A051904 -id:A051904 - cp's OEIS frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

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

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

A051903 Maximum exponent in the prime factorization of n.

Original entry on oeis.org

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

A071625 Number of distinct exponents when n is factorized as a product of primes.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, May 29 2002

nonn

First term greater than 2 is a(360) = 3.
From Michel Marcus, Apr 24 2016: (Start)
A006939(n) gives the least m such that a(m) = n.
A062770 is the sequence of integers m such that a(m) = 1. (End)
We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first two omegas are A001222 and A001221, while this sequence is the third, and A323022 is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices). - Gus Wiseman, Jan 02 2019
Sanna (2020) proved that for each k>=1, the sequence of numbers n with A071625(n) = k has an asymptotic density A_k = (6/Pi^2) * Sum_{n>=1, n squarefree} rho_k(n)/psi(n), where psi is the Dedekind psi function (A001615), and rho_k(n) is defined by rho_1(n) = 1 if n = 1 and 0 otherwise, rho_{k+1}(n) = 0 if n = 1 and (1/(n-1)) * Sum_{d|n, dAmiram Eldar, Oct 18 2020

			n = 5040 = 2^4*(3*5)^2*7, three different exponents arise:4,2 and 1; so a(5040)=3.

Giovanni Resta, Table of n, a(n) for n = 1..10000
E. T. Bell, Functions of coprime divisors of integers, Bull. Amer. Math. Soc. 43 (1937), 818-822.
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link, arXiv preprint, arXiv:1902.09224 [math.NT], 2019.

Cf. A051903, A051904, A006939, A124010.

Cf. A001221, A001222, A001615, A059404, A062770, A118914, A181819, A323014, A323022, A323023, A323024, A323025.

# Using function 'PrimeSignature' from A124010.
a := n -> nops(convert(PrimeSignature(n), set)):
seq(a(n), n = 1..105); # Peter Luschny, Jun 15 2025

ffi[x_] := Flatten[FactorInteger[x]];
lf[x_] := Length[FactorInteger[x]];
ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}];
Table[Length[Union[ep[w]]], {w, 1, 256}]
(* Second program: *)
{0}~Join~Array[Length@ Union@ FactorInteger[#][[All, -1]] &, 104, 2] (* Michael De Vlieger, Apr 10 2019 *)

a(n) = #Set(factor(n)[,2]); \\ Michel Marcus, Mar 12 2015

from sympy import factorint
def a(n): return len(set(factorint(n).values()))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Sep 01 2022

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

A071626 Number of distinct exponents in the prime factorization of n!.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Labos Elemer, May 29 2002

Erdős proved that there exist two constants c1, c2 > 0 such that c1 (n / log(n))^(1/2) < a(n) < c2 (n / log(n))^(1/2). - Carlo Sanna, May 28 2019

R. Heyman and R. Miraj proved that the cardinality of the set { floor(n/p) : p <= n, p prime } is same as the number of distinct exponents in the prime factorization of n!. - Md Rahil Miraj, Apr 05 2024

			n=7: 7! = 5040 = 2*2*2*2*3*3*5*7; three different exponents arise: 4, 2 and 1; a(7)=3.
n=7: { floor(7/p) : p <= 7, p prime } = {3,2,1}. So, its cardinality is 3. - _Md Rahil Miraj_, Apr 05 2024

David A. Corneth, Table of n, a(n) for n = 1..10000
Thomas Bloom, Problem 912, Erdős Problems. See Terence Tao's 31 August 2025 comment.
Paul Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congressus Numerantium 34 (1982), 25-45.
Randell Heyman and Md Rahil Miraj, On some floor function sets, arXiv:2309.16072 [math.NT], 2023-2024.
Terence Tao, Erdős problem database, see no. 912.

Cf. A051903, A051904, A071625, A240751.
Cf. A000142, A001221, A001222, A011371, A022559, A076934, A115627, A135291.
Cf. A325272, A325273, A325276, A325508.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] Table[Length[Union[ep[w! ]]], {w, 1, 100}]
Table[Length[Union[Last/@If[n==1,{},FactorInteger[n!]]]],{n,30}] (* Gus Wiseman, May 15 2019 *)

a(n) = #Set(factor(n!)[, 2]); \\ Michel Marcus, Sep 05 2017

a(n) = A071625(n!) = A323023(n!,3). - Gus Wiseman, May 15 2019

A059404 Numbers with different exponents in their prime factorizations.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200

Offset: 1

Labos Elemer, Jul 18 2001

Former name: Numbers k such that k/(largest power of squarefree kernel of k) is larger than 1.

Also numbers k = p(1)^alpha(1)* ... * p(r)^alpha(r) such that RootMeanSquare(alpha(1), ..., alpha(r)) is not an integer. - Ctibor O. Zizka, Sep 19 2008

			440 is in the sequence because 440 = 2^3*5*11 and it has two distinct exponents, 3 and 1.

Donald Alan Morrison, Table of n, a(n) for n = 1..10000
Donald Alan Morrison, Sage program

Complement of A072774 (powers of squarefree numbers).

Cf. A003557, A007947, A051904, A062759, A062760, A071625.

isA := n -> 1 < nops({seq(padic:-ordp(n, p), p in NumberTheory:-PrimeFactors(n))}): select(isA, [seq(1..190)]);  # Peter Luschny, Apr 14 2025

A059404Q[n_] := Length[Union[FactorInteger[n][[All, 2]]]] > 1;
Select[Range[200], A059404Q] (* Paolo Xausa, Jan 07 2025 *)

is(n)=#Set(factor(n)[,2])>1 \\ Charles R Greathouse IV, Sep 18 2015

from sympy import factorint
def ok(n): return len(set(factorint(n).values())) > 1
print([k for k in range(190) if ok(k)]) # Michael S. Branicky, Sep 01 2022

from math import isqrt
from sympy import mobius, integer_nthroot
def A059404(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def f(x): return n+1-(y:=x.bit_length())+sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

def isA(n): return 1 < len(set(valuation(n, p) for p in prime_divisors(n)))
print([n for n in range(1, 190) if isA(n)])  # Peter Luschny, Apr 14 2025

A062760(a(n)) > 1, i.e., a(n)/(A007947(a(n))^A051904(a(n))) = a(n)/A062759(a(n)) > 1.
A071625(a(n)) > 1. - Michael S. Branicky, Sep 01 2022
Sum_{n>=1} 1/a(n)^s = zeta(s) - 1 - Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025

A072411 LCM of exponents in prime factorization of n, a(1) = 1.

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

Offset: 1

Labos Elemer, Jun 17 2002

The sums of the first 10^k terms, for k = 1, 2, ..., are 14, 168, 1779, 17959, 180665, 1808044, 18084622, 180856637, 1808585068, 18085891506, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.8085... . - Amiram Eldar, Sep 10 2022

			n = 288 = 2*2*2*2*2*3*3; lcm(5,2) = 10; Product(5,2) = 10, max(5,2) = 5;
n = 180 = 2*2*3*3*5; lcm(2,2,1) = 2; Product(2,2,1) = 4; max(2,2,1) = 2; it deviates both from maximum of exponents (A051903, for the first time at n=72), and product of exponents (A005361, for the first time at n=36).
For n = 36 = 2*2*3*3 = 2^2 * 3^2 we have a(36) = lcm(2,2) = 2.
For n = 72 = 2*2*2*3*3 = 2^3 * 3^2 we have a(72) = lcm(2,3) = 6.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(2,4) = 4.
For n = 360 = 2^3 * 3^2 * 5^1 we have a(360) = lcm(1,2,3) = 6.

Cf. A028234, A067029, A072412-A072414, A273058, A284569, A290103.
Similar sequences: A001222 (sum of exponents), A005361 (product), A051903 (maximal exponent), A051904 (minimal exponent), A052409 (gcd of exponents), A267115 (bitwise-and), A267116 (bitwise-or), A268387 (bitwise-xor).
Cf. also A055092, A060131.
Differs from A290107 for the first time at n=144.
After the initial term, differs from A157754 for the first time at n=360.

Table[LCM @@ Last /@ FactorInteger[n], {n, 2, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n) = lcm(factor(n)[,2]); \\ Michel Marcus, Mar 25 2017

from sympy import lcm, factorint
def a(n):
    l=[]
    f=factorint(n)
    for i in f: l+=[f[i],]
    return lcm(l)
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 25 2017

a(1) = 1; for n > 1, a(n) = lcm(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 09 2016
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A284569(A156552(n)).
a(n) = A290103(A181819(n)).
a(A289625(n)) = A002322(n).
a(A290095(n)) = A055092(n).
a(A275725(n)) = A060131(n).
a(A260443(n)) = A277326(n).
a(A283477(n)) = A284002(n). (End)

a(1) = 1 prepended and the data section filled up to 120 terms by Antti Karttunen, Aug 09 2016

A382525 Number of times n appears in A048767 (rank of Look-and-Say partition of prime indices). Number of ordered set partitions whose block-sums are the prime signature of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 05 2025

nonn

The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.
Also the number of ways to choose a set of disjoint strict integer partitions, one of each nonzero multiplicity in the prime factorization of n.

			The a(27) = 2 partitions with Look-and-Say partition (2,2,2) are: (3,3), (2,2,1,1).
The prime indices of 3456 are {1,1,1,1,1,1,1,2,2,2}, and the partitions with Look-and-Say partition (2,2,2,1,1,1,1,1,1,1) are:
  (7,3,3)
  (7,2,2,1,1)
  (6,3,3,1)
  (5,3,3,2)
  (4,3,3,2,1)
  (4,3,2,2,1,1)
so a(3456) = 6.

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
Positions of 1 are A381540, conjugate A381434.
Positions of terms > 1 are A381541, conjugate A381435.
Positions of first appearances are A382775.
A000670 counts ordered set partitions.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381436 lists the section-sum partition of prime indices, ranks A381431.
A381440 lists the Look-and-Say partition of prime indices, ranks A048767.
Cf. A003557, A047966, A048768, A050361, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964.

stp[y_]:=Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
Table[Length[stp[Last/@FactorInteger[n]]],{n,100}]

a(2^n) = A000009(n).

a(prime(n)) = 1.

A090699 Decimal expansion of the Erdos-Szekeres constant zeta(3/2)/zeta(3).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 14 2004

Let N(x) denotes the number of 2-full integers not exceeding x. Then lim_{x->oo} N(x)/sqrt(x) = zeta(3/2)/zeta(3). Also related to Niven's constant.

			zeta(3/2)/zeta(3) = 2.17325431251955413823708984...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 112-114.

S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.

Cf. A001694 (powerful numbers), A102834 (nonsquare powerful numbers).

Cf. A033150, A051904.

RealDigits[N[Zeta[3/2]/Zeta[3],150]][[1]] (* T. D. Noe, May 03 2006 *)

zeta(3/2)/zeta(3) \\ Michel Marcus, Oct 06 2017

Product_{p prime} (1+1/p^(3/2)) = zeta(3/2)/zeta(3). - T. D. Noe, May 03 2006

Equals lim_{n->oo} (Sum_{k=1..n} A051904(k) - n)/sqrt(n) (Niven, 1969). - Amiram Eldar, Jul 11 2020

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 16 2007

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 07 2025

nonn

The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The a(96) = 4 permutations are:
  (1,1,1,1,1,2)
  (1,1,1,2,1,1)
  (1,1,2,1,1,1)
  (2,1,1,1,1,1)
The a(216) = 4 permutations are:
  (1,1,2,2,2,1)
  (1,2,2,2,1,1)
  (2,1,1,1,2,2)
  (2,2,1,1,1,2)
The a(360) = 6 permutations are:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)

Positions of 1 are A000961.
Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433, equal A382879.
Sorted positions of first appearances are A382772, equal A382878.
For prescribed signature we have A382773, equal A382858.
The restriction to factorials is A382774, equal A335407.
For equal instead of distinct run-lengths we have A382857.
For run-sums instead of run-lengths we have A382876, equal A382877.
Positions of terms > 1 are A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, complement A351293.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]

a(A181821(n)) = a(A304660(n)) = A382773(n).

a(n!) = A382774(n).

cp's OEIS Frontend

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

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A051903 Maximum exponent in the prime factorization of n.

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A071625 Number of distinct exponents when n is factorized as a product of primes.

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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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A071626 Number of distinct exponents in the prime factorization of n!.

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A059404 Numbers with different exponents in their prime factorizations.

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