A055231 -id:A055231 - cp's OEIS frontend

A207359 Indices n, not squarefree, where A055231(n) = A055231(n-A055231(n)).

9, 45, 63, 99, 117, 153, 171, 207, 261, 279, 289, 315, 333, 369, 387, 423, 477, 495, 531, 549, 585, 603, 639, 657, 676, 693, 711, 747, 765, 801, 819, 855, 873, 909, 927, 963, 981, 1017, 1035, 1071, 1143, 1179, 1197, 1233, 1251, 1287, 1305, 1341, 1359, 1395

Offset: 1

Michel Lagneau, Feb 17 2012

nonn

A055231(n) is the powerfree part of n. This sequence is infinite because all numbers of the form n = 9p, where p is a prime > 3, are in the sequence : A055231(9p) = p and A055231(9p - p) = A055231(8p) = p. The positive numbers of A055792 are also in the sequence because A055792(n) are squares and A055792(n)-1 are also squares.

			63 is in the sequence because A055231(63) = A055231(7*3^2) = 7, A055231(63 - 7) = A055231(56) = A055231(7*2^3) = 7.

Cf. A055231, A140394.

isA013929 := proc(n)
    n>3 and not numtheory[issqrfree](n) ;
end proc:
isA207359 := proc(n)
    isA013929(n) and (A055231(n)- A055231(n- A055231(n))=0);
end proc:
for n from 1 to 5000 do
      if isA207359(n) then
          printf(`%d, `,n);
      end if;
end do: #  adapted from A140394.

A207360 Numbers n, not squarefree, satisfying A055231(n) = A055231(n + A055231(n)).

Original entry on oeis.org

8, 40, 56, 88, 104, 136, 152, 184, 232, 248, 280, 288, 296, 328, 344, 376, 424, 440, 472, 488, 520, 536, 568, 584, 616, 632, 664, 675, 680, 712, 728, 760, 776, 808, 824, 856, 872, 904, 920, 952, 1016, 1048, 1064, 1096, 1112, 1144, 1160, 1192, 1208, 1240, 1256

Offset: 1

Michel Lagneau, Feb 17 2012

nonn

A055231(n) is the powerfree part of n.
This sequence is infinite because the numbers of the form n = 8p, where p is prime, are in the sequence : A055231(8p) = p and A055231(8p + p) = A055231(9p)  = p.
The numbers such that n and n+1 are a pair of consecutive powerful numbers (the again infinite A060355) are also in the sequence because A055231 (A060355(n)) = A055231(A060355 (n+1)) = 1.

			136 is in the sequence because A055231(136) = A055231(17*2^3) = 17, A055231(136 + 17) = A055231(153) = A055231(17*3^2) = 17.

Cf. A055231, A060355, A140394.

isA013929 := proc(n)
        n>3 and not numtheory[issqrfree](n) ;
end proc:
isA207360 := proc(n)
       isA013929(n)  and (A055231(n)- A055231(n+ A055231(n))=0);
end proc:
for n from 1 to 5000 do
        if isA207360(n) then
            printf(`%d, `,n);
        end if;
end do: # (adapted from A140394).

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A055231[n_] := Denominator[n/rad[n]^2];
Select[Range[2000], !SquareFreeQ[#] && A055231[#] == A055231[# + A055231[#]]&] (* Jean-François Alcover, Jun 18 2024 *)

isA013929(n)={
    (n>3) && !issquarefree(n)
}
isA207360(n)={
    isA013929(n) && ( A055231(n)-A055231(n+A055231(n)) ==0)
}
{ for(n=1,1300, if(isA207360(n), print1(n" ") ) ; ) ;
} /* R. J. Mathar, Mar 12 2012 */

A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

Original entry on oeis.org

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

Offset: 1

R. Muller, Mar 15 1996

Also called core(n). [Not to be confused with the squarefree kernel of n, A007947.]
Sequence read mod 4 gives A065882. - Philippe Deléham, Mar 28 2004
This is an arithmetic function and is undefined if n <= 0.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), lcm(A007947(b),c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n. [Corrected by M. F. Hasler, Mar 01 2018]
If n > 1, the quantity f(n) = log(n/core(n))/log(n) satisfies 0 <= f(n) <= 1; f(n) = 0 when n is squarefree and f(n) = 1 when n is a perfect square. One can define n as being "epsilon-almost squarefree" if f(n) < epsilon. - Kurt Foster (drsardonicus(AT)earthlink.net), Jun 28 2008
a(n) is the smallest natural number m such that product of geometric mean of the divisors of n and geometric mean of the divisors of m are integers. Geometric mean of the divisors of number n is real number b(n) = Sqrt(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A000290: a(A000290(n)) = 1. For n = 8; b(8) = sqrt(8), a(n) = 2 because b(2) = sqrt(2); sqrt(8) * sqrt(2) = 4 (integer). - Jaroslav Krizek, Apr 26 2010
Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011
Booker, Hiary, & Keating outline a method for bounding (on the GRH) a(n) for large n using L-functions. - Charles R Greathouse IV, Feb 01 2013
According to the formula a(n) = n/A000188(n)^2, the scatterplot exhibits the straight lines y=x, y=x/4, y=x/9, ..., i.e., y=x/k^2 for all k=1,2,3,... - M. F. Hasler, May 08 2014
The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - Álvar Ibeas, Mar 19 2015
a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - Zak Seidov, Jan 30 2016
All solutions of the Diophantine equation n*x=y^2 or, equivalently, G(n,x)=y, with G being the geometric mean, are of the form x=k^2*a(n), y=k*sqrt(n*a(n)), where k is a positive integer. - Stanislav Sykora, Feb 03 2016
If f is a multiplicative function then Sum_{d divides n} f(a(d)) is also multiplicative. For example, A010052(n) = Sum_{d divides n} mu(a(d)) and A046951(n) = Sum_{d divides n} mu(a(d)^2). - Peter Bala, Jan 24 2024

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Krassimir T. Atanassov, On the 22nd, 23rd, and the 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.
Andrew Booker, Ghaith Hiary, and Jon Keating, Detecting squarefree numbers, CNTA XII (2012).
Henry Bottomley, Some Smarandache-type multiplicative sequences.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Vlad Copil and Laurenţiu Panaitopol, Properties of a sequence generated by positive integers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 2 (2007), pp. 131-137; alternative link.
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Eric Weisstein's World of Mathematics, Squarefree Part.

See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.
See A027746, A027748, A124010 for factorization data for n.
Analogous sequences: A050985, A053165, A055231.
Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums), A090699, A350389.
Related to A006519 via A225546.

a007913 n = product $
            zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)
-- Reinhard Zumkeller, Jul 06 2012

[ Squarefree(n) : n in [1..256] ]; // N. J. A. Sloane, Dec 23 2006

A007913 := proc(n) local f,a,d; f := ifactors(n)[2] ; a := 1 ; for d in f do if type(op(2,d),'odd') then a := a*op(1,d) ; end if; end do: a; end proc: # R. J. Mathar, Mar 18 2011
# second Maple program:
a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 20 2015
seq(n / expand(numtheory:-nthpow(n, 2)), n=1..77);  # Peter Luschny, Jul 12 2022

data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[] -> 1; sfp = data/sp /. Sqrt[x] -> x (* Artur Jasinski, Nov 03 2008 *)
Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,100}] (* Zak Seidov, Apr 08 2009 *)
Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* T. D. Noe, May 20 2013 *)
Sqrt[#] /. (c_:1)*a_^(b_:0) -> (c*a^b)^2& /@ Range@100 (* Bill Gosper, Jul 18 2015 *)

PARI
```
a(n)=core(n)
```

from sympy import factorint, prod
def A007913(n):
    return prod(p for p, e in factorint(n).items() if e % 2)
# Chai Wah Wu, Feb 03 2015

[squarefree_part(n) for n in (1..77)] # Peter Luschny, Feb 04 2015

Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson, Aug 01 2001
a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - Philippe Deléham, Mar 28 2004
Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - Anthony Browne, Jun 06 2016
a(n) = rad(n)/a(n/rad(n)), where rad = A007947. This recurrence relation together with a(1) = 1 generate the sequence. - Velin Yanev, Sep 19 2017
From Peter Munn, Nov 18 2019: (Start)
a(k*m) = A059897(a(k), a(m)).
a(n) = n / A008833(n).
(End)
a(A225546(n)) = A225546(A006519(n)). - Peter Munn, Jan 04 2020
From Amiram Eldar, Mar 14 2021: (Start)
Theorems proven by Copil and Panaitopol (2007):
Lim sup_{n->oo} a(n+1)-a(n) = oo.
Lim inf_{n->oo} a(n+1)-a(n) = -oo.
Sum_{k=1..n} 1/a(k) ~ c*sqrt(n) + O(log(n)), where c = zeta(3/2)/zeta(3) (A090699). (End)
a(n) = A019554(n)^2/n. - Jianing Song, May 08 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Oct 25 2022
a(n) = A007947(A350389(n)). - Amiram Eldar, Jan 20 2024

More terms from Michael Somos, Nov 24 2001

Definition reformulated by Daniel Forgues, Mar 24 2009

A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=bc^2d^3 then a(n)=c^2*d^3 when b is minimized.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 8, 1, 9, 1, 4, 1, 1, 1, 32, 1

Offset: 1

Henry Bottomley, Sep 01 2000

			a(40) = 8 since 40 = 2^3 * 5 so the powerful part is 2^3 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..16383 (first 1000 terms from T. D. Noe)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000000 terms).
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seq., Vol. 6 (2003), Article 03.3.4.
Index entries for sequences related to powerful numbers.

Cf. A001694, A003415, A003557, A055231, A064549.

A057521 := proc(n)
    local a,d,e,p;
    a := 1;
    for d in ifactors(n)[2] do
        e := d[1] ;
        p := d[2] ;
        if e > 1 then
            a := a*p^e ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Jun 09 2016

rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := n/Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 97}] (* Jean-François Alcover, Jun 20 2013 *)
f[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

a(n)=my(f=factor(n));prod(i=1,#f~,if(f[i,2]>1,f[i,1]^f[i,2],1)) \\ Charles R Greathouse IV, Aug 13 2013

a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); factorback(f); \\ Michel Marcus, Jan 29 2021

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod(1 if e==1 else p**e for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

from math import prod
from sympy import factorint
def A057521(n): return n//prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

a(n) = n / A055231(n).
Multiplicative with a(p)=1 and a(p^e)=p^e for e>1. - Vladeta Jovovic, Nov 01 2001
From Antti Karttunen, Nov 22 2017: (Start)
a(n) = A064549(A003557(n)).
A003557(a(n)) = A003557(n).
(End)
a(n) = gcd(n, A003415(n)^k), for all k >= 2. [This formula was found in the form k=3 by Christian Krause's LODA miner. See Ufnarovski and Åhlander paper, Theorem 5 on p. 4 for why this holds] - Antti Karttunen, Mar 09 2021
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/ p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1)). - Amiram Eldar, Sep 18 2023
From Vaclav Kotesovec, Apr 09 2025, simplified May 11 2025: (Start)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 - 1/p^(3*s-2) + 1/p^(3*s-3) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^(5/2)) = 3.51955505841710664719752940369857817... = A328013. (End)

A056169 Number of unitary prime divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 27 2000

The zeros of this sequences are the powerful numbers (A001694). There are no arbitrarily long subsequences with a given upper bound; for example, every sequence of 4 values includes one divisible by 2 but not 4, so there are no more than 3 consecutive zeros. Similarly, there can be no more than 23 consecutive values with none divisible by both 2 and 3 but neither 4 nor 9 (so a(n) >= 2), etc. In general, this gives an upper bound that is a (relatively) small multiple of the k-th primorial number (prime(k)#). One suspects that the actual upper bounds for such subsequences are quite a bit lower; e.g., Erdős conjectured that there are no three consecutive powerful numbers. - Franklin T. Adams-Watters, Aug 08 2006
In particular, for every A048670(k)*A002110(k) consecutive terms, at least one is greater than or equal to k. - Charlie Neder, Jan 03 2019
Following Catalan's conjecture (which became Mihăilescu's theorem in 2002), the first case of two consecutive zeros in this sequence is for a(8) and a(9), because 8 = 2^3 and 9 = 3^2, and there are no other consecutive zeros for consecutive powers. However, there are other pairs of consecutive zeros at powerful numbers (A001694, A060355). The next example is a(288) = a(289) = 0, because 288 = 2^5 * 3^2 and 289 = 17^2, then also a(675) and a(676). - Bernard Schott, Jan 06 2019
a(2k-1) is the number of primes p such that p || x + y and p^2 || x^(2k-1) + y^(2k-1) for some positive integers x and y. For any positive integers x, y and k > 1, there is no prime p such that p || x + y and p^2 || x^(2k) + y^(2k). - Jinyuan Wang, Apr 08 2020

			9 = 3^2 so a(9) = 0; 10 = 2 * 5 so a(10) = 2; 11 = 11^1 so a(11) = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001694, A002110, A034444, A056170, A055231, A076445, A162642, A275812, A295659, A295662, A295664, A001694.
See also A060355.
Cf. A077761, A085548.

a056169 = length . filter (== 1) . a124010_row
-- Reinhard Zumkeller, Sep 10 2013

a:= n-> nops(select(i-> i[2]=1, ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 27 2017

Join[{0},Table[Count[Transpose[FactorInteger[n]][[2]],1],{n,2,110}]] (* Harvey P. Dale, Mar 15 2012 *)
Table[DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]==1) \\ Charles R Greathouse IV, Apr 29 2015

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017

;; With memoization-macro definec.
(definec (A056169 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 1 0) (A056169 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

A prime factor of n is unitary iff its exponent is 1 in prime factorization of n. In general, gcd(p, n/p) = 1 or = p.
Additive with a(p^e) = 1 if e = 1, 0 otherwise.
a(n) = #{k: A124010(n,k) = 1, k = 1..A001221}. - Reinhard Zumkeller, Sep 10 2013
From Antti Karttunen, Nov 28 2017: (Start)
a(1) = 0; for n > 1, a(n) = A063524(A067029(n)) + a(A028234(n)).
a(n) = A001221(A055231(n)) = A001222(A055231(n)).
a(n) = A001221(n) - A056170(n) = A001221(n) - A001221(A000188(n)).
a(n) = A001222(n) - A275812(n).
a(n) = A162642(n) - A295662(n).
a(n) <= A162642(n) <= a(n) + A295659(n).
a(n) <= A295664(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} (1/p^2) = 0.452247... (A085548). - Amiram Eldar, Sep 28 2023

A106177 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 5, 2, 9, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 7, 1, 25, 1, 3, 1, 1, 1, 1, 1, 36, 1, 2, 1, 8, 1, 1, 1, 1, 49, 1, 5, 1, 27, 1, 1, 1, 10, 3, 1, 1, 6, 1, 1, 1, 2, 1, 1, 11, 1, 1, 2, 7, 1, 125, 4, 3, 1, 1, 1, 3, 1, 100, 1, 1, 1, 216, 1, 1, 1, 4, 1, 1, 13

Offset: 1

Jon Awbrey, May 23 2005

The right diagonal labeled by the prime power of the form j:k = (prime(j))^k contains the j^th power primes in the factorization raised to the k^th power. For example, the right diagonal labeled by the number 2 = 1:1 = (prime(1))^1 contains the power-free parts of each positive integer, specifically A055231 and the right diagonal labeled by the number 4 = 1:2 = (prime(1))^2 contains the squares of the squarefree parts of positive integers.
In general, then the right diagonal labeled by m = (j_i : k_i)_i = Product_i prime(j_i)^(k_i) contains the product over i of the (j_i)th power primes in the factorization raised to the (k_i)th powers.
For example, the operator 5 = 3:1 extracts the 3rd power primes in the factorization of each n and raises them to the first power, thus sending 8 = 1:3 to 2 = 1:1, 27 = 2:3 to 3 = 2:1 and so on.

			` ` ` ` ` ` ` ` ` ` `n o m
` ` ` ` ` ` ` ` ` ` ` \ /
` ` ` ` ` ` ` ` ` ` `1 . 1
` ` ` ` ` ` ` ` ` ` \ / \ /
` ` ` ` ` ` ` ` ` `2 . 1 . 2
` ` ` ` ` ` ` ` ` \ / \ / \ /
` ` ` ` ` ` ` ` `3 . 1 . 1 . 3
` ` ` ` ` ` ` ` \ / \ / \ / \ /
` ` ` ` ` ` ` `4 . 1 . 2 . 1 . 4
` ` ` ` ` ` ` \ / \ / \ / \ / \ /
` ` ` ` ` ` `5 . 1 . 3 . 1 . 1 . 5
` ` ` ` ` ` \ / \ / \ / \ / \ / \ /
` ` ` ` ` `6 . 1 . 1 . 1 . 4 . 1 . 6
` ` ` ` ` \ / \ / \ / \ / \ / \ / \ /
` ` ` ` `7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
` ` ` ` \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` `8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
` ` ` \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` `9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
` ` \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
Primal codes of finite partial functions on positive integers:
1 = { }
2 = 1:1
3 = 2:1
4 = 1:2
5 = 3:1
6 = 1:1 2:1
7 = 4:1
8 = 1:3
9 = 2:2
10 = 1:1 3:1
11 = 5:1
12 = 1:2 2:1
13 = 6:1
14 = 1:1 4:1
15 = 2:1 3:1
16 = 1:4
17 = 7:1
18 = 1:1 2:2
19 = 8:1
20 = 1:2 3:1
From _Antti Karttunen_, Nov 16 2019: (Start)
When the sequence is viewed as a square array read by falling antidiagonals, the top left 15 X 15 corner looks like this:
k=  | 1  2   3  4    5    6    7  8  9    10    11  12    13    14    15
----+--------------------------------------------------------------------
n= 1| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
   2| 1, 2,  3, 1,   5,   6,   7, 1, 1,   10,   11,  3,   13,   14,   15,
   3| 1, 1,  1, 2,   1,   1,   1, 1, 3,    1,    1,  2,    1,    1,    1,
   4| 1, 4,  9, 1,  25,  36,  49, 1, 1,  100,  121,  9,  169,  196,  225,
   5| 1, 1,  1, 1,   1,   1,   1, 2, 1,    1,    1,  1,    1,    1,    1,
   6| 1, 2,  3, 2,   5,   6,   7, 1, 3,   10,   11,  6,   13,   14,   15,
   7| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
   8| 1, 8, 27, 1, 125, 216, 343, 1, 1, 1000, 1331, 27, 2197, 2744, 3375,
   9| 1, 1,  1, 4,   1,   1,   1, 1, 9,    1,    1,  4,    1,    1,    1,
  10| 1, 2,  3, 1,   5,   6,   7, 2, 1,   10,   11,  3,   13,   14,   15,
  11| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
  12| 1, 4,  9, 2,  25,  36,  49, 1, 3,  100,  121, 18,  169,  196,  225,
  13| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
  14| 1, 2,  3, 1,   5,   6,   7, 1, 1,   10,   11,  3,   13,   14,   15,
  15| 1, 1,  1, 2,   1,   1,   1, 2, 3,    1,    1,  2,    1,    1,    1,
(End)

Antti Karttunen, Table of n, a(n) for n = 1..25425; the first 225 antidiagonals of the array
J. Awbrey, Riffs and Rotes

Cf. A000040, A061396, A062504, A062537, A062860, A106178, A181819, A286561.
Rows 1 - 3: A000012, A055231, A329376.
Columns 1 - 2: A000012, A006519.

up_to = 105;
A106177sq(n,k) = { my(f = factor(k)); prod(i=1,#f~,f[i, 1]^valuation(n, prime(f[i, 2]))); };
A106177list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A106177sq(col,(a-(col-1))))); (v); };
v106177 = A106177list(up_to);
A106177(n) = v106177[n]; \\ Antti Karttunen, Nov 16 2019

If k = Product p_i^e_i, A(n,k) = p_i^A286561(n, A000040(e_i)), where A286561(x,y) gives the y-valuation of x. - Antti Karttunen, Nov 16 2019

A332785 Nonsquarefree numbers that are not squareful.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204, 207, 208, 212, 220, 224

Offset: 1

Bernard Schott, Feb 24 2020

Sometimes nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false. This sequence = A013929 \ A001694 and consists of these counterexamples.
This sequence is not a duplicate: the first 16 terms (<= 68) are the same first 16 terms of A059404, A323055, A242416 and A303946, then 72 is the 17th term of these 4 sequences. Also, the first 37 terms (<= 140) are the same first 37 terms of A317616 then 144 is the 38th term of this last sequence.
From Amiram Eldar, Sep 17 2023: (Start)
Called "hybrid numbers" by Jakimczuk (2019).
These numbers have a unique representation as a product of two numbers > 1, one is squarefree (A005117) and the other is powerful (A001694).
Equivalently, numbers k such that A055231(k) > 1 and A057521(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is equal to 1 and at least one exponent that is larger than 1.
The asymptotic density of this sequence is 1 - 1/zeta(2) (A229099). (End)

			18 = 2 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is not squareful because 2 divides 18 but 2^2 does not divide 18, hence 18 is a term.
72 = 2^3 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is also squareful because primes 2 and 3 divide 72, and 2^2 and 3^2 divide also 72, so 72 is not a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Powerful Numbers Multiple of a Set of Primes and Hybrid Numbers, 2019.

Cf. A005117 (squarefree), A013929 (nonsquarefree), A001694 (squareful), A052485 (not squareful).

Cf. A059404, A126706, A229099, A242416, A286708, A303946, A317616, A323055 (first terms are the same).

filter:= proc(n) local F;
 F:= ifactors(n)[2][..,2];
 max(F) > 1 and min(F) = 1
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2024

Select[Range[225], Max[(e = FactorInteger[#][[;;,2]])] > 1 && Min[e] == 1 &] (* Amiram Eldar, Feb 24 2020 *)

isok(m) = !issquarefree(m) && !ispowerful(m); \\ Michel Marcus, Feb 24 2020

from math import isqrt
from sympy import mobius, integer_nthroot
def A332785(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l, j = n-1+squarefreepi(integer_nthroot(x,3)[0])+squarefreepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

This sequence is A126706 \ A286708.

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(s)/zeta(2*s) - zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1. - Amiram Eldar, Sep 17 2023

A092261 Sum of unitary, squarefree divisors of n, including 1.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 1, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 4, 1, 42, 1, 8, 30, 72, 32, 1, 48, 54, 48, 1, 38, 60, 56, 6, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 3, 72, 8, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 1, 74, 114, 4, 20, 96, 168, 80

Offset: 1

Steven Finch, Feb 20 2004

Unitary convolution of the sequence of n*mu^2(n) (absolute values of A055615) and A000012. - R. J. Mathar, May 30 2011

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960) 66-80, sequence sigma'(n).
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

Cf. A000203, A003557, A007947, A048250, A055231, A056671, A057521, A065465, A295294, A295295, A329728.

Table[Plus @@ Select[Divisors@ n, Max @@ Last /@ FactorInteger@ # == 1 && GCD[#, n/#] == 1 &], {n, 1, 79}] (* Michael De Vlieger, Mar 08 2015 *)
f[p_, e_] := If[e==1, p+1, 1]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 79] (* Amiram Eldar, Mar 01 2019 *)

a(n) = sumdiv(n, d, d*issquarefree(d)*(gcd(d, n/d) == 1)); \\ Michel Marcus, Mar 06 2015

for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X^3 - p*X^2 - p^2*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

;; This implementation utilizes the memoization-macro definec for which an implementation is available at http://oeis.org/wiki/Memoization#Scheme
;; The other functions, A020639, A067029 and A028234 can be found under the respective entries, and should likewise defined with definec:
(definec (A092261 n) (if (= 1 n) 1 (* (+ 1 (if (> (A067029 n) 1) 0 (A020639 n))) (A092261 (A028234 n))))) ;; Antti Karttunen, Nov 25 2017

Multiplicative with a(p) = p+1 and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Feb 22 2004
From Álvar Ibeas, Mar 06 2015: (Start)
a(n) = a(A055231(n)) = A000203(A055231(n)).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(1-2s)).
(End)
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(A055231(n)).
a(n) = A000203(n) / A295294(n).
a(n) = A048250(n) / A295295(n) = A048250(n) / A048250(A057521(n)), where A057521(n) = A064549(A003557(n)).
(End)
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Jun 10 2020
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(2-3*s) - p^(1-2*s) - p^(2-2*s)). - Vaclav Kotesovec, Aug 20 2021
a(n) = Sum_{d|n, gcd(d,n/d)=1} d * mu(d)^2. - Wesley Ivan Hurt, May 26 2023

A108352 a(n) = primal code characteristic of n, which is the least positive integer, if any, such that (n o)^k = 1, otherwise equal to 0. Here "o" denotes the primal composition operator, as illustrated in A106177 and A108371 and (n o)^k = n o ... o n, with k occurrences of n.

Original entry on oeis.org

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

Offset: 1

Jon Awbrey, May 31 2005, revised Jun 01 2005

nonn

			a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
Detail of calculation for compositional powers of 12:
(12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
(12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
Detail of calculation for compositional powers of 20:
(20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
(20 o)^3 = (3:2) o (1:2 3:1) = 1.
From _Antti Karttunen_, Nov 20 2019: (Start)
For n=718, because 718 = prime(1)^1 * prime(72)^1, its partial function primal code is (1:1 72:1), which, when composed with itself stays same (that is, A106177(718,718) = 718), thus, as 1 is never reached, a(718) = 0, like is true for all even nonsquare semiprimes.
For n=1804, as 1804 = prime(1)^2 * prime(5)^1 * prime(13)^1, its primal code is (1:2 5:1 13:1), which, when composed with itself yields 203401 = prime(5)^2 * prime(13)^2, i.e., primal code (5:2 13:2), which when composed with (1:2 5:1 13:1) yields 1, which happened on the second iteration, thus a(1804) = 2+1 = 3.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..100000
Jon Awbrey, Riffs and Rotes
Jon Awbrey, Primal Code Characteristic, n = 1 to 3000 (Note: values given here differ at n = 718, 746, 1156, 1449, 1734 and 1804 from those computed in b-file). - _Antti Karttunen_, Nov 23 2019

Cf. A001747, A046315, A055231, A061396, A062504, A062537, A062860, A065091, A106177, A106178.

Cf. A108353, A108370, A108371, A108372, A108373, A108374, A111801.

A106177sq(n,k) = { my(f = factor(k)); prod(i=1,#f~,f[i, 1]^valuation(n, prime(f[i, 2]))); }; \\ As in A106177.
A108352(n) = { my(orgn=n,xs=Set([]), k=1); while(n>1, if(vecsearch(xs,n), return(0)); xs = setunion([n],xs); n = A106177sq(n,orgn); k++); (k); }; \\ Antti Karttunen, Nov 20 2019

a(A065091(n)) = 2 for all n, a(A001747(n)) = 0 for all n, except n=2, and a(A046315(n)) = 2 for n > 1. - Antti Karttunen, Nov 20 2019

Links and cross-references added, Aug 19 2005

Term a(63) corrected and five more terms added (up to a(105)) by Antti Karttunen, Nov 20 2019

A055773 a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.

Original entry on oeis.org

1, 1, 2, 6, 3, 15, 5, 35, 35, 35, 7, 77, 77, 1001, 143, 143, 143, 2431, 2431, 46189, 46189, 46189, 4199, 96577, 96577, 96577, 7429, 7429, 7429, 215441, 215441, 6678671, 6678671, 6678671, 392863, 392863, 392863, 14535931, 765049, 765049, 765049

Offset: 0

Labos Elemer, Jul 12 2000

nonn

Old name: Product of primes p for which p divides n! but p^2 does not (i.e. ord_p(n!)=1). - Dion Gijswijt (gijswijt(AT)science.uva.nl), Jan 07 2007
Squarefree part of n! divided by gcd(Q,F), where Q is the largest square divisor and F is the squarefree part of n!. - Labos Elemer, Jul 12 2000
a(1) = 1, a(n) = n*a(n-1) if n is a prime else a(n) = least integer multiple of a(n-1)/n. - Amarnath Murthy, Apr 29 2004
Let P(i) denote the primorial number A034386(i). Then a(n) = P(n)/P(floor(n/2)). - Peter Luschny, Mar 05 2011
Letting H(n) = 1 + 1/2 + ... + 1/n denote the n-th harmonic number, it is known that a(n) is equal to the denominator (in lowest terms) of H(n)^2*n! for n >= 6 (see below example). - John M. Campbell, Mar 27 2016
For all n satisfying 6 <= n < 897, a(n) = A130087(n). - John M. Campbell, Mar 27 2016
It is also known that a(n) is equal to lcm^2(1, 2, ..., n)/gcd(lcm^2(1, 2, ..., n), n!). - John M. Campbell, Apr 04 2016

			n = 13, P_n = {7, 11, 13}, a(13) = 7*11*13 = 1001.
Letting n = 14, the denominator (in lowest terms) of H(n)^2*n! = 131803989435744/143 is a(14)=143. - _John M. Campbell_, Mar 27 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200 (corrected by Michel Marcus, Jan 19 2019)
J. M. Campbell et al., A problem involving the product prod_{k=1..n} k^mu(k), where mu denotes the Möbius function, Mathematics Stack Exchange (2016).
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
Index entries for sequences related to Gijswijt's sequence

Cf. A000188, A008833, A007913, A055229, A055231 (for n), A055071, A055204, A055230, A094299, A094302, A193477, A130087.

a := n -> mul(k,k=select(isprime,[$iquo(n,2)+1..n])); # Peter Luschny, Jun 20 2009
A055773 := n -> numer(n!/iquo(n,2)!^4); # Peter Luschny, Jul 30 2011

Table[Numerator[n!/Floor[n/2]!^4], {n, 0, 40}] (* Michael De Vlieger, Mar 27 2016 *)

q=1;for(n=2,41,print1(q,",");q=if(isprime(n),q*n,q/gcd(q,n))) \\ Klaus Brockhaus, May 02 2004

a(n) = k=1;forprime(p=nextprime(n\2+1),precprime(n),k=k*p);k \\ Klaus Brockhaus, May 02 2004

a(n) = prod(i=primepi(n/2)+1,primepi(n),prime(i)) \\ John M. Campbell, Mar 27 2016

from math import prod
from sympy import primerange
def A055773(n): return prod(primerange((n>>1)+1,n+1)) # Chai Wah Wu, Apr 13 2024

a(n) = numerator(A056040(n)^2/n!).
a(n) = numerator(A056040(n)/floor(n/2)!^2).
a(n) = numerator(n!/floor(n/2)!^4). - Peter Luschny, Jul 30 2011
a(n) = product of primes p such that n/2 < p <= n. - Klaus Brockhaus, May 02 2004
a(n) = A055204(n)/A055230(n) = A055231(n!) = A007913(n!)/A055229(n!).
a(n) = Product_{i=pi(n/2)+1..pi(n)} prime(i), where pi denotes the prime counting function and prime(i) denotes the i-th prime number. - John M. Campbell, Mar 27 2016

Entry revised by N. J. A. Sloane, Jan 07 2007

Simpler definition based on a comment of Klaus Brockhaus, set offset to 0 and prepended 1 to data. - Peter Luschny, Mar 09 2013

cp's OEIS Frontend

A207359 Indices n, not squarefree, where A055231(n) = A055231(n-A055231(n)).

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A207360 Numbers n, not squarefree, satisfying A055231(n) = A055231(n + A055231(n)).

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A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

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A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=b*c^2*d^3 then a(n)=c^2*d^3 when b is minimized.

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A056169 Number of unitary prime divisors of n.

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A106177 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

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A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=bc^2d^3 then a(n)=c^2*d^3 when b is minimized.