cp's OEIS Frontend

A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

%I A375400 #7 Aug 17 2024 21:50:24
%S A375400 1,2,3,4,5,2,7,8,9,2,11,4,13,2,3,16,17,6,19,4,3,2,23,8,25,2,27,4,29,2,
%T A375400 31,32,3,2,5,12,37,2,3,8,41,2,43,4,9,2,47,16,49,10,3,4,53,18,5,8,3,2,
%U A375400 59,4,61,2,9,64,5,2,67,4,3,2,71,24,73,2,15,4,7
%N A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.
%C A375400 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 A375400 An anti-run is a sequence with no adjacent equal parts. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
%C A375400 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.
%e A375400 The prime indices of 540 are (1,1,2,2,2,3), with maximal anti-runs ((1),(1,2),(2),(2,3)), with minima (1,1,2,2), with Heinz number 36, so a(540) = 36.
%e A375400 The prime indices of 990 are (1,2,2,3,5), with maximal anti-runs ((1,2),(2,3,5)), with minima (1,2), with Heinz number 6, so a(990) = 6.
%t A375400 Table[Times@@Prime/@If[n==1,{},Min /@ Split[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],UnsameQ]],{n,100}]
%Y A375400 bigomega is A001222(a(n)) = A375136(n).
%Y A375400 Least prime factor is A020639(a(n)) = A020639(n).
%Y A375400 Least prime index is A055396(a(n)) = A055396(n).
%Y A375400 Heinz weights are A056239(a(n)) = A374706(n).
%Y A375400 The greatest prime index A061395(a(n)) is the maximum of row n of A375128.
%Y A375400 Firsts for omega (except first term) are half A061742.
%Y A375400 Prime indices A112798(a(n)) are row n of A375128.
%Y A375400 Positions of prime-powers are A375396, counted by A115029.
%Y A375400 Positions of squarefree numbers are A375398, counted by A375134.
%Y A375400 A000041 counts integer partitions, strict A000009.
%Y A375400 A027748 lists distinct prime factors, sum A008472.
%Y A375400 A304038 lists distinct prime indices, sum A066328.
%Y A375400 A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
%Y A375400 A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
%Y A375400 Both have length A001222, distinct A001221.
%Y A375400 Cf. A000040, A046660, A124768, A320324, A358836, A374683, A374700, A375133.
%K A375400 nonn
%O A375400 1,2
%A A375400 _Gus Wiseman_, Aug 17 2024