cp's OEIS Frontend

A385216 Greatest Heinz number of a sparse submultiset of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

%I A385216 #5 Jul 06 2025 17:49:16
%S A385216 1,2,3,4,5,3,7,8,9,10,11,4,13,14,5,16,17,9,19,20,21,22,23,8,25,26,27,
%T A385216 28,29,10,31,32,33,34,7,9,37,38,39,40,41,21,43,44,9,46,47,16,49,50,51,
%U A385216 52,53,27,55,56,57,58,59,20,61,62,63,64,65,33,67,68,69
%N A385216 Greatest Heinz number of a sparse submultiset of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.
%C A385216 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A385216 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.
%F A385216 a(n) = n iff n belongs to A319630.
%e A385216 The prime indices of 12 are {1,1,2}, with sparse submultisets {{},{1},{2},{1,1}}, with Heinz numbers {1,2,3,4}, so a(12) = 4.
%e A385216 The prime indices of 36 are {1,1,2,2}, with sparse submultisets {{},{1},{2},{1,1},{2,2}}, with Heinz numbers {1,2,3,4,9}, so a(36) = 9.
%e A385216 The prime indices of 462 are {1,2,4,5}, with sparse submultisets {{},{1},{2},{4},{5},{1,4},{2,4},{1,5},{2,5}}, with Heinz numbers {1,2,3,7,11,14,21,22,33}, so a(462) = 33.
%t A385216 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A385216 Table[Max@@Select[Divisors[n],FreeQ[Differences[prix[#]],1]&],{n,100}]
%Y A385216 Sparse submultisets are counted by A166469, maximal A385215.
%Y A385216 The union is A319630 (Heinz numbers of sparse multisets), complement A104210.
%Y A385216 For binary instead of prime indices we have A374356, see A245564, A384883.
%Y A385216 A000005 counts divisors (or submultisets of prime indices).
%Y A385216 A001222 counts prime factors, distinct A001221.
%Y A385216 A051903 gives greatest prime exponent, least A051904, counted by A091602.
%Y A385216 A055396 gives least prime index, greatest A061395, counted by A008284.
%Y A385216 A056239 adds up prime indices, row sums of A112798.
%Y A385216 A212166 ranks partitions with max multiplicity = length, counted by A239964.
%Y A385216 A381542 ranks partitions with max part = max multiplicity, counted by A240312.
%Y A385216 Cf. A000720, A005117, A122111, A130091, A268193, A316476.
%K A385216 nonn
%O A385216 1,2
%A A385216 _Gus Wiseman_, Jul 05 2025