cp's OEIS Frontend

A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

%I A326847 #14 Aug 09 2019 12:35:51
%S A326847 2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,30,31,32,37,41,43,47,49,53,
%T A326847 59,61,64,67,71,73,79,81,83,84,89,97,101,103,107,109,113,121,125,127,
%U A326847 128,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197
%N A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.
%C A326847 First differs from A071139, A089352 and A086486 in lacking 60. First differs from A326837 in lacking 268.
%C A326847 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A326847 The enumeration of these partitions by sum is given by A326842.
%H A326847 R. J. Mathar, <a href="/A326847/b326847.txt">Table of n, a(n) for n = 1..489</a>
%F A326847 Intersection of A326841 and A316413.
%e A326847 The sequence of terms together with their prime indices begins:
%e A326847    2: {1}
%e A326847    3: {2}
%e A326847    4: {1,1}
%e A326847    5: {3}
%e A326847    7: {4}
%e A326847    8: {1,1,1}
%e A326847    9: {2,2}
%e A326847   11: {5}
%e A326847   13: {6}
%e A326847   16: {1,1,1,1}
%e A326847   17: {7}
%e A326847   19: {8}
%e A326847   23: {9}
%e A326847   25: {3,3}
%e A326847   27: {2,2,2}
%e A326847   29: {10}
%e A326847   30: {1,2,3}
%e A326847   31: {11}
%e A326847   32: {1,1,1,1,1}
%e A326847   37: {12}
%p A326847 isA326847 := proc(n)
%p A326847     psigsu := A056239(n) ;
%p A326847     for ifs in ifactors(n)[2] do
%p A326847         p := op(1,ifs) ;
%p A326847         psig := numtheory[pi](p) ;
%p A326847         if modp(psigsu,psig) <> 0 then
%p A326847             return false;
%p A326847         end if;
%p A326847     end do:
%p A326847     psigle := numtheory[bigomega](n) ;
%p A326847     if modp(psigsu,psigle) = 0 then
%p A326847         true;
%p A326847     else
%p A326847         false;
%p A326847     end if;
%p A326847 end proc:
%p A326847 n := 1:
%p A326847 for i from 2 to 3000 do
%p A326847     if isA326847(i) then
%p A326847         printf("%d %d\n",n,i);
%p A326847         n := n+1 ;
%p A326847     end if;
%p A326847 end do: # _R. J. Mathar_, Aug 09 2019
%t A326847 Select[Range[2,100],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Divisible[Total[y],Length[y]]&&And@@IntegerQ/@(Total[y]/y)]&]
%Y A326847 Intersection of A326841 and A316413.
%Y A326847 Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.
%K A326847 nonn
%O A326847 1,1
%A A326847 _Gus Wiseman_, Jul 26 2019