A181062 -id:A181062 - cp's OEIS frontend

A341154 Number of partitions of 2*n into exactly n prime powers (including 1).

1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 71, 94, 116, 151, 182, 234, 282, 354, 424, 528, 627, 773, 914, 1113, 1311, 1585, 1854, 2227, 2599, 3095, 3597, 4262, 4931, 5811, 6704, 7855, 9035, 10542, 12080, 14036, 16047, 18561, 21161, 24397, 27736, 31866

Offset: 0

Ilya Gutkovskiy, Feb 06 2021

nonn

Cf. A000961, A023893, A181062, A280954, A341153.

nmax = 48; CoefficientList[Series[Product[1/(1 - Boole[PrimePowerQ[k + 1]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Boole[PrimePowerQ[d + 1]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

G.f.: Product_{p prime, k>=1} 1 / (1 - x^(p^k-1)).

A281947 Smallest prime p such that p^i - 1 is a totient (A002202) for all i = 1 to n, or 0 if no such p exists.

Original entry on oeis.org

2, 3, 7, 7, 37, 37, 113, 113, 241, 241, 241, 241, 241, 241, 241, 241, 241, 241, 2113, 2113, 2113, 2113, 2113, 2113, 3121, 3121, 3121, 3121

Offset: 1

Altug Alkan, Feb 03 2017

p - 1 = phi(p) is a totient for all primes p.

If A281909(n) is prime, then a(n) = A281909(n).

			a(3) = 7 because 7^2 - 1 = 48, 7^3 - 1 = 342 are both totient numbers (A002202) and 7 is the least prime number with this property.

Cf. A000010, A002202, A181062, A281909.

isok(p, n)=for (i=1, n, if (! istotient(p^i-1), return(0));); 1;
a(n) = {my(p=2); while (! isok(p, n), p = nextprime(p+1)); p;} \\ Michel Marcus, Feb 04 2017

a(19) from Michel Marcus, Feb 04 2017

a(20)-a(28) from Ray Chandler, Feb 08 2017

A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

Original entry on oeis.org

6, 48, 168, 180, 480, 2016, 3528, 5760, 11232, 13200, 20160, 26208, 61200, 78336, 123120, 181440, 267168, 374400, 511056, 682080, 892800, 1014816, 1488000, 1822176, 2755200, 3337488, 4773696, 5644800, 7738848, 9999360, 11908560, 13615200, 16511040, 19845936, 24261120, 25048800, 28003968

Offset: 1

Bernard Schott, Jun 04 2020

nonn

GL(m,q) is the general linear group, the group of invertible m X m matrices over the finite field F_q with q = p^k elements.
By definition, all fields must contain at least two distinct elements, so q >= 2. As GL(1,q) is isomorphic to F_q*, the multiplicative group [whose order is p^k-1 (A181062)] of finite field F_q, data begins with m >= 2.
Some isomorphisms (let "==" denote "isomorphic to"):
a(1) = 6 for GL(2,2) == PSL(2,2) == S_3.
a(2) = 48 for GL(2,3) that has 55 subgroups.
a(3) = 168 for GL(3,2) == PSL(2,7) [A031963].
a(11) = 20160 for GL(4,2) == PSL(4,2) == Alt(8).
Array for order of GL(m,q) begins:
=============================================================
m\q  |      2         3       4=2^2             5         7
-------------------------------------------------------------
  2  |      6        48         180           480      2016
  3  |    168     11232      181440       1488000  33784128
  4  |  20160  24261120  2961100800  116064000000   #GL(4,7)
  5  |9999360  #GL(5,3)      ...           ...         ...

			a(1) = #GL(2,2) = (2^2-1)*(2^2-2) = 3*2 = 6 and the 6 elements of GL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 invertible matrices with entries in F_2:
  (1 0)   (1 1)   (1 0)   (0 1)   (0 1)   (1 1)
  (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.

Wikipedia, General linear group

Cf. A059238 [GL(2,q)].
Cf. A002884 [GL(m,2)], A053290 [GL(m,3)], A053291 [GL(m,4)], A053292 [GL(m,5)], A053293 [GL(m,7)], A052496 [GL(m,8)], A052497 [GL(m,9)], A052498 [GL(m,11)].
Cf. A316622 [GL(n,Z_k)].

#GL(m,q) = Product_{k=0..m-1}(q^m-q^k).

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A341154 Number of partitions of 2*n into exactly n prime powers (including 1).

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A281947 Smallest prime p such that p^i - 1 is a totient (A002202) for all i = 1 to n, or 0 if no such p exists.

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A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

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