A124010 -id:A124010 - cp's OEIS frontend

A051904 Minimal exponent in prime factorization of n.

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

A048103 Numbers not divisible by p^p for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

N. J. A. Sloane

If a(n) = Product p_i^e_i then p_i > e_i for all i.
Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012
Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

			6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement: A100716.
Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).
Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).
Cf. A276086 (permutation of this sequence, see also A376411, A376413).
Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A328387, A328832, A359547, A370114, A371083, A373848, A377871, A377992.
Disjoint union of {1}, A327934 and A358215.
Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)

isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,1] <= f[i,2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020

A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024

from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]A048103_list = list(islice(A048103_gen(),30)) # Chai Wah Wu, Jan 05 2023

;; With Antti Karttunen's IntSeq-library.
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012

For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

More terms from James Sellers, Apr 22 2000

A067824 a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 16, 2, 6, 6, 16, 2, 16, 2, 16, 6, 6, 2, 40, 4, 6, 8, 16, 2, 26, 2, 32, 6, 6, 6, 52, 2, 6, 6, 40, 2, 26, 2, 16, 16, 6, 2, 96, 4, 16, 6, 16, 2, 40, 6, 40, 6, 6, 2, 88, 2, 6, 16, 64, 6, 26, 2, 16, 6, 26, 2, 152, 2, 6, 16, 16, 6, 26, 2, 96, 16, 6, 2, 88, 6, 6, 6, 40, 2, 88, 6, 16, 6, 6, 6, 224, 2, 16, 16, 52

Offset: 1

Reinhard Zumkeller, Feb 08 2002

nonn

By a result of Karhumaki and Lifshits, this is also the number of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1}.
The number of tiles of a discrete interval of length n (an interval of Z). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007
Bodini and Rivals proved this is the number of tiles of a discrete interval of length n and also is the number (A107067) of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1} (Bodini, Rivals, 2006). This structure of such tiles is also known as Krasner's factorization (Krasner and Ranulac, 1937). The proof also gives an algorithm to recognize if a set is a tile in optimal time and in this case, to compute the smallest interval it can tile (Bodini, Rivals, 2006). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007
Number of lone-child-avoiding rooted achiral (or generalized Bethe) trees with positive integer leaves summing to n, where a rooted tree is lone-child-avoiding if all terminal subtrees have at least two branches, and achiral if all branches directly under any given node are equal. For example, the a(6) = 6 trees are 6, (111111), (222), ((11)(11)(11)), (33), ((111)(111)). - Gus Wiseman, Jul 13 2018. Updated Aug 22 2020.
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of strict chains of divisors starting with n. For example, the a(n) chains for n = 1, 2, 4, 6, 8, 12 are:
  1  2    4      6      8        12
     2/1  4/1    6/1    8/1      12/1
          4/2    6/2    8/2      12/2
          4/2/1  6/3    8/4      12/3
                 6/2/1  8/2/1    12/4
                 6/3/1  8/4/1    12/6
                        8/4/2    12/2/1
                        8/4/2/1  12/3/1
                                 12/4/1
                                 12/4/2
                                 12/6/1
                                 12/6/2
                                 12/6/3
                                 12/4/2/1
                                 12/6/2/1
                                 12/6/3/1
(End)
a(n) is the number of chains including n of the divisor lattice of divisors of n, which is to say, a(n) is the number of (d_1,d_2,...,d_k) such that d_1 < d_2 < ... < d_k = n and d_i divides d_{i+1} for 1 <= i <= k-1. Using this definition, the recurrence a(n) = 1 + Sum_{0 < d < n, d|n} a(d) is evident by enumerating the preceding element of n in the chains. If we count instead the chains whose LCM of members is n, then a(1) would be 2 because the empty chain is included, and we would obtain 2*A074206(n). - Jianing Song, Aug 21 2024

			a(12) = 1 + a(6) + a(4) + a(3) + a(2) + a(1)
= 1+(1+a(3)+a(2)+a(1))+(1+a(2)+a(1))+(1+a(1))+(1+a(1))+(1)
= 1+(1+(1+a(1))+(1+a(1))+1)+(1+(1+a(1))+1)+(1+1)+(1+1)+(1)
= 1+(1+(1+1)+(1+1)+1)+(1+(1+1)+1)+(1+1)+(1+1)+(1)
= 1 + 6 + 4 + 2 + 2 + 1 = 16.

Olivier Bodini and Eric Rivals. Tiling an Interval of the Discrete Line. In M. Lewenstein and G. Valiente, editors, Proc. of the 17th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 4009 of Lecture Notes in Computer Science, pages 117-128. Springer Verlag, 2006.
Juhani Karhumaki, Yury Lifshits and Wojciech Rytter, Tiling Periodicity, in Combinatorial Pattern Matching, Lecture Notes in Computer Science, Volume 4580/2007, Springer-Verlag.

Reinhard Zumkeller, Table = of n, a(n) for n = 1..10000
Olivier Bodini and Eric Rivals, Tiling an Interval of the Discrete Line
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See Table 1 p. 8.
T. M. A. Fink, Properties of the recursive divisor function and the number of ordered factorizations, arXiv:2307.09140 [math.NT], 2023.
Michael Greene and Robin Michaels, One Dimensional Tilings, Eureka (Cambridge) 54 (1996), 4-13.
G. Hajos, Sur le problème de factorisation des groupes cycliques, Acta Math. Acad. Sci. Hung., 1:189-195, 1950.
J. Karhumaki and Y. Lifshits, Tiling periodicity.
M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
Eric H. Rivals, Tiling
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001678, A003238, A107067, A107748, A167865, A316782.
Cf. A122408 (fixed points).
Inverse Möbius transform of A074206.
A001055 counts factorizations.
A008480 counts maximal chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A253249 counts nonempty chains of divisors.
A337070 counts chains of divisors starting with A006939(n).
A337071 counts chains of divisors starting with n!.
A337256 counts chains of divisors.
Cf. A001221, A001222, A002033, A124010, A337074, A337105, A378223, A378225 (Dirichlet inverse).

a067824 n = 1 + sum (map a067824 [d | d <- [1..n-1], mod n d == 0])
-- Reinhard Zumkeller, Oct 13 2011

a:= proc(n) option remember;
      1+add(a(d), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 17 2021

a[1]=1; a[n_] := a[n] = 1+Sum[If[Mod[n,d]==0, a[d], 0], {d, 1, n-1}]; Array[a,100] (* Jean-François Alcover, Apr 28 2011 *)

A=vector(100);A[1]=1; for(n=2,#A,A[n]=1+sumdiv(n,d,A[d])); A \\ Charles R Greathouse IV, Nov 20 2012

a(n) = 2*A074206(n), n>1. - Vladeta Jovovic, Jul 03 2005
a(p^k) = 2^k for primes p. - Reinhard Zumkeller, Sep 03 2006
a(n) = Sum_{d|n} A002033(d-1) = Sum_{d|n} A074206(d). - Gus Wiseman, Jul 13 2018
Dirichlet g.f.: zeta(s) / (2 - zeta(s)). - Álvar Ibeas, Dec 30 2018
G.f. A(x) satisfies: A(x) = x/(1 - x) + Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, May 18 2019

Entry revised by N. J. A. Sloane, Aug 27 2006

A049417 a(n) = isigma(n): sum of infinitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90

Offset: 1

Yasutoshi Kohmoto, Dec 11 1999

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.
This sequence is an infinitary analog of the Dedekind psi function A001615. Indeed, a(n) = Product_{q in Q_n}(q+1) =  n*Product_{q in Q_n} (1+1/q), where {q} are terms of A050376 and Q_n is the set of distinct q's whose product is n. - Vladimir Shevelev, Apr 01 2014
1/a(n) is the asymptotic density of numbers that are infinitarily divided by n (i.e., numbers whose set of infinitary divisors includes n). - Amiram Eldar, Jul 23 2025

			If n = 8: 8 = 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) = 1+2+4+8 = 15.
n = 90 = 2*5*9, where 2, 5, 9 are in A050376; so a(n) = 3*6*10 = 180. - _Vladimir Shevelev_, Feb 19 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7417 from R. J. Mathar)
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.
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, pp. 49-56.
J. O. M. Pedersen, Tables of Aliquot Cycles.
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]
Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, Vol. 47 (2017), pp. 211-218; arXiv preprint, arXiv:1705.10933 [math.NT], 2017.

Cf. A004607, A037445, A050376, A077609, A091732, A127661, A293355.
Cf. A049418 (3-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).
Cf. A000079, A001615, A027748, A030308, A124010, A126168.

a049417 1 = 1
a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)
           (map (subtract 1 . (p ^)) a000079_list)
-- Reinhard Zumkeller, Sep 18 2015

isidiv := proc(d, n)
    local n2, d2, p, j;
    if n mod d <> 0 then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        padic[ordp](n,p) ;
        n2 := convert(%, base, 2) ;
        padic[ordp](d,p) ;
        d2 := convert(%, base, 2) ;
        for j from 1 to nops(d2) do
            if op(j, n2) = 0 and op(j, d2) <> 0 then
                return false;
            end if;
        end do:
    end do;
    return true;
end proc:
idivisors := proc(n)
    local a, d;
    a := {} ;
    for d in numtheory[divisors](n) do
        if isidiv(d, n) then
            a := a union {d} ;
        end if;
    end do:
    a ;
end proc:
A049417 := proc(n)
    local d;
    add(d, d=idivisors(n)) ;
end proc:
seq(A049417(n),n=1..100) ; # R. J. Mathar, Feb 19 2011

bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]; Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
(* Second program: *)
a[n_] := If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
Array[a, 100] (* Jean-François Alcover, Mar 23 2020, after Paul Abbott in A077609 *)

A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[,2], b = binary(f[k,2]); prod(j=1, #b, if(b[j], 1+f[k,1]^(2^(#b-j)), 1)))} \\ Andrew Lelechenko, Apr 22 2014

isigma(n)=vecprod([vecprod([f[1]^2^k+1|k<-[0..exponent(f[2])], bittest(f[2],k)])|f<-factor(n)~]) \\ M. F. Hasler, Oct 20 2022

from math import prod
from sympy import factorint
def A049417(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

Multiplicative: If e = Sum_{k >= 0} d_k 2^k (binary representation of e), then a(p^e) = Product_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005 [This means there is a factor p^2^k + 1 if d_k = 1, otherwise the factor is 1. - M. F. Hasler, Oct 20 2022]
Let n = Product(q_i) where {q_i} is a set of distinct terms of A050376. Then a(n) = Product(q_i + 1). - Vladimir Shevelev, Feb 19 2011
If n is squarefree, then a(n) = A001615(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{k>=1} A077609(n,k). - R. J. Mathar, Oct 04 2017
a(n) = A126168(n)+n. - R. J. Mathar, Oct 05 2017
Multiplicative with a(p^e) = Product{k >= 0, e_k = 1} p^2^k + 1, where e = Sum e_k 2^k, i.e., e_k is bit k of e. - M. F. Hasler, Oct 20 2022
a(n) = iphi(n^2)/iphi(n), where iphi(n) = A091732(n). - Amiram Eldar, Sep 21 2024

More terms from Wouter Meeussen, Sep 02 2001

A010057 a(n) = 1 if n is a cube, else 0.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Mar 15 1996

Multiplicative with a(p^e) = 1 if 3 divides e, 0 otherwise. - Mitch Harris, Jun 09 2005
a(A000578(n)) = 1; a(A007412(n)) = 0. - Reinhard Zumkeller, Oct 22 2011
a(n) = A000007(sum(A010872(A124010(n,k))): k = 1..A001221(n)) for n > 0. - Reinhard Zumkeller, Jun 21 2013
If n has 4 divisors, a(n) = bigomega(n) - 2. - Wesley Ivan Hurt, Jun 06 2014

E. Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen ueber Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for characteristic functions

Cf. A000578.

Cf. A003215. - Reinhard Zumkeller, Sep 27 2008

a010057 0 = 1
a010057 n = fromEnum $ all ((== 0) . (`mod` 3)) $ a124010_row n
a010057_list = concatMap (\x -> 1 : replicate (a003215 x - 1) 0) [0..]
-- Reinhard Zumkeller, Jun 21 2013, Oct 22 2011

A010057 := proc(n)
    if n = 0 then
        1;
    else
        for pe in ifactors(n)[2] do
            if modp(op(2,pe),3) <> 0 then
                return 0 ;
            end if;
        end do:
    end if;
    1 ;
end proc: # R. J. Mathar, Feb 07 2023

Table[ Boole[ IntegerQ[n^(1/3)]], {n, 0, 80}] (* Jean-François Alcover, Jun 10 2013 *)

a(n) = ispower(n, 3); \\ Michel Marcus, Feb 24 2015

from sympy import integer_nthroot
def A010057(n): return int(integer_nthroot(n,3)[1]) # Chai Wah Wu, Apr 02 2021

Dirichlet generating function: zeta(3s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = f(n,0) with f(x,y) = if x>0 then f(x-3*y*(y+1),y+1) else 0^(-x). - Reinhard Zumkeller, Sep 27 2008
a(n) = 1 + floor(n^(1/3)) - ceiling(n^(1/3)). - Wesley Ivan Hurt, Jun 06 2014
a(n) = floor(n^(1/3)) - floor((n-1)^(1/3)). - Mikael Aaltonen, Feb 24 2015

A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

Original entry on oeis.org

45, 117, 153, 245, 261, 325, 333, 369, 405, 425, 477, 549, 605, 637, 657, 725, 801, 833, 845, 873, 909, 925, 981, 1017, 1025, 1053, 1233, 1325, 1341, 1377, 1413, 1421, 1445, 1525, 1557, 1573, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2057, 2061, 2097, 2169

Offset: 1

T. D. Noe, Aug 13 2013

nonn

It has been proved that if an odd perfect number exists, it belongs to this sequence. The first term of the form p^5 * n^2 is 28125 = 5^5 * 3^2, occurring in position 520.
Sequence A228059 lists the subsequence of these numbers that are closer to being perfect than smaller numbers. - T. D. Noe, Aug 15 2013
Sequence A326137 lists terms with at least five distinct prime factors. See further comments there. - Antti Karttunen, Jun 13 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A191218, and also of A228056 and A228057 (simpler versions of this sequence).
For various subsequences with additional conditions, see A228059, A325376, A325380, A325822, A326137 (with omega(n)>=5), A324898 (conjectured, subsequence if it does not contain any prime powers), A354362, A386425 (conjectured), A386427 (nondeficient terms), A386428 (powerful terms), A386429 U A351574.
Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.

import Data.List (partition)
a228058 n = a228058_list !! (n-1)
a228058_list = filter f [1, 3 ..] where
   f x = length us == 1 && not (null vs) &&
         fst (head us) `mod` 4 == 1 && snd (head us) `mod` 4 == 1
         where (us,vs) = partition (odd . snd) $
                         zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Aug 14 2013

nn = 100; n = 1; t = {}; While[Length[t] < nn, n = n + 2; {p, e} = Transpose[FactorInteger[n]]; od = Select[e, OddQ]; If[Length[e] > 1 && Length[od] == 1 && Mod[od[[1]], 4] == 1 && Mod[p[[Position[e, od[[1]]][[1,1]]]], 4] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Aug 15 2013 *)

up_to = 1000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, Apr 22 2019

From Antti Karttunen, Apr 22 2019 & Jun 03 2019: (Start)
A325313(a(n)) = -A325319(n).
A325314(a(n)) = -A325320(n).
A001065(a(n)) = A325377(n).
A033879(a(n)) = A325379(n).
A034460(a(n)) = A325823(n).
A325814(a(n)) = A325824(n).
A324213(a(n)) = A325819(n).
(End)

Note in parentheses added to the definition by Antti Karttunen, Jun 03 2019

A056170 Number of non-unitary prime divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 27 2000

A prime factor of n is unitary iff its exponent is 1 in the prime factorization of n. (Of course for any prime p, GCD(p, n/p) is either 1 or p. For a unitary prime factor it must be 1.)
Number of squared primes dividing n. - Reinhard Zumkeller, May 18 2002
a(A005117(n)) = 0; a(A013929(n)) > 0; a(A190641(n)) = 1. - Reinhard Zumkeller, Dec 29 2012
First differences of A013940. - Jason Kimberley, Feb 01 2017
Number of exponents larger than 1 in the prime factorization of n. - Antti Karttunen, Nov 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A001221, A003557, A013940, A034444, A046660, A048105, A056169, A085548, A124010, A162641, A212177, A275812, A295659, A295666.

Cf. A057427(a(n)) = 1 - A008966(n).

a056170 = length . filter (> 1) . a124010_row
-- Reinhard Zumkeller, Dec 29 2012

A056170:=func;
[A056170(n):n in[1..105]];
// Jason Kimberley, Jan 22 2017

A056170 := n -> nops(select(t -> (t[2]>1), ifactors(n)[2]));
seq(A056170(n),n=1..100); # Robert Israel, Jun 03 2014

a[n_] := Count[FactorInteger[n], {, k /; k > 1}]; Table[a[n], {n, 105}]  (* Jean-François Alcover, Mar 23 2011 *)
Table[Count[FactorInteger[n][[All,2]],?(#>1&)],{n,110}] (* _Harvey P. Dale, Jul 08 2019 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]>1) \\ Charles R Greathouse IV, May 18 2015

from sympy import factorint
def a(n):
    f = factorint(n)
    return sum([1 for i in f if f[i]!=1]) # Indranil Ghosh, Apr 24 2017

Additive with a(p^e) = 0 if e = 1, 1 otherwise.
G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Jan 01 2017
a(n) = log_2(A000005(A071773(n))). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
From Antti Karttunen, Nov 28 2017: (Start)
a(n) = A001221(n) - A056169(n).
a(n) = omega(A000188(n)) = omega(A003557(n)) = omega(A057521(n)) = omega(A295666(n)), where omega = A001221.
For all n >= 1 it holds that:
a(A003557(n)) = A295659(n).
a(n) >= A162641(n).
(End)
Dirichlet g.f.: primezeta(2s)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Sum_{p prime} 1/p^2 = 0.452247... (A085548). - Amiram Eldar, Nov 01 2020
a(n) = A275812(n) - A046660(n). - Amiram Eldar, Jan 09 2024

Minor edits by Franklin T. Adams-Watters, Mar 23 2011

A188999 Bi-unitary sigma: sum of the bi-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 27, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 63, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 108, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 119, 84, 144, 68, 90, 96, 144, 72, 150, 74, 114, 104, 100

Offset: 1

R. J. Mathar, Apr 15 2011

The sequence of bi-unitary perfect numbers obeying a(n) = 2*n consists of only 6, 60, 90 [Wall].

Row sum of row n of the irregular table of the bi-unitary divisors, A222266.

			The divisors of n=16 are d=1, 2, 4, 8 and 16. The greatest common unitary divisor of (1,16) is 1, of (2,8) is 1, of (4,4) is 4, of (8,2) is 1, of (16,1) is 1 (see A165430). So 1, 2, 8 and 16 are bi-unitary divisors of 16, which sum to a(16) = 1 + 2 + 8 + 16 = 27.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On arithmetic functions and divisors of higher order, J. Austral. Math. Soc. 23 (series A) (1977), 9-27.
József Sándor and Borislav Crstici, Perfect numbers: Old and new issues; perspectives, in Handbook of number theory, II, p. 45.
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function J. Int. Seq. 12 (2009), Article 09.5.2.
Charles R. Wall, Bi-unitary perfect numbers, Proc. Am. Math. Soc. 33 (1) (1972), 39-42.
Eric Weisstein's World of Mathematics, Biunitary Divisor.
Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017.

Cf. A222266, A027748, A124010, A034448, A319072.

a188999 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = (p ^ (e + 1) - 1) `div` (p - 1) - (1 - m) * p ^ e' where
           (e', m) = divMod e 2
-- Reinhard Zumkeller, Mar 04 2013

A188999 := proc(n) local a,e,p,f; a :=1 ; for f in ifactors(n)[2] do e := op(2,f) ; p := op(1,f) ; if type(e,'odd') then a := a*(p^(e+1)-1)/(p-1) ; else a := a*((p^(e+1)-1)/(p-1)-p^(e/2)) ; end if; end do: a ; end proc:
seq( A188999(n),n=1..80) ;

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; Table[DivisorSum[n, # &, Last@ Intersection[f@ #, f[n/#]] == 1 &], {n, 76}] (* Michael De Vlieger, May 07 2017 *)
a[n_] := If[n==1, 1, Product[{p, e} = pe; If[OddQ[e], (p^(e+1)-1)/(p-1), ((p^(e+1)-1)/(p-1)-p^(e/2))], {pe, FactorInteger[n]}]]; Array[a, 80] (* Jean-François Alcover, Sep 22 2018 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
a(n) = vecsum(biudivs(n)); \\ Michel Marcus, May 07 2017

a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; e = f[i,2]; f[i,1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 09 2017

from math import prod
from sympy import factorint
def A188999(n): return prod((p**(e+1)-1)//(p-1)-(0 if e&1 else p**(e>>1)) for p,e in factorint(n).items()) # Chai Wah Wu, Dec 28 2024

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd, a(p^e) = (p^(e+1)-1)/(p-1) -p^(e/2) if e is even.
a(n) = A000203(n) - A319072(n). - Omar E. Pol, Sep 29 2018
Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 2/p^(2*s-1) + 1/p^(3*s-2) + 1/p^(3*s-1) - 1/p^(4*s-2)). - Amiram Eldar, Aug 28 2023

A181796 a(n) = number of divisors of n whose canonical prime factorizations contain no repeated positive exponents (cf. A130091).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 4, 2, 6, 3, 3, 3, 7, 2, 3, 3, 7, 2, 4, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 7, 3, 4, 2, 5, 3, 4, 2, 10, 2, 3, 5, 5, 3, 4, 2, 9, 5, 3, 2, 7, 3, 3, 3, 7, 2, 7, 3, 5, 3, 3, 3, 11, 2, 5, 5, 7, 2, 4, 2, 7, 4

Offset: 1

Matthew Vandermast, Nov 22 2010

nonn

The canonical factorization of n into prime powers can be written as Product p(i)^e(i), for example. A host of equivalent notations can also be used (for another example, see Weisstein link). a(n) depends only on prime signature of n (cf. A025487).
a(n) >= A085082(n). (A085082(n) equals the number of members of A025487 that divide A046523(n), and each member of A025487 is divisible by at least one member of A130091 that divides no smaller member of A025487.) a(n) > A085082(n) iff n has in its canonical prime factorization at least two exponents greater than 1.
a(n) = number of such divisors of n that in their prime factorization all exponents are unique. - Antti Karttunen, May 27 2017
First differs from A335549 at a(90) = 7, A335549(90) = 8. First differs from A335516 at a(180) = 9, A335516(180) = 10. - Gus Wiseman, Jun 28 2020

			12 has a total of six divisors (1, 2, 3, 4, 6 and 12). Of those divisors, the number 1 has no prime factors, hence, no positive exponents at all (and no repeated positive exponents) in its canonical prime factorization. The lists of positive exponents for 2, 3, 4, 6 and 12 are (1), (1), (2), (1,1) and (2,1) respectively (cf. A124010). Of all six divisors, only the number 6 (2^1*3^1) has at least one positive exponent repeated (namely, 1). The other five do not; hence, a(12) = 5.
For n = 90 = 2 * 3^2 * 5, the divisors that satisfy the condition are: 1, 2, 3, 3^2, 5, 2 * 3^2, 3^2 * 5, altogether 7, (but for example 90 itself is not included), thus a(90) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.
Gus Wiseman, Sequences counting and encoding certain classes of multisets

Diverges from A088873 at n=24 and from A085082 at n=36. a(36) = 7, while A085082(36) = 6.
Partitions with distinct multiplicities are A098859.
Sorted prime signature is A118914.
Unsorted prime signature is A124010.
a(n) is the number of divisors of n in A130091.
Factorizations with distinct multiplicities are A255231.
The largest of the counted divisors is A327498.
Factorizations using the counted divisors are A327523.
Cf. A000005, A007916, A045778, A056239, A212168, A327498, A335519, A335549.

Table[DivisorSum[n, 1 &, Length@ Union@ # == Length@ # &@ FactorInteger[#][[All, -1]] &], {n, 105}] (* Michael De Vlieger, May 28 2017 *)

no_repeated_exponents(n) = { my(es = factor(n)[, 2]); if(length(Set(es)) == length(es),1,0); }
A181796(n) = sumdiv(n,d,no_repeated_exponents(d)); \\ Antti Karttunen, May 27 2017

from sympy import factorint, divisors
def ok(n):
    f=factorint(n)
    ex=[f[i] for i in f]
    for i in ex:
        if ex.count(i)>1: return 0
    return 1
def a(n): return sum([1 for i in divisors(n) if ok(i)]) # Indranil Ghosh, May 27 2017

a(A000079(n)) = a(A002110(n)) = n+1.
a(A006939(n)) = A000110(n+1).
a(A181555(n)) = A002720(n).

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 22, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 66, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102, 69, 70, 71

Offset: 1

Yasutoshi Kohmoto

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

a(n) = n if and only if n is squarefree. - Jon Perry, Nov 13 2012

			a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.
Nicussor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.
Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A051378, A049419 (number of e-divisors).
Cf. A027748, A124010, A027750.
Row sums of A322791.
See A307042 and A275480 where the formula and constant appear.

A051377:=n->Product(List(Collected(Factors(n)), p -> Sum(DivisorsInt(p[2]),d->p[1]^d))); List([1..10^4], n -> A051377(n)); # Muniru A Asiru, Oct 29 2017

a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where
   sum_e p e = sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051377 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        add(p^d,d=numtheory[divisors](e)) ;
        a := a*% ;
    end do:
    a ;
end proc:
seq(A051377(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d,f[i,1]^d)) \\ Charles R Greathouse IV, Nov 22 2011

ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
a(n)=vecsum(ediv(n)) \\ Charles R Greathouse IV, Oct 29 2018

Multiplicative with a(p^e) = Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = A126164(n)+n. - R. J. Mathar, Oct 05 2017
The average order of a(n) is Dn + O(n^e) for any e > 0, due to Fabrykowski & Subbarao, where D is about 0.568. (D >= 0.5 since a(n) >= n.) - Charles R Greathouse IV, Sep 22 2023

More terms from Jud McCranie, May 29 2000

Definition corrected by Jaroslav Krizek, Feb 27 2009

cp's OEIS Frontend

A051904 Minimal exponent in prime factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Scheme

Formula

A048103 Numbers not divisible by p^p for any prime p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Scheme

Formula

Extensions

A067824 a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A049417 a(n) = isigma(n): sum of infinitary divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A010057 a(n) = 1 if n is a cube, else 0.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI