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 A319171 Square array, read by antidiagonals, upwards: T(n,k) is the number of groups of order prime(k+1)^n. 0
 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, 504, 77, 15, 5, 2, 1, 1, 56092, 9310, 684, 83, 15, 5, 2, 1, 1, 10494213, 1396077, 34297, 860, 87, 15, 5, 2, 1, 1, 49487365422 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS H. U. Besche, B. Eick and E. A. O'Brien. A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644. Heiko Dietrich, Computational aspects of finite p-groups Groupprops, Groups of prime power order Groupprops, Higman's PORC conjecture Groupprops, PORC function Graham Higman, Enumerating p-Groups. I: Inequalities, Proc. London Math. Soc. Vol. 10 (1960), 24-30. Graham Higman, Enumerating p-Groups. II: Problem whose solution is PORC, Proc. London Math. Soc. Vol. 10 (1960), 566-582. Eamonn O'Brien, Polycyclic groups Gordon Royle, Numbers of Small Groups Michael Vaughan-Lee, Graham Higmanâ€™s PORC Conjecture, Jahresbericht der Deutschen Mathematiker-Vereinigung Vol. 114 (2012), 89-16. Michael Vaughan-Lee, Groups of order p^8 and exponent p, International Journal of Group Theory Vol. 4 (2015), 25-42. Brett E. Witty, Enumeration of groups of prime-power order, PhD thesis, 2006. FORMULA T(n,0) = A000679(n). T(n,1) = A090091(n). T(n,2) = A090130(n). T(n,3) = A090140(n). T(0,n) = 1, T(1,n) = 1, T(2,n) = 2 and T(3,n) = 5. T(4,0) = 14 and T(4,n) = 15, n > 0. T(5,n) = A232105(n+1). T(6,n) = A232106(n+1). T(7,n) = A232107(n+1). EXAMPLE Array begins:   (p = 2) (p = 3) (p = 5) (p = 7) (p = 11) (p = 13) ...        1       1       1       1        1        1  ...        1       1       1       1        1        1  ...        2       2       2       2        2        2  ...        5       5       5       5        5        5  ...       14      15      15      15       15       15  ...       51      67      77      83       87       97  ...      267     504     684     860     1192     1476  ...     2328    9310   34297  113147   750735  1600573  ...      ... MAPLE 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 MATHEMATICA (* This program uses Higman's PORC functions to compute the rows 0 to 7 *) f[0, p_] := 1; f[1, p_] := 1; f[2, p_] := 2; f[3, p_] := 5; f[4, p_] := If[p == 2, 14, 15]; f[5, p_] := If[p == 2, 51, If[p == 3, 67, 61 + 2*p + 2*GCD[p - 1, 3] + GCD[p - 1, 4]]]; 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]]]; 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]]]]; tabl[kk_] := TableForm[Table[f[n, Prime[k+1]], {n, 0, 7}, {k, 0, kk}]]; PROG (GAP) # This program computes the first 45 terms, rows 0..8. P:=Filtered([1..300], IsPrime);; T1:=List([0..7], n->List([0..15], k->NumberSmallGroups(P[k+1]^n)));; T2:=[Flat(Concatenation(List([8], n->List([0], k->NumberSmallGroups(P[k+1]^n))), List([1..14], i->0)))];; T:=Concatenation(T1, T2);; b:=List([2..10], n->OrderedPartitions(n, 2));; 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 CROSSREFS Inspired by A158106. Cf. A000001, A000679, A090091, A090130, A090140, A128604, A232105, A232106, A232107. Sequence in context: A128604 A098885 A106270 * A047888 A330964 A128704 Adjacent sequences:  A319168 A319169 A319170 * A319172 A319173 A319174 KEYWORD tabl,nonn,hard,more AUTHOR Franck Maminirina Ramaharo, Sep 12 2018 STATUS approved

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Last modified August 15 00:52 EDT 2020. Contains 336484 sequences. (Running on oeis4.)