A326847 - OEIS

%I #14 Aug 09 2019 12:35:51

%S 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 59,61,64,67,71,73,79,81,83,84,89,97,101,103,107,109,113,121,125,127,

%U 128,131,137,139,149,151,157,163,167,169,173,179,181,191,193,197

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

%C First differs from A071139, A089352 and A086486 in lacking 60. First differs from A326837 in lacking 268.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C The enumeration of these partitions by sum is given by A326842.

%H R. J. Mathar, <a href="/A326847/b326847.txt">Table of n, a(n) for n = 1..489</a>

%F Intersection of A326841 and A316413.

%e The sequence of terms together with their prime indices begins:

%e    2: {1}

%e    3: {2}

%e    4: {1,1}

%e    5: {3}

%e    7: {4}

%e    8: {1,1,1}

%e    9: {2,2}

%e   11: {5}

%e   13: {6}

%e   16: {1,1,1,1}

%e   17: {7}

%e   19: {8}

%e   23: {9}

%e   25: {3,3}

%e   27: {2,2,2}

%e   29: {10}

%e   30: {1,2,3}

%e   31: {11}

%e   32: {1,1,1,1,1}

%e   37: {12}

%p isA326847 := proc(n)

%p     psigsu := A056239(n) ;

%p     for ifs in ifactors(n)[2] do

%p         p := op(1,ifs) ;

%p         psig := numtheory[pi](p) ;

%p         if modp(psigsu,psig) <> 0 then

%p             return false;

%p         end if;

%p     end do:

%p     psigle := numtheory[bigomega](n) ;

%p     if modp(psigsu,psigle) = 0 then

%p         true;

%p     else

%p         false;

%p     end if;

%p end proc:

%p n := 1:

%p for i from 2 to 3000 do

%p     if isA326847(i) then

%p         printf("%d %d\n",n,i);

%p         n := n+1 ;

%p     end if;

%p end do: # _R. J. Mathar_, Aug 09 2019

%t 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 Intersection of A326841 and A316413.

%Y Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 26 2019