A038538 -id:A038538 - cp's OEIS frontend

A369763 Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).

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

Offset: 0

Amiram Eldar, Jan 31 2024

The asymptotic mean of the ratio between the number of non-isomorphic abelian groups and the number of non-isomorphic semisimple rings of the same order.
The constant A in Kühleitner's paper (1995).
The ratio is 1 for all biquadratefree numbers (whose asymptotic density is A215267 = 0.923..., see A046100), and smaller than 1 otherwise.

			0.98771484004493763774023068670639349351901075670395...

Manfred Kühleitner, Comparing the number of Abelian groups and of semisimple rings of a given order, Mathematica Slovaca, Vol. 45, No. 5 (1995), pp. 509-518.

Cf. A000041, A000688, A004101, A038538, A046100, A215267.

default(realprecision, 120); my(N=512, x='x+O('x^N), v); v = Vec(1/prod(k=1, sqrtint(N)+1, prod(j=1, 1+N\k^2, 1-x^(j*k^2)))); prodeulerrat((1-1/p)*vecsum(vector(N, i, numbpart(i-1)/(v[i]*p^(i-1))))) \\ after Vaclav Kotesovec at A004101

Equals Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} A000041(k)/(A004101(k)*p^k)).

A046951 a(n) is the number of squares dividing n.

Original entry on oeis.org

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

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Rediscovered by the HR automatic theory formation program.
a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001
We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009
Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]
Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015
The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020
The number of unordered factorizations of n into cubefree powers of primes (1, primes and squares of primes, A166684). - Amiram Eldar, Jun 12 2025

			a(16) = 3 because the squares 1, 4, and 16 divide 16.
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Antonio Amariti, Claudius Klare, Domenico Orlando and Susanne Reffert, The M-theory origin of global properties of gauge theories, Nuclear Physics B, Vol. 901 (2015), pp. 318-337, arXiv preprint, arXiv:1507.04743 [hep-th], 2015 (see (A.13)).
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics
Ian G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).
Andrew V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.
Werner Georg Nowak and László Tóth, On the average number of subgroups of the group Z_m X Z_n, International Journal of Number Theory, Vol. 10, No. 2 (2014), pp. 363-374, arXiv preprint, arXiv:1307.1414 [math.NT], 2013.
N. J. A. Sloane, Transforms.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000188, A004101, A005117 (positions of 1's), A008619, A008833, A013936 (partial sums), A038538, A046952, A052304, A056595, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A130279, A156552, A166684, A278161, A307445 (Dirichlet inverse).
One more than A071325.
Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).
Sum of the k-th powers of the square divisors of n for k=0..10: this sequence (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: this sequence (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
Cf. A082293 (a(n)==2), A082294 (a(n)==3).

a046951 = sum . map a010052 . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[#[d: d in Divisors(n)|IsSquare(d)]:n in [1..120]]; // Marius A. Burtea, Jan 21 2020

A046951 := proc(n)
    local a,s;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*(1+floor(op(2,p)/2)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 17 2012
# Alternatively:
isbidivisible := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d:
a := n -> nops(select(k -> isbidivisible(n, k), [seq(1..n)])): # Peter Luschny, Jun 13 2025

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)
Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)
f[p_, e_] := 1 + Floor[e/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n)=my(f=factor(n));for(i=1,#f[,1],f[i,2]\=2);numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

a(n)=factorback(apply(e->e\2+1, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 17 2015

from math import prod
from sympy import factorint
def A046951(n): return prod((e>>1)+1 for e in factorint(n).values()) # Chai Wah Wu, Aug 04 2024

def is_bidivisible(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == d
def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_bidivisible(n, k)]
print([len(aList(n)) for n in range(1, 126)])  # Peter Luschny, Jun 13 2025

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))
(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))
;; Antti Karttunen, Nov 14 2016

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016
a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.
Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007
a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007
Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).
G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013
a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014
From Antti Karttunen, Nov 14 2016: (Start)
a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).
a(n) = A278161(A156552(n)). (End)
G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016
From  Antti Karttunen, Nov 12 2017: (Start)
a(n) = A000005(n) - A056595(n).
a(n) = 1 + A071325(n).
a(n) = 1 + A001222(A293515(n)). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A000005(d) * A008836(n/d). - Torlach Rush, Jan 21 2020
a(n) = A000005(sqrt(A008833(n))). - Amiram Eldar, Jul 07 2020
a(n) = Sum_{d divides n} mu(core(d)^2), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n).
For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796.
Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).

			First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.

Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.
Additive: A046660, A056170.
Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.
Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

&cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012

row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)

For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

A004101 Number of partitions of n of the form a_1b_1^2 + a_2b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197, 1549, 2025, 2600, 3377, 4306, 5523, 7000, 8922, 11235, 14196, 17777, 22336, 27825, 34720, 43037, 53446, 65942, 81423, 100033, 122991, 150481, 184149, 224449, 273614, 332291

Offset: 0

N. J. A. Sloane

The number of semisimple rings with p^n elements does not depend on the prime number p. - Paul Laubie, Mar 05 2024

			4 = 4*1^2 = 1*2^2 = 3*1^2 + 1*1^2 = 2*1^2 + 2*1^2 = 2*1^2 + 1*1^2 + 1*1^2 = 1*1^2 + 1*1^2 + 1*1^2 + 1*1^2.

J. Knopfmacher, Abstract Analytic Number Theory. North-Holland, Amsterdam, 1975, p. 293.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006 (section 6).
I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34.
I. G. Connell, Letter to N. J. A. Sloane, no date
N. J. A. Sloane, Transforms

Cf. A006171, A038538, A280451, A280661, A280662.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d* mul(1+iquo(i[2], 2),
      i=ifactors(d)[2]), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 26 2013
sqd:=proc(n) local t1,d; t1:=0; for d in divisors(n) do if (n mod d^2) = 0 then t1:=t1+1; fi; od; t1; end; # A046951
t2:=mul( 1/(1-x^n)^sqd(n),n=1..65); series(t2,x,60); seriestolist(%); # N. J. A. Sloane, Jun 24 2015

max = 45; A046951 = Table[Sum[Floor[n/k^2], {k, n}], {n, 0, max}] // Differences; f = Product[1/(1-x^n)^A046951[[n]], {n, 1, max}]; CoefficientList[Series[f, {x, 0, max}], x] (* Jean-François Alcover, Feb 11 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2017 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^(j^2))); Vec(gf) /* Joerg Arndt, May 03 2008 */

EULER transform of A046951.

a(n) ~ exp(Pi^2 * sqrt(n) / 3 + sqrt(3/(2*Pi)) * Zeta(1/2) * Zeta(3/2) * n^(1/4) - 9 * Zeta(1/2)^2 * Zeta(3/2)^2 / (16*Pi^3)) * Pi^(3/4) / (sqrt(2) * 3^(1/4) * n^(5/8)) [Almkvist, 2006]. - Vaclav Kotesovec, Jan 03 2017

More terms, formula and better description from Christian G. Bower, Nov 15 1999

Name clarified by Paul Laubie, Mar 05 2024

A339948 Number of non-isomorphic generalized quaternion rings over Z/nZ.

Original entry on oeis.org

1, 1, 4, 7, 4, 16, 4, 16, 10, 16, 4, 40, 4, 16, 16, 36, 4, 40, 4, 40, 16, 16, 4, 80, 10, 16, 20, 40, 4, 64, 4, 52, 16, 16, 16

Offset: 1

José María Grau Ribas, Dec 24 2020

Generalized quaternion rings over Z/nZ are of the form Z_n/(x^2-a, y^2-b, xy+yx).

			For n=2 all such rings are isomorphic to Z_n<x,y>/(x^2, y^2, xy+yx), so a(2)=1.
For n=4:
  Z_n<x,y>/(x^2,   y^2,   xy+yx),
  Z_n<x,y>/(x^2,   y^2-1, xy+yx),
  Z_n<x,y>/(x^2,   y^2-2, xy+yx),
  Z_n<x,y>/(x^2,   y^2-3, xy+yx),
  Z_n<x,y>/(x^2-1, y^2-1, xy+yx),
  Z_n<x,y>/(x^2-1, y^2-2, xy+yx),
  Z_n<x,y>/(x^2-3, y^2-3, xy+yx),
so a(4)=7.

José María Grau Ribas, Generalized quaternion rings over Z/nZ for an odd n
Jose María Grau, C. Miguel and A. M. Oller-Marcen, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017.
J. M. Grau, C. Miguel and A. M. Oller-Marcén, Generalized quaternion rings over Z/nZ for an odd n, Advances in Applied Clifford Algebras, 28(1), (2018) article 17.

Cf. A286779, A027623, A037221, A127708, A037291, A127707, A038538, A037289, A037290, A038036.

Clear[phi]; phi[1] = phi[2] = 1; phi[4] = 7; phi[8] = 16;
phi[16] = 36; phi[p_, s_] := 2 s^2 + 2;
phi[n_] :=  Module[{aux = FactorInteger[n]},Product[phi[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];
Table[phi[i], {i,1, 35}]

If n is odd then a(n) = A286779(n).

A070932 Possible number of units in a finite (commutative or non-commutative) ring.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 36, 40, 42, 44, 45, 46, 48, 49, 52, 54, 56, 58, 60, 62, 63, 64, 66, 70, 72, 78, 80, 81, 82, 84, 88, 90, 92, 93, 96, 98, 100

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002

This is a list of the numbers of units in R where R ranges over all finite commutative or non-commutative rings.
By considering the ring Z_n and the finite fields GF(q) this sequence contains the values of the Euler function phi(n) (A000010) and prime powers - 1 (A181062). By taking direct product of rings, if n and m belong to the sequence then so does m*n.
Eric M. Rains has shown that these rules generate all terms of this sequence. More precisely, he shows this sequence (with 0 removed) is the multiplicative monoid generated by all numbers of the form q^n-q^{n-1} for n >= 1 and q a prime power (see Rains link).
Since the number of units of F_q[X]/(X^n) is q^n - q^(n-1), restricting to finite commutative rings gives the same sequence. A296241, which is a proper supersequence, allows the ring R to be infinite. - Jianing Song, Dec 24 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. M. Rains, Comments on A070932

Cf. A000010, A002202, A000252, A000961, A181062, A221178.
A000252 is a subsequence.
A282572 is the subsequence of odd terms.
Proper subsequence of A296241.
The main entries concerned with the enumeration of rings are A027623, A037234, A037291, A037289, A038538, A186116.

max = 100; A000010 = EulerPhi[ Range[2*max]] // Union // Select[#, # <= max &] &; A181062 = Select[ Range[max], Length[ FactorInteger[#]] == 1 &] - 1; FixedPoint[ Select[ Outer[ Times, #, # ] // Flatten // Union, # <= max &] &, Union[A000010, A181062] ] (* Jean-François Alcover, Sep 10 2013 *)

list(lim)=my(P=1, q, v, u=List()); forprime(p=2, default(primelimit), if(eulerphi(P*=p)>=lim, q=p; break)); v=vecsort(vector(P/q*lim\eulerphi(P/q), k, eulerphi(k)), , 8); v=select(n->n<=lim, v); forprime(p=2, sqrtint(lim\1+1), P=p; while((P*=p) <= lim+1, listput(u, P-1))); v=vecsort(concat(v, Vec(u)), , 8); u=List([0]); while(#u, v=vecsort(concat(v, Vec(u)),,8); u=List(); for(i=3,#v, for(j=i,#v,P=v[i]*v[j]; if(P>lim,break); if(!vecsearch(v, P), listput(u, P))))); v \\ Charles R Greathouse IV, Jan 08 2013

Entry revised by N. J. A. Sloane, Jan 06 2013, Jan 08 2013

Definition clarified by Jianing Song, Dec 24 2021

A244285 Decimal expansion of A1B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{rm^2 > 1} zeta(r*m^2).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 25 2014

The asymptotic mean of A038538. - Amiram Eldar, Jan 31 2024

			2.499616112936298274932373821756498034570402258807595443206210948121224368...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.

Cf. A021002 (A1), A038538, A123030.

Digits := 200: z:=product(Zeta(1.0*j), j = 2..1000): for k from 10 by 10 to 50 do print(z*product(product(Zeta(1.0*r*m^2), r = 1..k^2), m = 2..k)); end do; # Vaclav Kotesovec, Jun 11 2020

digits = 20; digitsPlus = 100; n0 = 50; dn = 1; A1 = NProduct[Zeta[m], {m, 2, Infinity}, WorkingPrecision -> digitsPlus]; Clear[B1]; B1[n_] := B1[n] = NProduct[Zeta[r*m^2], {r, 1, n}, {m, 2, n}, WorkingPrecision -> digitsPlus]; B1[n0]; B1[n = n0 + dn]; While[ RealDigits[B1[n], 10, digitsPlus] != RealDigits[B1[n - dn], 10, digitsPlus], Print["n = ", n]; n = n + dn]; RealDigits[A1*B1[n], 10, digits] // First

prodinf(m = 2, zeta(m)) * prodinf(r = 1, prodinf(m = 2, zeta(r*m^2))) \\ Amiram Eldar, Jan 31 2024

More digits from Vaclav Kotesovec, Jun 11 2020

A370360 Number of labeled semisimple rings with n elements.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 24409921536000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000

Offset: 1

Paul Laubie, Mar 05 2024

nonn

Using the Artin-Wedderburn theorem, a finite semisimple ring is a product of matrix algebras over finite field. In particular, if n is squarefree then any semisimple ring of cardinal n is commutative. One can be more precise, indeed all semisimple rings with n elements are commutative if and only if the only 4th power that divides n is 1.
The analogous sequences for abelian groups and cyclic groups are A034382 and A034381, respectively.
In the case of commutative semisimple rings, we get the factorial numbers.

			For n=4, we have two possible rings: F_4 and F_2 X F_2. We use the notation F_q to denote the finite field with q elements. To compute a(4) we need to know how many ring automorphisms F_4 and F_2 X F_2 admit. For F_4, we have that Aut(F_4) is generated by the Frobenius morphism, hence we have 2 automorphisms. For F_2 X F_2, the only nontrivial automorphism is exchanging the two coordinates, hence we also have 2 automorphisms. Hence:
a(4) = 24/2 + 24/2 = 24.
We can compute a(2^k) for some small values of k:
a(4) = 4! = 24,
a(8) = 8!,
a(16) = 16! + 16!/6,
a(32) = 32! + 32!/6,
a(64) = 64! + 64!/12 + 64!/12,
a(128) = 128! + 128!/36 + 128!/18 + 128!/12,
...

Paul Laubie, A Github repository with a code to compute the terms of the form a(p^n).

Cf. A000142, A005117, A038538.

Cf. A034382, A034381.

If n is squarefree then we have a(n) = n!. More precisely, a(n) = n! if and only if the only 4th power that divides n is 1. In particular, n=16 is the smallest n such that a(n) is different from n!.

If n and m are relatively prime, then a(n*m) = (n*m)!*a(n)*a(m)/(n!*m!).

cp's OEIS Frontend

A123030 Partial sums of A038538.

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A052305 Number of semisimple rings with A025487(n) elements: a(n) = A038538(A025487(n)).

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A369763 Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).

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A046951 a(n) is the number of squares dividing n.

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A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

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A004101 Number of partitions of n of the form a_1*b_1^2 + a_2*b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

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A339948 Number of non-isomorphic generalized quaternion rings over Z/nZ.

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A004101 Number of partitions of n of the form a_1b_1^2 + a_2b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

A244285 Decimal expansion of A1B1, the average number of non-isomorphic semisimple rings of any order, where A1 is Product_{m>1} zeta(m) and B1 is Product_{rm^2 > 1} zeta(r*m^2).