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Heinz numbers of integer partitions of m >= 0 using divisors of m.
22

%I #7 Aug 09 2019 12:15:31

%S 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,25,27,29,30,31,32,36,37,40,41,

%T 43,47,48,49,53,59,61,63,64,67,71,73,79,81,83,84,89,97,101,103,107,

%U 108,109,112,113,121,125,127,128,131,137,139,144,149,151,157,163

%N Heinz numbers of integer partitions of m >= 0 using divisors of m.

%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 A018818.

%H R. J. Mathar, <a href="/A326841/b326841.txt">Table of n, a(n) for n = 1..543</a>

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

%e 1: {}

%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 12: {1,1,2}

%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}

%p isA326841 := proc(n)

%p local ifs,psigsu,p,psig ;

%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 true;

%p end proc:

%p for i from 1 to 3000 do

%p if isA326841(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[100],With[{y=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},And@@IntegerQ/@(Total[y]/y)]&]

%Y The case where the length also divides m is A326847.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 26 2019