller, Jan 09 2012

A005361 := proc(n)
    local a, p ;
    a := 1 ;
    for p in ifactors(n)[2] do
       a := a*op(2, p) ;
    end do:
    a ;
end proc:
seq(A005361(n),n=1..30) ; # R. J. Mathar, Nov 20 2012
# second Maple program:
a:= n-> mul(i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 18 2020

Prepend[ Array[ Times @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
Array[Times@@Transpose[FactorInteger[#]][[2]]&,80] (* Harvey P. Dale, Aug 15 2012 *)

for(n=1,100, f=factor(n); print1(prod(i=1,omega(f), f[i,2]),",")) \\ edited by M. F. Hasler, Feb 18 2020

a(n)=factorback(factor(n)[,2]) \\ Charles R Greathouse IV, Nov 07 2014

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

from math import prod
from sympy import factorint
def a(n): return prod(factorint(n).values())
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Jul 04 2022

(define (A005361 n) (if (= 1 n) 1 (* (A067029 n) (A005361 (A028234 n))))) ;; Antti Karttunen, Mar 06 2017

n = Product (p_j^k_j) -> a(n) = Product (k_j).
Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/zeta(6s).
Multiplicative with a(p^e) = e. - David W. Wilson, Aug 01 2001
a(n) = Sum_{d dividing n} floor(rad(d)/rad(n)) where rad(n) is A007947. - Enrique Pérez Herrero, Nov 06 2009
For n > 1: a(n) = Product_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(n) = tau(n/rad(n)), where tau is A000005 and rad is A007947. - Anthony Browne, May 11 2016
a(n) = Sum_{k=1..n}(floor(cos^2(Pi*k^n/n))*floor(cos^2(Pi*n/k))). - Anthony Browne, May 11 2016
From Antti Karttunen, Mar 06 2017: (Start)
For all n >= 1, a(prime^n) = n, a(A002110(n)) = a(A005117(n)) = 1. [From Crossrefs section.]
a(1) = 1; for n > 1, a(n) = A067029(n) * a(A028234(n)).
(End)
Let (b(n)) be multiplicative with b(p^e) = -1 + ( (floor((e-1)/3)+floor(e/3)) mod 4 ) for p prime and e > 0, then b(n) is the Dirichlet inverse of (a(n)). - Werner Schulte, Feb 23 2018
Sum_{i=1..k} a(i) ~ (zeta(2)*zeta(3)/zeta(6)) * k (Suryanarayana and Sitaramachandra Rao, 1972). - Amiram Eldar, Apr 13 2020
More precise asymptotics: Sum_{k=1..n} a(k) ~ 315*zeta(3)*n / (2*Pi^4) + zeta(1/2)*zeta(3/2)*sqrt(n) / zeta(3) + 6*zeta(1/3)*zeta(2/3)*n^(1/3) / Pi^2 [Knopfmacher, 1973]. - Vaclav Kotesovec, Jun 13 2020

A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

Offset: 1

Antti Karttunen, Aug 21 2016

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)-1).
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

A left inverse of A276086.
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).
Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.
Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).
Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].
Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

from sympy import primorial, primepi, factorint
def a002110(n):
    return 1 if n<1 else primorial(n)
def a(n):
    f=factorint(n)
    return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).
a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.
Other identities.
For all n >= 0:
a(A276086(n)) = n.
a(A000040(1+n)) = A002110(n).
a(A002110(1+n)) = A143293(n).
From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)
a(A283477(n)) = A283985(n).
a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]
When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:
a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).
a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).
In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:
a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).
a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).
a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).
a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).
The sum or difference of the rhs-sequences is A108951:
a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).
a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).
Here the two sequences are inverse permutations of each other:
a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).
a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).
a(A346101(n)) = A289234(n). [Self-inverse]
Other correspondences:
a(A324350(x,y)) = A324351(x,y).
a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]
a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]
(End)

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024

A023900 Dirichlet inverse of Euler totient function (A000010).

Original entry on oeis.org

1, -1, -2, -1, -4, 2, -6, -1, -2, 4, -10, 2, -12, 6, 8, -1, -16, 2, -18, 4, 12, 10, -22, 2, -4, 12, -2, 6, -28, -8, -30, -1, 20, 16, 24, 2, -36, 18, 24, 4, -40, -12, -42, 10, 8, 22, -46, 2, -6, 4, 32, 12, -52, 2, 40, 6, 36, 28, -58, -8, -60, 30, 12, -1, 48, -20, -66, 16, 44, -24, -70, 2, -72, 36, 8, 18, 60, -24, -78, 4, -2

Offset: 1

Olivier Gérard

Also called reciprocity balance of n.
Apart from different signs, same as Sum_{d divides n} core(d)*mu(n/d), where core(d) (A007913) is the squarefree part of d. - Benoit Cloitre, Apr 06 2002
Main diagonal of A191898. - Mats Granvik, Jun 19 2011

			x - x^2 - 2*x^3 - x^4 - 4*x^5 + 2*x^6 - 6*x^7 - x^8 - 2*x^9 + 4*x^10 - ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 125.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
K. Dohmen and M. Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv preprint arXiv:1404.5480 [math.CO], 2014.
R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

Cf. A000010, A007913, A023898, A117209, A134842.
Moebius transform is A055615.
Cf. A027748, A173557 (gives the absolute values), A295876.
Cf. A253905 (Dgf at s=3).

a023900 1 = 1
a023900 n = product $ map (1 -) $ a027748_row n
-- Reinhard Zumkeller, Jun 01 2015

A023900 := n -> mul(1-i,i=numtheory[factorset](n)); # Peter Luschny, Oct 26 2010

a[ n_] := If[ n < 1, 0, Sum[ d MoebiusMu @ d, { d, Divisors[n]}]] (* Michael Somos, Jul 18 2011 *)
Array[ Function[ n, 1/Plus @@ Map[ #*MoebiusMu[ # ]/EulerPhi[ # ]&, Divisors[ n ] ] ], 90 ]
nmax = 81; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* Stuart Clary, Apr 15 2006 *)
t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] :=  t[n, k] = If[n < k, If[n > 1 && k > 1, Sum[-t[k - i, n], {i, 1, n - 1}], 0], If[n > 1 && k > 1, Sum[-t[n - i, k], {i, 1, k - 1}], 0]]; Table[t[n, n], {n, 36}] (* Mats Granvik, Robert G. Wilson v, Jun 25 2011 *)
Table[DivisorSum[m, # MoebiusMu[#] &], {m, 90}] (* Jan Mangaldan, Mar 15 2013 *)
f[p_, e_] := (1 - p); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

{a(n) = direuler( p=2, n, (1 - p*X) / (1 - X))[n]}

{a(n) = if( n<1, 0, sumdiv( n, d, d * moebius(d)))} /* Michael Somos, Jul 18 2011 */

a(n)=sumdivmult(n,d, d*moebius(d)) \\ Charles R Greathouse IV, Sep 09 2014

from sympy import divisors, mobius
def a(n): return sum([d*mobius(d) for d in divisors(n)]) # Indranil Ghosh, Apr 29 2017

from math import prod
from sympy import primefactors
def A023900(n): return prod(1-p for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

;; With memoization-macro definec.
(definec (A023900 n) (if (= 1 n) 1 (* (- 1 (A020639 n)) (A023900 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = Sum_{ d divides n } d*mu(d) = Product_{p|n} (1-p).
a(n) = 1 / (Sum_{ d divides n } mu(d)*d/phi(d)).
Dirichlet g.f.: zeta(s)/zeta(s-1). - Michael Somos, Jun 04 2000
a(n+1) = det(n+1)/det(n) where det(n) is the determinant of the n X n matrix M_(i, j) = i/gcd(i, j) = lcm(i, j)/j. - Benoit Cloitre, Aug 19 2003
a(n) = phi(n)*moebius(A007947(n))*A007947(n)/n. Logarithmic g.f.: Sum_{n >= 1} a(n)*x^n/n = log(F(x)) where F(x) is the g.f. of A117209 and satisfies: 1/(1-x) = Product_{n >= 1} F(x^n). - Paul D. Hanna, Mar 03 2006
G.f.: A(x) = Sum_{k >= 1} mu(k) k x^k/(1 - x^k) where mu(k) is the Moebius (Mobius) function, A008683. - Stuart Clary, Apr 15 2006
G.f.: A(x) is x times the logarithmic derivative of A117209(x). - Stuart Clary, Apr 15 2006
Row sums of triangle A134842. - Gary W. Adamson, Nov 12 2007
G.f.: x/(1-x) = Sum_{n >= 1} a(n)*x^n/(1-x^n)^2. - Paul D. Hanna, Aug 16 2008
a(n) = phi(rad(n)) *(-1)^omega(n) = A000010(A007947(n)) *(-1)^A001221(n). - Enrique Pérez Herrero, Aug 24 2010
a(n) = Product_{i = 2..n} (1-i)^( (pi(i)-pi(i-1)) * floor( (cos(n*Pi/i))^2 ) ), where pi = A000720, Pi = A000796. - Wesley Ivan Hurt, May 24 2013
a(n) = -limit of zeta(s)*(Sum_{d divides n} moebius(d)/exp(d)^(s-1)) as s->1 for n>1. - Mats Granvik, Jul 31 2013
a(n) = Sum_{d divides n} mu(d)*rad(d), where rad is A007947. - Enrique Pérez Herrero, May 29 2014
Conjecture for n>1: Let n = 2^(A007814(n))*m = 2^(ruler(n))*odd_part(n), where m = A000265(n), then a(n) = (-1)^(m=n)*(0+Sum_{i=1..m and gcd(i,m)=1} (4*min(i,m-i)-m)) = (-1)^(m1} (4*min(i,m-i)-m)). - I. V. Serov, May 02 2017
a(n) = (-1)^A001221(n) * A173557(n). - R. J. Mathar, Nov 02 2017
a(1) = 1; for n > 1, a(n) = (1-A020639(n)) * a(A028234(n)), because multiplicative with a(p^e) = (1-p). - Antti Karttunen, Nov 28 2017
a(n) = 1 - Sum_{d|n, d > 1} d*a(n/d). - Ilya Gutkovskiy, Apr 26 2019
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k)) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)) = 0. (End)
a(n) = rad(n)*(-1)^omega(n)*phi(n)/n = A062953(n)*A000010(n)/n. - Amrit Awasthi, Jan 30 2022
a(n) = mu(n)*phi(n) = A008683(n)*A000010(n) whenever n is squarefree. - Amrit Awasthi, Feb 03 2022
From Peter Bala, Jan 24 2024: (Start)
a(n) = Sum_{d divides n} core(d)*mu(d). Cf. Comment by Benoit Cloitre, dated Apr 06 2002.
a(n) = Sum_{d|n, e|n}  n/gcd(d, e) * mu(n/d) * mu(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value 1 - p for n = p^e, a prime power). (End)
From Peter Bala, Feb 01 2024: (Start)
G.f. Sum_{n >= 1} (2*n-1)*moebius(2*n-1)*x^(2*n-1)/(1 + x^(2n-1)).
a(n) = (-1)^(n+1) * Sum_{d divides n, d odd} d*moebius(d). (End)

A067029 Exponent of least prime factor in prime factorization of n, a(1)=0.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 17 2002

Even bisection is A001511: a(2n) = A007814(n) + 1. - Ralf Stephan, Jan 31 2004
Number of occurrences of the smallest part in the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: a(24)=3 because the partition with Heinz number 24 = 3*2*2*2 is [2,1,1,1]. - Emeric Deutsch, Oct 02 2015
Together with A028234 is useful for defining sequences that are multiplicative with a(p^e) = f(e), as recurrences of the form: a(1) = 1 and for n > 1, a(n) = f(A067029(n)) * a(A028234(n)). - Antti Karttunen, May 29 2017

			a(18) = a(2^1 * 3^2) = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Project Euler, Problem 779: Prime factor and exponent, (2022).

Cf. A051903, A020639, A028233, A034684, A071178, first column of A124010, A247180.

a067029 = head . a124010_row
-- Reinhard Zumkeller, Jul 05 2013, Jun 04 2012

A067029 := proc(n)
    local f,lp,a;
    a := 0 ;
    lp := n+1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        if p < lp then
            a := op(2,f) ;
            lp := p;
        fi;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 08 2015
seq(ifelse(n = 1, 0, ifactors(n)[2][1][2]), n = 1..90); # Peter Luschny, Jun 15 2025

Join[{0},Table[FactorInteger[n][[1,2]],{n,2,100}]] (* Harvey P. Dale, Oct 14 2011 *)

a(n) = if (n==1, 0, factor(n)[1,2]); \\ Michel Marcus, May 15 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)] # Indranil Ghosh, May 15 2017

;; Naive implementation of A020639 is given under that entry. All of these functions could be also defined with definec to make them faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(define (A067029 n) (if (< n 2) 0 (let ((mp (A020639 n))) (let loop ((e 0) (n (/ n mp))) (cond ((integer? n) (loop (+ e 1) (/ n mp))) (else e)))))) ;;  Antti Karttunen, May 29 2017

a(n) = A124010(n,1). - Reinhard Zumkeller, Aug 27 2011
A028233(n) = A020639(n)^a(n). - Reinhard Zumkeller, May 13 2006
a(A247180(n)) = 1. - Reinhard Zumkeller, Nov 23 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (Product_{i=1..k-1} (1 - 1/prime(i)))/(prime(k)-1) = 1/(prime(1)-1) + (1-1/prime(1))*(1/(prime(2)-1) + (1-1/prime(2))*(1/(prime(3)-1) + (1-1/prime(3))*( ... ))) = 1.6125177915... - Amiram Eldar, Oct 26 2021

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

A037445 Number of infinitary divisors (or i-divisors) of n.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.

The smallest number m with exactly 2^n infinitary divisors is A037992(n); for these values m, a(m) increases also to a new record. - Bernard Schott, Mar 09 2023

			For n = 8, n = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8, so a(8) = 4.
For n = 90, n = 2*5*9 where 2,5,9 are in A050376, so a(90) = 2^3 = 8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
Eric Weisstein's World of Mathematics, Infinitary Divisor
Index entries for sequences computed from exponents in factorization of n

Cf. A000120, A001316, A004607, A007358, A007357, A037992, A038148, A049417, A064547, A074848, A077609, A124010, A156552, A286575.

a037445 = product . map (a000079 . a000120) . a124010_row
-- Reinhard Zumkeller, Mar 19 2013

A037445 := proc(n)
    local a,p;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*2^wt(p[2]) ;
    end do:
    a ;
end proc: # R. J. Mathar, May 16 2016

Table[Length@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@
Flatten[Outer[z, Sequence @@ bitty /@
Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 240}]
bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]
y[n_] := Select[Range[0, n], BitOr[n, # ] == n & ] divisors[Infinity][1] := {1}
divisors[Infinity][n_] := Sort[Flatten[Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^y[m])]]] Length /@ divisors[Infinity] /@ Range[105] (* Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 29 2005 *)
a[1] = 1; a[n_] := Times @@ Flatten[ 2^DigitCount[#, 2, 1]&  /@ FactorInteger[n][[All, 2]] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Aug 19 2013, after Reinhard Zumkeller *)

A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[,2])) \\ Andrew Lelechenko, May 10 2014

from sympy import factorint
def wt(n): return bin(n).count("1")
def a(n):
    f=factorint(n)
    return 2**sum([wt(f[i]) for i in f]) # Indranil Ghosh, May 30 2017

(define (A037445 n) (if (= 1 n) n (* (A001316 (A067029 n)) (A037445 (A028234 n))))) ;; Antti Karttunen, May 28 2017

Multiplicative with a(p^e) = 2^A000120(e). - David W. Wilson, Sep 01 2001
Let n = q_1*...*q_k, where q_1,...,q_k are different terms of A050376. Then a(n) = 2^k (the number of subsets of a set with k elements is 2^k). - Vladimir Shevelev, Feb 19 2011.
a(n) = Product_{k=1..A001221(n)} A000079(A000120(A124010(n,k))). - Reinhard Zumkeller, Mar 19 2013
From Antti Karttunen, May 28 2017: (Start)
a(n) = A286575(A156552(n)). [Because multiplicative with a(p^e) = A001316(e).]
a(n) = 2^A064547(n). (End)
a(A037992(n)) = 2^n. - Bernard Schott, Mar 10 2023

Corrected and extended by Naohiro Nomoto, Jun 21 2001

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

A064547 Sum of binary digits (or count of 1-bits) in the exponents of the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3

Offset: 1

Wouter Meeussen, Oct 09 2001

This sequence is different from A058061 for n containing 6th, 8th, ..., k-th powers in its prime decomposition, where k runs through the integers missing from A064548.
For n > 1, n is a product of a(n) distinct members of A050376. - Matthew Vandermast, Jul 13 2004
For n > 1: a(n) = length of n-th row in A213925. - Reinhard Zumkeller, Mar 20 2013
Number of Fermi-Dirac factors of n. - Peter Munn, Dec 27 2019

			For n = 54, n = 2^1 * 3^3 with exponents (1) and (11) in binary, so a(54) = A000120(1) + A000120(3) = 1 + 2 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..32768 (terms 1..2000 from Harry J. Smith)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to binary expansion of n.

Cf. A000028 (positions of odd terms), A000379 (of even terms).
Cf. A050376 (positions of ones), A268388 (terms larger than ones).
Row lengths of A213925.
A000120, A007814, A028234, A037445, A052331, A064989, A067029, A156552, A223491, A286574 are used in formulas defining this sequence.
Cf. A005117, A058061 (to which A064548 relates), A138302.
Cf. other sequences counting factors of n: A001221, A001222.
Cf. other sequences where a(n) depends only on the prime signature of n: A181819, A267116, A268387.
A003961, A007913, A008833, A059895, A059896, A059897, A225546 are used to express relationship between terms of this sequence.
Cf. A077761, A176699, A037992, A382294.

a064547 1 = 0
a064547 n = length $ a213925_row n  -- Reinhard Zumkeller, Mar 20 2013

expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end;
A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
LamMos:= proc(n) local t1,t2,t3,i; t1:=expts(n); add( A000120(t1[i]),i=1..nops(t1)); end; # N. J. A. Sloane, Dec 20 2007
# alternative Maple program:
A064547:= proc(n) local F;
F:= ifactors(n)[2];
add(convert(convert(f[2],base,2),`+`),f=F)
end proc:
map(A064547,[$1..100]); # Robert Israel, May 17 2016

Table[Plus@@(DigitCount[Last/@FactorInteger[k], 2, 1]), {k, 105}]

a(n) = {my(f = factor(n)[,2]); sum(k=1, #f, hammingweight(f[k]));} \\ Michel Marcus, Feb 10 2016

from sympy import factorint
def wt(n): return bin(n).count("1")
def a(n):
    f=factorint(n)
    return sum([wt(f[i]) for i in f]) # Indranil Ghosh, May 30 2017

;; uses memoizing-macro definec
(definec (A064547 n) (cond ((= 1 n) 0) (else (+ (A000120 (A067029 n)) (A064547 (A028234 n))))))
;; Antti Karttunen, Feb 09 2016

;; uses memoizing-macro definec
(definec (A064547 n) (if (= 1 n) 0 (+ (A000120 (A007814 n)) (A064547 (A064989 n)))))
;; Antti Karttunen, Feb 09 2016

a(m*n) <= a(m)*a(n). - Reinhard Zumkeller, Mar 20 2013
From Antti Karttunen, Feb 09 2016: (Start)
a(1) = 0, and for n > 1, a(n) = A000120(A067029(n)) + a(A028234(n)).
a(1) = 0, and for n > 1, a(n) = A000120(A007814(n)) + a(A064989(n)).
(End)
a(n) = log_2(A037445(n)). - Vladimir Shevelev, May 13 2016
a(n) = A286574(A156552(n)). - Antti Karttunen, May 28 2017
Additive with a(p^e) = A000120(e). - Jianing Song, Jul 28 2018
a(n) = A000120(A052331(n)). - Peter Munn, Aug 26 2019
From Peter Munn, Dec 18 2019: (Start)
a(A000379(n)) mod 2 = 0.
a(A000028(n)) mod 2 = 1.
A001221(n) <= a(n) <= A001222(n).
A001221(n) < a(n) => a(n) < A001222(n).
a(n) = A001222(n) if and only if n is in A005117.
a(n) = A001221(n) if and only if n is in A138302.
a(n^2) = a(n).
a(A003961(n)) = a(n).
a(A225546(n)) = a(n).
a(n) = a(A007913(n)) + a(A008833(n)).
a(A050376(n)) = 1.
a(A059897(n,k)) + 2 * a(A059895(n,k)) = a(n) + a(k).
a(A059896(n,k)) + a(A059895(n,k)) = a(n) + a(k).
Alternative definition: a(1) = 0; a(n * m) = a(n) + 1 for m = A050376(k) > A223491(n).
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.13605447049622836522... (A382294), where f(x) = -x + Sum_{k>=0} x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 28 2023
a(n) << log n/log log n. - Charles R Greathouse IV, Nov 29 2024

cp's OEIS Frontend

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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A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

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A005361 Product of exponents of prime factorization of n.

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A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

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A023900 Dirichlet inverse of Euler totient function (A000010).

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