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%I #55 Sep 02 2023 21:38:52
%S 1,1,1,2,1,1,5,2,1,1,14,5,2,1,1,51,15,5,2,1,1,267,67,15,5,2,1,1,2328,
%T 504,77,15,5,2,1,1,56092,9310,684,83,15,5,2,1,1,10494213,1396077,
%U 34297,860,87,15,5,2,1,1,49487367289,5937876645
%N Square array, read by antidiagonals, upwards: T(n,k) is the number of groups of order prime(k+1)^n.
%C In 1960, Higman conjectured that the function f(n,p) giving the number of groups of prime-power order p^n, for fixed n and varying p, is a "Polynomial in Residue Classes" (PORC), i.e., there exist an integer M and polynomials q_i(x) in Z[x] (i = 1, 2, ..., M) such that if p = i mod M, then f(n,p) = q_i(p). The conjecture is confirmed for n <= 7.
%H H. U. Besche, B. Eick, and E. A. O'Brien. <a href="http://dx.doi.org/10.1142/S0218196702001115">A Millennium Project: Constructing Small Groups</a>, Internat. J. Algebra and Computation, 12 (2002), 623-644.
%H David Burrell, <a href="https://doi.org/10.1080/00927872.2021.2006680">On the number of groups of order 1024</a>, Communications in Algebra, 2021, 1-3.
%H David Burrell, <a href="https://doi.org/10.1080/00927872.2023.2169706">The number of p-groups of order 19,683 and new lists of p-groups</a>, Communications in Algebra, Vol. 51 - Issue 6 (2023), 2673-2679.
%H Heiko Dietrich, <a href="http://users.monash.edu/~heikod/icts2016/CPGmain.pdf">Computational aspects of finite p-groups</a>
%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Groups_of_prime_power_order">Groups of prime power order</a>
%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Higman%27s_PORC_conjecture">Higman's PORC conjecture</a>
%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/PORC_function">PORC function</a>
%H Graham Higman, <a href="https://doi.org/10.1112/plms/s3-10.1.24">Enumerating p-Groups. I: Inequalities</a>, Proc. London Math. Soc. Vol. 10 (1960), 24-30.
%H Graham Higman, <a href="https://doi.org/10.1112/plms/s3-10.1.566">Enumerating p-Groups. II: Problem whose solution is PORC</a>, Proc. London Math. Soc. Vol. 10 (1960), 566-582.
%H Eamonn O'Brien, <a href="https://www.icts.res.in/sites/default/files/1284624569notes.pdf">Polycyclic groups</a>
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubcay/">Numbers of Small Groups</a>
%H Michael Vaughan-Lee, <a href="https://dx.doi.org/10.1365/s13291-012-0039-x">Graham Higman’s PORC Conjecture</a>, Jahresbericht der Deutschen Mathematiker-Vereinigung Vol. 114 (2012), 89-16.
%H Michael Vaughan-Lee, <a href="http://dx.doi.org/10.22108/ijgt.2015.5758">Groups of order p^8 and exponent p</a>, International Journal of Group Theory Vol. 4 (2015), 25-42.
%H Brett E. Witty, <a href="https://www.brettwitty.net/pages/phd.html">Enumeration of groups of prime-power order</a>, PhD thesis, 2006.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F T(n,0) = A000679(n).
%F T(n,1) = A090091(n).
%F T(n,2) = A090130(n).
%F T(n,3) = A090140(n).
%F T(0,n) = 1, T(1,n) = 1, T(2,n) = 2 and T(3,n) = 5.
%F T(4,0) = 14 and T(4,n) = 15, n > 0.
%F T(5,n) = A232105(n+1).
%F T(6,n) = A232106(n+1).
%F T(7,n) = A232107(n+1).
%e Array begins:
%e (p = 2) (p = 3) (p = 5) (p = 7) (p = 11) (p = 13) ...
%e 1 1 1 1 1 1 ...
%e 1 1 1 1 1 1 ...
%e 2 2 2 2 2 2 ...
%e 5 5 5 5 5 5 ...
%e 14 15 15 15 15 15 ...
%e 51 67 77 83 87 97 ...
%e 267 504 684 860 1192 1476 ...
%e 2328 9310 34297 113147 750735 1600573 ...
%e ...
%p with(GroupTheory): T:=proc(n,k) NumGroups(ithprime(k+1)^n); end proc: seq(seq(T(n-k,k),k=0..n),n=0..10); # _Muniru A Asiru_, Oct 03 2018
%t (* This program uses Higman's PORC functions to compute the rows 0 to 7 *)
%t f[0, p_] := 1; f[1, p_] := 1; f[2, p_] := 2; f[3, p_] := 5;
%t f[4, p_] := If[p == 2, 14, 15];
%t f[5, p_] := If[p == 2, 51, If[p == 3, 67, 61 + 2*p + 2*GCD[p - 1, 3] + GCD[p - 1, 4]]];
%t f[6, p_] := If[p == 2, 267, If[p == 3, 504, 3*p^2 + 39*p + 344 + 24*GCD[p - 1, 3] + 11*GCD[p - 1, 4] + 2*GCD[p - 1, 5]]];
%t f[7, p_] := If[p == 2, 2328, If[p == 3, 9310, If[p == 5, 34297, 3*p^5 + 12*p^4 + 44*p^3 + 170*p^2 + 707*p + 2455 + (4*p^2 + 44*p + 291)*GCD[p - 1, 3] + (p^2 + 19*p + 135)*GCD[p - 1, 4] + (3*p + 31)*GCD[p - 1, 5] + 4*GCD[p - 1, 7] + 5*GCD[p - 1, 8] + GCD[p - 1, 9]]]];
%t tabl[kk_] := TableForm[Table[f[n, Prime[k+1]], {n, 0, 7}, {k, 0, kk}]];
%o (GAP) # This program computes the first 45 terms, rows 0..8.
%o P:=Filtered([1..300],IsPrime);;
%o T1:=List([0..7],n->List([0..15],k->NumberSmallGroups(P[k+1]^n)));;
%o T2:=[Flat(Concatenation(List([8],n->List([0],k->NumberSmallGroups(P[k+1]^n))),List([1..14],i->0)))];;
%o T:=Concatenation(T1,T2);;
%o b:=List([2..10],n->OrderedPartitions(n,2));;
%o a:=Flat(List([1..Length(b)],i->List([1..Length(b[i])],j->T[b[i][j][2]][b[i][j][1]]))); # _Muniru A Asiru_, Oct 01 2018
%Y Inspired by A158106.
%Y Cf. A000001, A000679, A090091, A090130, A090140, A128604, A232105, A232106, A232107.
%K tabl,nonn,hard,more
%O 0,4
%A _Franck Maminirina Ramaharo_, Sep 12 2018
%E a(55)=T(10,0) corrected by _David Burrell_, Jun 07 2022
%E a(56)=T(9,1) from _David Burrell_, Sep 01 2023
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