A000188 -id:A000188 - cp's OEIS frontend

A059592 Square-full part of n^2 + 1.

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 17, 1, 1, 29, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 13, 1, 1, 1

Offset: 0

Marc LeBrun, Jan 25 2001

Related to period-1 continued fractions [z,z,z,...].

A124808 gives number of numbers m <= n with a(m)=1. - Reinhard Zumkeller, Nov 08

			a(7)=5 since 7^2 + 1 = 50 = 25*2 = (5^2)*2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 (corrected by _Sean A. Irvine_)

Cf. A000188, A059591, A007913, A002522.

a(n) = core(n^2+1, 1)[2]; \\ Michel Marcus, Nov 12 2023

a(n) = A000188(A002522(n)).

a(n)^2 * A059591(n) = n^2 + 1.

A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

From Gus Wiseman, Aug 05 2021 and Antti Karttunen, Oct 13 2021: (Start)
Also the product of primes at even positions in the weakly decreasing list (with multiplicity) of prime factors of n. For example, the prime factors of 108 are (3,3,3,2,2), with even bisection (3,2), with product 6, so a(108) = 6.
Proof: A108951(n) gives a number with the same largest prime factor (A006530) and its exponent (A071178) as in n, and with each smaller prime p = 2, 3, 5, 7, ... < A006530(n) having as its exponent the partial sum of the exponents of all prime factors >= p present in n (with primes not present in n having the exponent 0). Then applying A000188 replaces each such "partial sum exponent" k with floor(k/2). Finally, A319626 replaces those halved exponents with their first differences (here the exponent of the largest prime present stays intact, because the next larger prime's exponent is 0 in n). It should be easy to see that if prime q is not present in n (i.e., does not divide it), then neither it is present in a(n). Moreover, if the partial sum exponent of q is odd and only one larger than the partial sum exponent of the next larger prime factor of n, then q will not be present in a(n), while in all other cases q is present in a(n). See also the last example.
(End)

			From _Gus Wiseman_, Aug 15 2021: (Start)
The list of all numbers with image 12 and their corresponding prime factors begins:
  144: (3,3,2,2,2,2)
  216: (3,3,3,2,2,2)
  240: (5,3,2,2,2,2)
  288: (3,3,2,2,2,2,2)
  336: (7,3,2,2,2,2)
  360: (5,3,3,2,2,2)
(End)
The positions from the left are indexed as 1, 2, 3, ..., etc, so e.g., for 240 we pick the second, the fourth and the sixth prime factor, 3, 2 and 2, to obtain a(240) = 3*2*2 = 12. For 288, we similarly pick the second (3), the fourth (2) and the sixth (2) to obtain a(288) = 3*2*2 = 12. - _Antti Karttunen_, Oct 13 2021
Consider n = 11945934 = 2*3*3*3*7*11*13*13*17. Its primorial inflation is A108951(11945934) = 96478365991115908800000 = 2^9 * 3^8 * 5^5 * 7^5 * 11^4 * 13^3 * 17^1. Applying A000188 to this halves each exponent (floored down if the exponent is odd), leaving the factors 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 = 2497294800. Then applying A319626 to this number retains the largest prime factor (and its exponent), and subtracts from the exponent of each of the rest of primes the exponent of the next larger prime, so from 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 we get 2^(4-4) * 3^(4-2) * 5^(2-2) * 7^(2-2) * 11^(2-1) * 13^1 = 3^2 * 11^1 * 13^1 = 1287 = a(11945934), which is obtained also by selecting every second prime from the list [17, 13, 13, 11, 7, 3, 3, 3, 2] and taking their product. - _Antti Karttunen_, Oct 15 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to Heinz numbers
Index entries for sequences related to primorial numbers

Cf. A000188, A108951, A319626, A329602, A329900.
A left inverse of A000290.
Positions of 1's are A008578.
Positions of primes are A168645.
The sum of prime indices of a(n) is A346700(n).
The odd version is A346701.
The odd non-reverse version is A346703.
The non-reverse version is A346704.
The version for standard compositions is A346705, odd A346702.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A001414 adds up prime factors, row sums of A027746.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A346633 adds up the even bisection of standard compositions.
A346698 adds up the even bisection of prime indices.
Cf. A000097, A035363, A102750, A236913, A253553, A344606, A344617, A344653, A345957, A345958, A345959.

Table[Times@@Last/@Partition[Reverse[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]],2],{n,100}] (* Gus Wiseman, Oct 13 2021 *)

PARI
```
A329888(n) = A329900(A329602(n));
```

A329888(n) = if(1==n,n,my(f=factor(n),m=1,p=0); forstep(k=#f~,1,-1,while(f[k,2], m *= f[k,1]^(p%2); f[k,2]--; p++)); (m)); \\ (After Wiseman's new interpretation) - Antti Karttunen, Sep 21 2021

a(n) = A329900(A329602(n)) = A329900(A000188(A108951(n))).
A108951(a(n)) = A329602(n).
a(n^2) = n for all n >= 1.
a(n) * A346701(n) = n. - Gus Wiseman, Aug 07 2021
A056239(a(n)) = A346700(n). - Gus Wiseman, Aug 07 2021
Antti Karttunen, Sep 21 2021
From Antti Karttunen, Oct 13 2021: (Start)
a(n) = A319626(A329602(n)) = A319626(A000188(A108951(n))).
For all x in A102750, a(x) = a(A253553(x)). (End)

Name amended with Gus Wiseman's new interpretation - Antti Karttunen, Oct 13 2021

A055993 Number of square divisors of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 6, 8, 12, 30, 30, 36, 36, 72, 96, 128, 128, 180, 180, 300, 300, 600, 600, 864, 1152, 2304, 2688, 4368, 4368, 5376, 5376, 6144, 6144, 13056, 16320, 19440, 19440, 38880, 43200, 48000, 48000, 64000, 64000, 100800, 133056, 278784, 278784

Offset: 1

Labos Elemer, Jul 21 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000142, A000005, A046951, A055772, A000188.

Table[Count[Divisors[n!], d_ /; IntegerQ@ Sqrt@ d], {n, 30}] (* Michael De Vlieger, Feb 05 2017 *)

a(n) = sumdiv(n!, d, issquare(d)); \\ Michel Marcus, Dec 26 2013

a(n) = A046951(n!) = A000005(A055772(n)) = A000005(A000188(A000142(n))).

A056060 The powerfree part of the central binomial coefficients.

Original entry on oeis.org

1, 2, 3, 6, 10, 5, 35, 70, 14, 7, 462, 231, 429, 429, 715, 1430, 24310, 12155, 92378, 46189, 88179, 88179, 1352078, 676039, 52003, 52003, 7429, 7429, 1077205, 1077205, 33393355, 66786710, 43214930, 21607465, 181502706, 90751353, 176726319, 176726319, 7658140490

Offset: 1

Labos Elemer, Jul 26 2000

nonn

			n=14, binomial(14,7) = 3432 = 8*3*11*13. The largest square divisor is 4, and the squarefree part is 858. So GCD(4,858) = 2 and a(14) = 858/2 = 429.

Amiram Eldar, Table of n, a(n) for n = 1..3388

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974, A056056, A056057, A056058, A056059, A056060, A056061.

a[n_] := Denominator[(b = Binomial[n, Floor[n/2]])/(Times @@ First /@ FactorInteger[b])^2]; Array[a, 36] (* Amiram Eldar, Sep 05 2020 *)

a(n) = A055231(A001405(n)).

New name and more terms from Amiram Eldar, Sep 05 2020

A078430 Sum of gcd(k^2,n) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 5, 10, 9, 15, 13, 28, 33, 27, 21, 50, 25, 39, 45, 88, 33, 99, 37, 90, 65, 63, 45, 140, 145, 75, 153, 130, 57, 135, 61, 240, 105, 99, 117, 330, 73, 111, 125, 252, 81, 195, 85, 210, 297, 135, 93, 440, 385, 435, 165, 250, 105, 459, 189, 364, 185, 171, 117, 450, 121

Offset: 1

Vladeta Jovovic, Dec 30 2002

a(n) is the number of non-congruent solutions to x^2*y = 0 mod n. - Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 17 2003

Row sums of triangle A245717. - Reinhard Zumkeller, Jul 30 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
E. Krätzel, W. G. Nowak, and L. Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 10 (2012), 761-774.
M. Kühleitner and W. G. Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions , arXiv: 1204.1146 [math.NT], 2012.
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.

Cf. A018804, A069097, A069193.

Cf. A245717.

a078430 = sum . a245717_row  -- Reinhard Zumkeller, Jul 30 2014

Table[Sum[GCD[k^2,n],{k,n}],{n,70}] (* Harvey P. Dale, Sep 29 2014 *)
f[p_, e_] := If[EvenQ[e], p^(3*e/2) + p^(3*e/2 - 1), 2*p^((3*e - 1)/2)] - p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sum(k=1,n, gcd(k^2, n)); \\ Michel Marcus, Aug 03 2016

a(n) is multiplicative. G.f. for a(p^n), p a prime, is given by (1+(p-1)*x-p^2*x^2)/(1-p*x)/(1-p^3*x^2).
a(n) = n*Sum_{d|n} phi(d)*N(d)/d, where phi is Euler's totient function A000010 and N(n) is sequence A000188. - Laszlo Toth, Apr 15 2012
Multiplicative with a(p^e) = p^(3*e/2) + p^(3*e/2-1) - p^(e-1) if e is even, and 2*p^((3*e-1)/2) - p^(e-1) if e is odd. - Amiram Eldar, Apr 28 2023

A240976 Decimal expansion of 3zeta(3)/(2Pi^2), a constant appearing in the asymptotic evaluation of the average LCM of two integers chosen independently from the uniform distribution [1..n].

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 07 2014

15*zeta(3)/Pi^2 = 10 * (this constant) equals the asymptotic mean of the abundancy index of the squares (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			0.18269074235035962468150918282692865988200290126984361751783...

Persi Diaconis and Paul Erdős, On the distribution of the greatest common divisor, Technical Report No. 12 (1977) U.S. Army Research Office.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 17.
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (4).
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv:1310.7053 [math.NT], 2013-2014, formula (47), p. 23.

Cf. A000188, A002117, A013662, A048250, A059956.

RealDigits[3*Zeta[3]/(2*Pi^2), 10, 102] // First

Equals zeta(3)/(4*zeta(2)) = 3*zeta(3)/(2*Pi^2).
From Amiram Eldar, Jan 25 2024: (Start)
Equals (1/10) * Sum_{k>=1} A000188(k)/k^2.
Equals (1/10) * Sum_{k>=1} A048250(k)/k^3. (End)

A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 4, 4, 2, 10, 11, 4, 12, 7, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 10, 2, 13, 2, 7, 9, 4, 2, 16, 17, 18, 4, 7, 2, 19, 4, 10, 4, 4, 2, 20, 2, 4, 9, 21, 4, 13, 2, 7, 4, 13, 2, 22, 2, 4, 18, 7, 4, 13, 2, 16, 23, 4, 2, 20, 4, 4, 4, 10, 2, 24, 4, 7, 4, 4, 4, 25, 2, 26, 9, 27, 2, 13, 2, 10, 13, 4, 2, 28, 2, 13

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

Restricted growth sequence transform of A291757, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291757(i) = A291757(j) <=> A003557(i) = A003557(j) and A046523(i) = A046523(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
Also the restricted growth sequence transform of A294876, Product_{d|n, d>1} prime(gcd(d,n/d)). (This was the original definition).
For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A319347(i) = A319347(j) => a(i) = a(j),
  a(i) = a(j) => A055155(i) = A055155(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. also A291751, A291757, A294876, A295300, A295888, A319347, A322021.

Cf. A000188, A055155, A294897, A295666, A322020 (a few of the matched sequences).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };
v294877 = rgs_transform(vector(up_to,n,A294876(n)));
A294877(n) = v294877[n];

A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v294877 = rgs_transform(vector(up_to,n,[A003557(n),A046523(n)]));
A294877(n) = v294877[n]; \\ Antti Karttunen, Nov 28 2018

Name changed and comments added by Antti Karttunen, Nov 28 2018

A337533 1 together with nonsquares whose square part's square root is in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68

Offset: 1

Peter Munn, Aug 31 2020

The appearance of a number is determined by its prime signature.
Every squarefree number is present, as the square root of the square part of a squarefree number is 1. Other 4th-power-free numbers are present if and only if they are nonsquare.
If the square part of nonsquarefree k is a 4th power, k does not appear.
Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k > 1 is in this sequence if and only if the members of S_k occur in consecutive columns of A329050, starting with column 0.
If the qualifying condition in the previous paragraph was based on the rows instead of the columns of A329050, we would get A055932. The self-inverse function defined by A225546 transposes A329050. A225546 also has multiplicative properties such that if we consider A055932 and this sequence as sets, A225546(.) maps the members of either set 1:1 onto the other set.

			4 is square and not 1, so 4 is not in the sequence.
12 = 3 * 2^2 is nonsquare, and has square part 4, whose square root (2) is in the sequence. So 12 is in the sequence.
32 = 2 * 4^2 is nonsquare, but has square part 16, whose square root (4) is not in the sequence. So 32 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square part
Index to sequences related to prime signature

Complement of A337534.
Closed under A000188(.).
A209229, A267116 are used in a formula defining this sequence.
Subsequences: A005117, A036537, A179646, A179666, A179669, A189987, A191554, A191555, A298207, A317090.
Subsequence of A164514.
A007913, A008833, A008835, A335324 give the squarefree, square and comparably related parts of a number.
Cf. A050376, A329050.
Related to A055932 via A225546.

S:= {1}:
for n from 2 to 100 do
  if not issqr(n) then
    F:= ifactors(n)[2];
    s:= mul(t[1]^floor(t[2]/2),t=F);
    if member(s,S) then S:= S union {n} fi
  fi
od:
sort(convert(S,list)); # Robert Israel, Jan 07 2025

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[100], # == 1 || pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)

Numbers m such that A209229(A267116(m) + 1) = 1.
If A008835(a(n)) > 1 then A335324(a(n)) > 1.
If A008833(a(n)) > 1 then A007913(a(n)) > 1.

A352780 Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14

Offset: 1

Antti Karttunen and Peter Munn, Apr 02 2022

This is well-defined because positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2.
Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.
For all k, column k is column k+1 of A060176 conjugated by A225546.

			The top left corner of the array:
  n/k |   0   1   2   3   4   5   6
------+------------------------------
    1 |   1,  1,  1,  1,  1,  1,  1,
    2 |   2,  1,  1,  1,  1,  1,  1,
    3 |   3,  1,  1,  1,  1,  1,  1,
    4 |   1,  4,  1,  1,  1,  1,  1,
    5 |   5,  1,  1,  1,  1,  1,  1,
    6 |   6,  1,  1,  1,  1,  1,  1,
    7 |   7,  1,  1,  1,  1,  1,  1,
    8 |   2,  4,  1,  1,  1,  1,  1,
    9 |   1,  9,  1,  1,  1,  1,  1,
   10 |  10,  1,  1,  1,  1,  1,  1,
   11 |  11,  1,  1,  1,  1,  1,  1,
   12 |   3,  4,  1,  1,  1,  1,  1,
   13 |  13,  1,  1,  1,  1,  1,  1,
   14 |  14,  1,  1,  1,  1,  1,  1,
   15 |  15,  1,  1,  1,  1,  1,  1,
   16 |   1,  1, 16,  1,  1,  1,  1,
   17 |  17,  1,  1,  1,  1,  1,  1,
   18 |   2,  9,  1,  1,  1,  1,  1,
   19 |  19,  1,  1,  1,  1,  1,  1,
   20 |   5,  4,  1,  1,  1,  1,  1,

Antti Karttunen, Table of n, a(n) for n = 1..33153; the first 257 antidiagonals

Sequences used in a formula defining this sequence: A000188, A007913, A060176, A225546.
Cf. A007913 (column 0), A335324 (column 1).
Range of values: {1} U A340682 (whole table), A005117 (column 0), A062503 (column 1), {1} U A113849 (column 2).
Row numbers of rows:
- with a 1 in column 0: A000290\{0};
- with a 1 in column 1: A252895;
- with a 1 in column 0, but not in column 1: A030140;
- where every 1 is followed by another 1: A337533;
- with 1's in all even columns: A366243;
- with 1's in all odd columns: A366242;
- where every term has an even number of distinct prime factors: A268390;
- where every term is a power of a prime: A268375;
- where the terms are pairwise coprime: A138302;
- where the last nonunit term is coprime to the earlier terms: A369938;
- where the last nonunit term is a power of 2: A335738.
Number of nonunit terms in row n is A331591(n); their positions are given (in reversed binary) by A267116(n); the first nonunit is in column A352080(n)-1 and the infinite run of 1's starts in column A299090(n).

up_to = 105;
A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);
A352780list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, forstep(col=a-1,0,-1, i++; if(i > up_to, return(v)); v[i] = A352780sq(a-col,col))); (v); };
v352780 = A352780list(up_to);
A352780(n) = v352780[n];

A(n,0) = A007913(n); for k > 0, A(n,k) = A(A000188(n), k-1)^2.
A(n,k) = A225546(A060176(A225546(n), k+1)).
A331591(A(n,k)) <= 1.

A055230 Greatest common divisor of largest square dividing n! and squarefree part of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 6, 10, 10, 10, 5, 5, 1, 21, 42, 42, 7, 7, 14, 42, 6, 6, 5, 5, 10, 330, 165, 231, 231, 231, 462, 2002, 5005, 5005, 4290, 4290, 390, 78, 39, 39, 13, 13, 26, 1326, 102, 102, 17, 935, 13090, 746130, 373065, 373065, 24871, 24871

Offset: 1

Labos Elemer, Jun 21 2000

nonn

			a(5) = 2 because 5! = 120; largest square divisor is 4, squarefree part is 30; GCD(4, 30) = 2.
a(7) = 1 because 7! = 5040; the largest square divisor is 144 and the squarefree part is 35 and these are coprime.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A008833, A007913, A000188, A000720.

Table[GCD[Times @@ Flatten@ Map[Table[#1, 2 Floor[#2/2]] & @@ # &, #], Times @@ Flatten@ Map[Table[#1, Floor[Mod[#2, 2]]] & @@ # &, #]] &@ FactorInteger[n!], {n, 61}] (* Michael De Vlieger, Jul 26 2016 *)

a(n) = my(fn=n!, cn=core(fn)); gcd(cn, fn/cn); \\ Michel Marcus, Dec 10 2013

a(n) = GCD(A008833(n!), A007913(n!)) = GCD(A055071(n), A055204(n)).

cp's OEIS Frontend

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A240976 Decimal expansion of 3*zeta(3)/(2*Pi^2), a constant appearing in the asymptotic evaluation of the average LCM of two integers chosen independently from the uniform distribution [1..n].

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A240976 Decimal expansion of 3zeta(3)/(2Pi^2), a constant appearing in the asymptotic evaluation of the average LCM of two integers chosen independently from the uniform distribution [1..n].