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A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).

%I A381075 #11 Apr 15 2025 08:25:33
%S A381075 1,4,8,9,16,25,32,49,64,81,121,128,169,256,289,361,512,529,625,841,
%T A381075 961,1024,1331,1369,1444,1681,1849,2048,2116,2197,2209,2809,3481,3721,
%U A381075 3844,4232,4489,4913,5041,5324,5329,5476,6241,6859,6889,7396,7569,7688,7921
%N A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).
%H A381075 Michel Marcus, <a href="/A381075/b381075.txt">Table of n, a(n) for n = 1..391</a>
%F A381075 Sorted positions of first appearances in A001414 - A008472.
%e A381075 The initial terms of A280292 are (0,0,0,2,0,0,0,4,3,0,0,2,0,0,0,6,0,3,0,2,0,0,0,4,5,0,6,2,...), wherein a value appears for the first time at positions 1, 4, 8, 9, 16, 25, ...
%t A381075 prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
%t A381075 q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,10000}];
%t A381075 Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]
%o A381075 (PARI) f(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2] - f[j, 1]); \\ A280292
%o A381075 lista(nn) = my(v=Set(vector(nn, i, f(i))), list=List()); for (i=1, #v, my(k=1); while(f(k) != v[i], k++); listput(list, k)); vecsort(Vec(list)); \\ _Michel Marcus_, Apr 15 2025
%Y A381075 For length instead of sum we have A151821.
%Y A381075 The unsorted version is A280286, firsts of A280292.
%Y A381075 For indices instead of factors we have A380957 (unsorted A380956), firsts of A380955.
%Y A381075 A multiplicative version is A380988 (unsorted A380987), firsts of A290106.
%Y A381075 For prime multiplicities instead of factors see A380989, firsts of A380958.
%Y A381075 For product instead of sum we have A381076, sorted firsts of A066503.
%Y A381075 A000040 lists the primes, differences A001223.
%Y A381075 A005117 lists squarefree numbers, complement A013929.
%Y A381075 A055396 gives least prime index, greatest A061395.
%Y A381075 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A381075 A364916 counts partitions by (sum minus sum of distinct parts).
%Y A381075 Cf. A000720, A001414, A008472, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A366528.
%K A381075 nonn
%O A381075 1,2
%A A381075 _Gus Wiseman_, Feb 18 2025