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A328188 Number of strict integer partitions of n with all pairs of consecutive parts relatively prime.

%I A328188 #15 May 10 2021 06:41:13
%S A328188 1,1,1,2,2,3,3,4,5,6,7,8,9,12,15,15,19,23,25,30,35,39,47,52,58,65,75,
%T A328188 86,95,109,124,144,165,181,203,221,249,285,316,352,392,438,484,538,
%U A328188 599,666,737,813,899,992,1102,1215,1335,1472,1621,1776,1946,2137,2336
%N A328188 Number of strict integer partitions of n with all pairs of consecutive parts relatively prime.
%H A328188 Alois P. Heinz, <a href="/A328188/b328188.txt">Table of n, a(n) for n = 0..1000</a>
%e A328188 The a(1) = 1 through a(15) = 15 partitions (A..F = 10..15):
%e A328188   1  2  3   4   5   6    7   8    9    A     B     C    D     E     F
%e A328188         21  31  32  51   43  53   54   73    65    75   76    95    87
%e A328188                 41  321  52  71   72   91    74    B1   85    B3    B4
%e A328188                          61  431  81   532   83    543  94    D1    D2
%e A328188                              521  432  541   92    651  A3    653   E1
%e A328188                                   531  721   A1    732  B2    743   654
%e A328188                                        4321  731   741  C1    752   753
%e A328188                                              5321  831  652   761   852
%e A328188                                                    921  751   851   951
%e A328188                                                         832   941   A32
%e A328188                                                         5431  A31   B31
%e A328188                                                         7321  B21   6531
%e A328188                                                               5432  7431
%e A328188                                                               6521  7521
%e A328188                                                               8321  54321
%p A328188 b:= proc(n, i, s) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
%p A328188       `if`(andmap(j-> igcd(i, j)=1, s), b(n-i, min(n-i, i-1),
%p A328188            numtheory[factorset](i)), 0)+b(n, i-1, s)))
%p A328188     end:
%p A328188 a:= n-> b(n$2, {}):
%p A328188 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 13 2019
%t A328188 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{___,x_,y_,___}/;GCD[x,y]>1]&]],{n,0,30}]
%t A328188 (* Second program: *)
%t A328188 b[n_, i_, s_] := b[n, i, s] = If[i(i + 1)/2 < n, 0, If[n == 0, 1, If[AllTrue[s,  GCD[i, #] == 1&], b[n - i, Min[n - i, i - 1], FactorInteger[i][[All, 1]]], 0] + b[n, i - 1, s]]];
%t A328188 a[n_] := b[n, n, {}];
%t A328188 a /@ Range[0, 60] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y A328188 The case of compositions is A167606.
%Y A328188 The non-strict case is A328172.
%Y A328188 The Heinz numbers of these partitions are given by A328335.
%Y A328188 Partitions with no pairs of consecutive parts relatively prime are A328187.
%Y A328188 Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328220.
%K A328188 nonn
%O A328188 0,4
%A A328188 _Gus Wiseman_, Oct 13 2019