cp's OEIS Frontend

A337694 Numbers with no two relatively prime prime indices.

%I A337694 #15 Oct 06 2020 14:46:25
%S A337694 1,2,3,5,7,9,11,13,17,19,21,23,25,27,29,31,37,39,41,43,47,49,53,57,59,
%T A337694 61,63,65,67,71,73,79,81,83,87,89,91,97,101,103,107,109,111,113,115,
%U A337694 117,121,125,127,129,131,133,137,139,147,149,151,157,159,163,167,169,171,173,179,181,183,185,189,191,193,197,199
%N A337694 Numbers with no two relatively prime prime indices.
%C A337694 First differs from A305078 in having 1 and lacking 195.
%C A337694 First differs from A305103 in having 1 and 169 and lacking 195.
%C A337694 First differs from A328336 in lacking 897, with prime indices (2,6,9).
%C A337694 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A337694 Also Heinz numbers of integer partitions in which no two parts are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A337694 Robert Israel, <a href="/A337694/b337694.txt">Table of n, a(n) for n = 1..10000</a>
%e A337694 The sequence of terms together with their prime indices begins:
%e A337694    1: {}      37: {12}     79: {22}      121: {5,5}
%e A337694    3: {2}     39: {2,6}    81: {2,2,2,2} 125: {3,3,3}
%e A337694    5: {3}     41: {13}     83: {23}      127: {31}
%e A337694    7: {4}     43: {14}     87: {2,10}    129: {2,14}
%e A337694    9: {2,2}   47: {15}     89: {24}      131: {32}
%e A337694   11: {5}     49: {4,4}    91: {4,6}     133: {4,8}
%e A337694   13: {6}     53: {16}     97: {25}      137: {33}
%e A337694   17: {7}     57: {2,8}   101: {26}      139: {34}
%e A337694   19: {8}     59: {17}    103: {27}      147: {2,4,4}
%e A337694   21: {2,4}   61: {18}    107: {28}      149: {35}
%e A337694   23: {9}     63: {2,2,4} 109: {29}      151: {36}
%e A337694   25: {3,3}   65: {3,6}   111: {2,12}    157: {37}
%e A337694   27: {2,2,2} 67: {19}    113: {30}      159: {2,16}
%e A337694   29: {10}    71: {20}    115: {3,9}     163: {38}
%e A337694   31: {11}    73: {21}    117: {2,2,6}   167: {39}
%p A337694 filter:= proc(n) local F,i,j,np;
%p A337694   if n::even and n>2 then return false fi;
%p A337694   F:= map(t -> numtheory:-pi(t[1]), ifactors(n)[2]);
%p A337694   np:= nops(F);
%p A337694   for i from 1 to np-1 do
%p A337694     for j from i+1 to np do
%p A337694       if igcd(F[i],F[j])=1 then return false fi
%p A337694   od od;
%p A337694   true
%p A337694 end proc:
%p A337694 select(filter, [$1..300]); # _Robert Israel_, Oct 06 2020
%t A337694 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A337694 stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
%t A337694 Select[Range[100],stabQ[primeMS[#],CoprimeQ]&]
%Y A337694 A200976 and A328673 count these partitions.
%Y A337694 A302696 and A302569 are pairwise coprime instead of pairwise non-coprime.
%Y A337694 A318719 is the squarefree case.
%Y A337694 A328867 looks at distinct prime indices.
%Y A337694 A337666 is the version for standard compositions.
%Y A337694 A101268 counts pairwise coprime or singleton compositions.
%Y A337694 A318717 counts strict pairwise non-coprime partitions.
%Y A337694 A327516 counts pairwise coprime partitions.
%Y A337694 A333227 ranks pairwise coprime compositions.
%Y A337694 A333228 ranks compositions whose distinct parts are pairwise coprime.
%Y A337694 A335236 ranks compositions neither a singleton nor pairwise coprime.
%Y A337694 A337462 counts pairwise coprime compositions.
%Y A337694 A337667 counts pairwise non-coprime compositions.
%Y A337694 Cf. A051185, A051424, A056239, A112798, A220377, A284825, A302797, A303282, A305843, A319752, A336737, A337599, A337604, A337605.
%K A337694 nonn
%O A337694 1,2
%A A337694 _Gus Wiseman_, Sep 23 2020