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A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.

%I A353841 #15 Jan 20 2025 16:22:37
%S A353841 0,1,1,2,1,1,1,2,2,1,1,3,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,
%T A353841 1,2,1,1,1,3,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,3,1,1,3,2,1,1,1,2,
%U A353841 1,1,1,2,1,1,2,2,1,1,1,2,2,1,1,4,1,1,1,2,1,2,1,2,1,1,1,2,1,2,2,2,1,1,1,2,1,1,1,2,1,1,1,3
%N A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.
%C A353841 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 A353841 Starting with n, this is one plus the number of times one must apply A353832 to reach a squarefree number.
%C A353841 Also Kimberling's depth statistic (defined in A237685 and A237750) plus one.
%H A353841 Antti Karttunen, <a href="/A353841/b353841.txt">Table of n, a(n) for n = 1..65537</a>
%H A353841 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A353841 a(1) = 0, and for n > 1, if A008966(n) = 1 [n is in A005117], a(n) = 1, otherwise a(n) = 1+a(A353832(n)). [See comments] - _Antti Karttunen_, Jan 20 2025
%e A353841 The trajectory for a(1080) = 4 is the following, with prime indices shown on the right:
%e A353841   1080: {1,1,1,2,2,2,3}
%e A353841    325: {3,3,6}
%e A353841    169: {6,6}
%e A353841     37: {12}
%e A353841 The trajectory for a(87780) = 5 is the following, with prime indices shown on the right:
%e A353841   87780: {1,1,2,3,4,5,8}
%e A353841   65835: {2,2,3,4,5,8}
%e A353841   51205: {3,4,4,5,8}
%e A353841   19855: {3,5,8,8}
%e A353841    2915: {3,5,16}
%e A353841 The trajectory for a(39960) = 5 is the following, with prime indices shown on the right:
%e A353841   39960: {1,1,1,2,2,2,3,12}
%e A353841   12025: {3,3,6,12}
%e A353841    6253: {6,6,12}
%e A353841    1369: {12,12}
%e A353841      89: {24}
%t A353841 Table[If[n==1,0,Length[NestWhileList[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&]]],{n,100}]
%o A353841 (PARI)
%o A353841 pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
%o A353841 A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum)));
%o A353841 A353841(n) = if(1==n,0,for(i=1,oo,if(issquarefree(n), return(i), n = A353832(n)))); \\ _Antti Karttunen_, Jan 20 2025
%Y A353841 Positions of 1's are A005117.
%Y A353841 The version for run-lengths instead of sums is A182850 or A323014.
%Y A353841 Positions of first appearances are A353743.
%Y A353841 These are the row-lengths of A353840.
%Y A353841 Other sequences pertaining to this trajectory are A353842-A353845.
%Y A353841 Counting partitions by this statistic gives A353846.
%Y A353841 The version for compositions is A353854, run-lengths of A353853.
%Y A353841 A001222 counts prime factors, distinct A001221.
%Y A353841 A005811 counts runs in binary expansion.
%Y A353841 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353841 A300273 ranks collapsible partitions, counted by A275870.
%Y A353841 A318928 gives runs-resistance of binary expansion.
%Y A353841 A353832 represents the operation of taking run-sums of a partition.
%Y A353841 A353833 ranks partitions with all equal run-sums, counted by A304442.
%Y A353841 A353835 counts distinct run-sums of prime indices, weak A353861.
%Y A353841 A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A353841 A353866 ranks rucksack partitions, counted by A353864.
%Y A353841 Cf. A008966, A071625, A073093, A181819, A182857, A325239, A325277, A325278, A353834, A353847, A353865, A353867.
%K A353841 nonn
%O A353841 1,4
%A A353841 _Gus Wiseman_, May 25 2022
%E A353841 More terms from _Antti Karttunen_, Jan 20 2025