A221048 The odd semiprime numbers (A046315) which are orders of a non-Abelian group.
21, 39, 55, 57, 93, 111, 129, 155, 183, 201, 203, 205, 219, 237, 253, 291, 301, 305, 309, 327, 355, 381, 417, 453, 471, 489, 497, 505, 543, 579, 597, 633, 655, 669, 687, 689, 723, 737, 755, 791, 813, 831, 849, 889, 905, 921, 939, 955, 979, 993, 1011, 1027, 1047
Offset: 1
Keywords
A192298 The number of sets of n positive integers strictly less than 2*n such that no integer in the set divides another.
1, 1, 2, 3, 2, 4, 6, 6, 10, 14, 13, 26, 34, 24, 48, 72, 60, 120, 168, 168, 264, 396, 312, 624, 816, 816, 1632, 2208
Offset: 1
Comments
Similar to A174094, but the maximum integer in each set must be strictly less than 2n here.
Examples
a(1) counts {1};
a(2) counts {2,3};
a(3) counts {2,3,5} and {3,4,5};
a(4) counts {2,3,5,7}, {3,4,5,7}, and {4,5,6,7};
a(5) counts {4,5,6,7,9} and {5,6,7,8,9}.
Crossrefs
Cf. A174094.
Programs
-
Maple
with(combstruct) ; A192298nodiv := proc(s) sl := sort(convert(s,list)) ; for i from 1 to nops(sl)-1 do for j from i+1 to nops(sl) do if op(j,sl) mod op(i,sl) = 0 then return false; end if; end do: end do:true ; end proc: A192298 := proc(n) a := 0 ; it := iterstructs(Subset({seq(i,i=1..2*n-1)},size=n)) : while not finished(it) do s := nextstruct(it) ; if nops(s) = n then if A192298nodiv(s) then a := a+1 ; end if; end if; end do: a ; end proc: # R. J. Mathar, Jul 12 2011
A174063 Triangular array (read by rows) containing the least n integers < 2n such that no integer divides another.
1, 2, 3, 2, 3, 5, 2, 3, 5, 7, 4, 5, 6, 7, 9, 4, 5, 6, 7, 9, 11, 4, 5, 6, 7, 9, 11, 13, 4, 6, 7, 9, 10, 11, 13, 15, 4, 6, 7, 9, 10, 11, 13, 15, 17, 4, 6, 7, 9, 10, 11, 13, 15, 17, 19, 4, 6, 9, 10, 11, 13, 14, 15, 17, 19, 21, 4, 6, 9, 10, 11, 13, 14, 15, 17, 19, 21, 23, 4, 6, 9, 10, 11, 13, 14
Offset: 1
Comments
The first number on each row is 2^k, where k is the greatest integer such that (3^k)/2 < n.
Examples
1; 2, 3; 2, 3, 5; 2, 3, 5, 7; 4, 5, 6, 7, 9; 4, 5, 6, 7, 9, 11; 4, 5, 6, 7, 9, 11, 13; 4, 6, 7, 9, 10, 11, 13, 15;
Programs
-
Mathematica
noneDivQ[L_] := NoneTrue[Subsets[L, {2}], Divisible[#[[2]], #[[1]]]&]; k1[n_] := For[k = Log[3, 2n]//Ceiling, True, k--, If[(3^k)/2
Extensions
Definition corrected by David Brown, Mar 20 2010
A174062 Least possible sum of exactly n positive integers less than 2n such that none of the n integers divides another.
1, 5, 10, 17, 31, 42, 55, 75, 92, 111, 139, 162, 187, 233, 262, 293, 337, 372, 409, 461, 502, 545, 615, 662, 711, 779, 832, 887, 963, 1022, 1083, 1181, 1246, 1313, 1405, 1476, 1549, 1649, 1726, 1805, 1951, 2034, 2119, 2235, 2324, 2415, 2539, 2634, 2731, 2885
Offset: 1
Keywords
Crossrefs
Row sums of triangle A174063. [David Brown, Mar 20 2010]
Programs
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Maple
f:= proc(N) local i,j,obj,cons; obj:= add(i*x[i],i=1..2*N-1); cons:= {seq(seq(x[i]+x[j]<=1, j=2*i..2*N-1, i),i=1..N), add(x[i],i=1..2*N-1)=N}; Optimization:-Minimize(obj,cons,assume=binary)[1] end proc: map(f, [$1..60]); # Robert Israel, May 06 2019 -
Mathematica
a[n_] := Module[{obj, cons}, obj = Sum[i*x[i], {i, 1, 2n-1}]; cons = Append[Flatten[Table[Table[x[i]+x[j] <= 1, {j, 2i, 2n-1, i}], {i, 1, n}], 1], AllTrue[Array[x, 2n-1], 0 <= # <= 1&] && Sum[x[i], {i, 1, 2n-1}] == n]; Minimize[{obj, cons}, Array[x, 2n-1], Integers][[1]]]; Reap[For[n = 1, n <= 50, n++, Print[n, " ", a[n]]; Sow[a[n]]]][[2, 1]]; (* Jean-François Alcover, May 17 2023, after Robert Israel *)
Extensions
Extended by Ray Chandler, Mar 19 2010
A174094 Number of ways to choose n positive integers less than or equal to 2n such that none of the n integers divides another.
1, 2, 2, 3, 5, 4, 6, 12, 10, 14, 26, 26, 34, 68, 48, 72, 120, 120, 168, 336, 264, 396, 792, 624, 816, 1632, 1632, 2208, 3616, 3616, 5056, 10112, 6592, 9888, 19776, 19776, 24384, 48768, 48768, 73152, 112320, 76032, 114048, 228096, 190080, 264960, 529920
Offset: 0
Keywords
Comments
a(n) >= 2^(1+floor((n-1)/3)). - Robert Israel, Aug 25 2015
Examples
a(1) = 2 because we can choose {1}, {2}.
a(2) = 2 because we can choose {2, 3}, {3, 4}.
a(3) = 3 because we can choose {2, 3, 5}, {3, 4, 5}, {4, 5, 6}.
References
- F. Caldarola, G. d'Atri, M.Pellegrini, Combinatorics on n-sets: Arithmetic properties and numerical results. In: Sergeyev Y., Kvasov D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science, vol. 11973. Springer, Cham, 389-401.
Links
- Bertram Felgenhauer, Table of n, a(n) for n = 0..3400 (terms 1..100 from Marco Pellegrini)
- C. Bindi, L. Bussoli, M. Grazzini, M. Pellegrini, G. Pirillo, M. A. Pirillo, and A. Troise, Su un risultato di uno studente di Erdös, Periodico di matematiche 1 (2016), Vol 8, No 1 (2016): Serie XII Anno CXXVI. [Broken link]
- Bertram Felgenhauer, Haskell program for computing A174094.
- Hong Liu, Péter Pál Pach, and Richárd Palincza, The number of maximum primitive sets of integers, arXiv:1805.06341 [math.CO], 2018.
- Richárd Palincza, Counting type and extremal problems from Arithmetic Combinatorics, Ph. D. Thesis, Budapest Univ. Tech. Econ. (Hungary, 2024).
- Sujith Vijay, On large primitive subsets of {1,2,...,2n}, arXiv:1804.01740 [math.CO], 2018.
Crossrefs
The smallest n integers possible is A174063.
Programs
-
Maple
F:= proc(S,m) option remember; local s,S1,S2; if nops(S) < m then return 0 fi; if m = 1 then return nops(S) fi; s:= min(S); S1:= S minus {s}; S2:= S minus {seq(j*s,j=1..floor(max(S)/s))}; F(S1, m) + F(S2, m-1); end proc; seq(F({$1..2*n},n), n=1..37); # Robert Israel, Aug 25 2015 -
Mathematica
F[S_List, m_] := F[S, m] = Module[{s, S1, S2}, If[Length[S]
Extensions
More terms from David Brown, Mar 14 2010
a(30)-a(46) from Ray Chandler, Mar 19 2010
a(0)=1 prepended by Alois P. Heinz, Jun 24 2022
Comments
Links
Crossrefs
Programs
Mathematica
Select[1 + 2*Range[500], (f = FactorInteger[#]; Last /@ f == {1, 1} && Mod @@ Reverse[First /@ f] == 1) &] (* Giovanni Resta, Apr 14 2013 *)PARI
lista(nn) = {forstep(n=1, nn, 2, my(f=factor(n)); if ((#f~ == 2) && (vecmax(f[,2]) == 1) && ((f[2,1] % f[1,1]) == 1), print1(n, ", ")););} \\ Michel Marcus, Sep 28 2017PARI
Extensions