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A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

%I A280286 #21 Feb 22 2025 09:56:34
%S A280286 4,9,8,25,16,49,32,81,64,121,128,169,256,625,512,289,1024,361,2048,
%T A280286 1444,1331,529,5324,2116,2197,4232,8788,841,17576,961,7569,3844,4913,
%U A280286 7688,19652,1369,6859,5476,12321,1681,34225,1849,15129,7396,12167,2209,46225,8836,19881
%N A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.
%H A280286 Michel Marcus, <a href="/A280286/b280286.txt">Table of n, a(n) for n = 2..500</a>
%F A280286 For p prime, a(p) = p^2 (see A001248).
%t A280286 prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
%t A280286 q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,1000}];
%t A280286 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A280286 Table[Position[q,k][[1,1]],{k,2,mnrm[q/.(0->1)]}] (* _Gus Wiseman_, Feb 20 2025 *)
%o A280286 (PARI) sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]*f[j,2]);
%o A280286 sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]);
%o A280286 a(n) = {my(k = 2); while (sopfr(k) - sopf(k) != n, k++); k;}
%Y A280286 Cf. A001414 (sopfr), A008472 (sopf), A001248, A280163.
%Y A280286 A multiplicative version is A064549 (sorted A001694), firsts of A003557.
%Y A280286 For length instead of sum we have A151821.
%Y A280286 These are the positions of first appearances in A280292 = A001414 - A008472.
%Y A280286 For indices instead of factors we have A380956 (sorted A380957), firsts of A380955.
%Y A280286 A multiplicative version for indices is A380987 (sorted A380988), firsts of A290106.
%Y A280286 For prime exponents instead of factors we have A380989, firsts of A380958.
%Y A280286 The sorted version is A381075.
%Y A280286 For product instead of sum see A381076, sorted firsts of A066503.
%Y A280286 A000040 lists the primes, differences A001223.
%Y A280286 A005117 lists squarefree numbers, complement A013929.
%Y A280286 A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
%Y A280286 A056239 adds up prime indices, row sums of A112798.
%Y A280286 Cf. A000720, A001222, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A364916, A366528.
%K A280286 nonn
%O A280286 2,1
%A A280286 _Michel Marcus_, Dec 31 2016