0,4

Table of n, a(n) for n=0..20.

The a(1) = 1 through a(7) = 13 compositions:

  (1)  (2)  (3)   (4)    (5)    (6)     (7)

            (12)  (13)   (14)   (15)    (16)

            (21)  (31)   (23)   (24)    (25)

                  (211)  (32)   (42)    (34)

                         (41)   (51)    (43)

                         (122)  (411)   (52)

                         (311)  (1221)  (61)

                                (2112)  (133)

                                        (322)

                                        (511)

                                        (2113)

                                        (3112)

                                        (12211)

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 10}]

Without equal relations we have A000213, equal only A027383.

Even-length opposite: A003242, ranked by A351010, partitions A035457.

The version for partitions is A351006.

The opposite version is A357643, partitions A351005.

A011782 counts compositions.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357645 counts compositions by half-alternating sum, skew A357646.

Cf. A001590, A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

nonn,more,new

Gus Wiseman, Oct 14 2022

approved

0,2

From David A. Corneth, Oct 05 2022: (Start)

a(11)-a(21) are assuming A061300(n+1)/A061300(n) is an integer for n = 11..21.

10^20 < a(22) <= 1583855315457687090000. (End)

Table of n, a(n) for n=0..21.

a(5) = 1105 as A061300(5+1) / A061300(5) = 61261200 / 55440 = 1105.

Cf. A061300.

nonn,hard,more,new

J. Lowell, Oct 04 2022

a(11)-a(21) from David A. Corneth, Oct 05 2022

approved

0,2

Table of n, a(n) for n=0..39.

1 has no divisors in A031443, so a(0) = 1;

2 has divisors 1 = 1_2, 2 = 10_2 and 2 = A031443(1), so a(1) = 2.

10 has divisors 2 = 10_2 and 10 = 1010_2 in A031443, so a(2) = 10.

digBalQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length[d]) && Count[d, 1] == m/2]; f[n_] := DivisorSum[n, 1 &, digBalQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[40, 10^7] (* Amiram Eldar, Sep 26 2022 *)

(Magma) bal:=func<n|Multiplicity(Intseq(n, 2), 1) eq Multiplicity(Intseq(n, 2), 0)>; a:=[]; for n in [0..38] do k:=1; while #[d:d in Divisors(k)|bal(d)] ne n  do k:=k+1; end while; Append(~a, k); end for; a;

Cf. A031443.

nonn,base,new

Marius A. Burtea, Sep 20 2022

approved

0,2

Table of n, a(n) for n=0..33.

1 has no divisors in A019506, so a(0) = 1;

22 has divisors 1, 2, 11, 22, and 22 = A019506(1), so a(1) = 22.

308 has divisors 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308 and 22 = A019506(1), 308 = A019506(14), so a(2) = 308.

digitSum[n_] := Total @ IntegerDigits[n]; hoaxQ[n_] := CompositeQ[n] && Total[digitSum /@ FactorInteger[n][[;; , 1]]] == digitSum[n]; f[n_] := DivisorSum[n, 1 &, hoaxQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[10, 10^5] (* Amiram Eldar, Sep 26 2022 *)

(Magma) hoax:=func<n| not IsPrime(n) and (&+Intseq(n, 10) eq &+[ &+Intseq(p, 10): p in PrimeDivisors(n)])>; a:=[]; for n in [0..33] do k:=1; while #[d:d in Set(Divisors(k)) diff {1}|hoax(d)] ne n do k:=k+1; end while; Append(~a, k); end for; a;

Cf. A019506.

nonn,base,new

Marius A. Burtea, Sep 20 2022

approved

0,2

Table of n, a(n) for n=0..32.

The divisors of 101 are 1 and 101. Of those, only 101 is a cyclops number; it is the smallest cyclops number, so a(1) = 101.

The divisors of 202 are 1, 2, 101, and 202, the cyclops numbers being 101 and 202, so a(2) = 202.

The divisors of 404 are 1, 2, 4, 101, 202, and 404, the cyclops numbers being 101, 202 and 404, so a(3) = 404.

L:= Vector(10^8):

C:= [0]:

for d from 3 to 7 by 2 do

  C:= [seq(seq(seq(a*10^(d-1)+10*b+c, c=1..9), b=C), a=1..9)];

  for x in C do

    Mx:= [seq(i, i=x..10^8, x)];

    L[Mx]:= map(`+`, L[Mx], 1)

  od;

od:

V:= Array(0..max(L)):

for n from 1 to 10^8 do

  if V[L[n]] = 0 then V[L[n]]:= n; fi

od:

if member(0, V, 'k') then convert(V[0..k-1], list)

else convert(V, list)

fi; # Robert Israel, Sep 20 2022

cyclopQ[n_] := Module[{d = IntegerDigits[n], len}, OddQ[len = Length[d]] && Position[d, 0] == {{(len + 1)/2}}]; f[n_] := DivisorSum[n, 1 &, cyclopQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[10, 10^5] (* Amiram Eldar, Sep 26 2022 *)

(Magma) ints:=func<n|n eq 0 select [0] else Intseq(n)>; cyc:=func<n|IsOdd(#ints(n)) and ints(n)[(#ints(n)+1) div 2] eq 0 and Multiplicity(ints(n), 0) eq 1>; a:=[]; for n in [0..32]  do k:=1; while #[s:s in Divisors(k)| cyc(s)] ne n do k:=k+1; end while; Append(~a, k); end for; a;

Cf. A134808.

nonn,base,new

Marius A. Burtea, Sep 20 2022

approved

0,3

Table of n, a(n) for n=0..48.

The a(1) = 1 through a(8) = 9 compositions:

  (1)  (2)   (3)  (4)    (5)    (6)     (7)      (8)

       (11)       (22)   (113)  (33)    (115)    (44)

                  (112)  (221)  (114)   (223)    (116)

                                (1122)  (331)    (224)

                                (2211)  (11221)  (332)

                                                 (1133)

                                                 (3311)

                                                 (22112)

                                                 (112211)

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 1, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 2, Length[#]-1, 2}]&]], {n, 0, 15}]

The even-length version is A003242, ranked by A351010, partitions A035457.

Without equal relations we have A016116, equal only A001590 (apparently).

The version for partitions is A351005.

The opposite version is A357644, partitions A351006.

A011782 counts compositions.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357645 counts compositions by half-alternating sum, skew A357646.

Cf. A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

nonn,new

Gus Wiseman, Oct 12 2022

More terms from Alois P. Heinz, Oct 12 2022

approved

5,2

Table of n, a(n) for n=5..10.

Firsching, Moritz Realizability and inscribability for simplicial polytopes via nonlinear optimization.  Math. Program. 166, No. 1-2 (A), 273-295 (2017). Table 3

nonn,hard,new

R. J. Mathar, Oct 14 2022

approved

2,2

Table of n, a(n) for n=2..71.

  n         A357753(n) a(n) A357659(n)    A357660(n)    A357754(n)

  bits  2^n  least sq  Ha w  k_min  ^2     k_max  ^2   largest sq

   2     4      4       1     2      4      2      4        4

   3     8      9       2     3      9      3      9        9

   4    16     16       3     5     25      5     25       25

   5    32     36       3     7     49      7     49       49

   6    64     64       5    11    121     11    121      121

   7   128    144       4    13    169     15    225      225

  12  4096   4096       9    75   5625     89   7921     8100

Cf. A000120, A000290, A356878, A357304, A357753, A357754.

A357659 and A357660 are the minimal and the maximal values of k producing a(n).

nonn,base,new

Hugo Pfoertner, Oct 09 2022

approved

2,1

Table of n, a(n) for n=2..45.

See A357658.

Cf. A000120, A000290, A357658, A357660.

nonn,base,new

Hugo Pfoertner, Oct 09 2022

approved

2,1

Table of n, a(n) for n=2..45.

See A357658.

Cf. A000120, A000290, A357658, A357659.

nonn,base,new

Hugo Pfoertner, Oct 09 2022

approved