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A325351 Heinz number of the augmented differences of the integer partition with Heinz number n.

%I A325351 #15 Nov 16 2019 20:05:20
%S A325351 1,2,3,4,5,6,7,8,6,10,11,12,13,14,9,16,17,12,19,20,15,22,23,24,10,26,
%T A325351 12,28,29,18,31,32,21,34,15,24,37,38,33,40,41,30,43,44,18,46,47,48,14,
%U A325351 20,39,52,53,24,25,56,51,58,59,36,61,62,30,64,35,42,67,68,57,30,71,48,73,74,18,76,21,66,79,80,24,82,83,60,55,86,69,88,89,36,35
%N A325351 Heinz number of the augmented differences of the integer partition with Heinz number n.
%C A325351 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325351 The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3). Note that aug preserves length so this sequence preserves omega (number of prime factors counted with multiplicity).
%H A325351 Antti Karttunen, <a href="/A325351/b325351.txt">Table of n, a(n) for n = 1..16384</a>
%H A325351 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A325351 <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%e A325351 The partition (3,2,2,1) with Heinz number 90 has augmented differences (2,1,2,1) with Heinz number 36, so a(90) = 36.
%t A325351 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325351 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325351 Table[Times@@Prime/@aug[primeptn[n]],{n,100}]
%o A325351 (PARI)
%o A325351 augdiffs(n) = { my(diffs=List([]), f=factor(n), prevpi, pi=0, i=#f~); while(i, prevpi=pi; pi = primepi(f[i, 1]); if(prevpi, listput(diffs, 1+(prevpi-pi))); if(f[i, 2]>1, f[i, 2]--, i--)); if(pi, listput(diffs,pi)); Vec(diffs); };
%o A325351 A325351(n) = factorback(apply(prime,augdiffs(n))); \\ _Antti Karttunen_, Nov 16 2019
%Y A325351 Number of appearances of n is A008480(n).
%Y A325351 Cf. A056239, A093641 (fixed points), A112798, A325350, A325352, A325355, A325366, A325389, A325394, A325395, A325396.
%K A325351 nonn,look
%O A325351 1,2
%A A325351 _Gus Wiseman_, Apr 23 2019
%E A325351 More terms from _Antti Karttunen_, Nov 16 2019