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%I #24 Sep 07 2023 07:15:36
%S 1,3,4,5,7,12,9,15,11,21,13,20,16,27,28,17,19,33,23,35,36,39,25,60,29,
%T 48,44,45,31,84,37,51,52,57,63,55,41,69,64,105,43,108,47,65,77,75,49,
%U 68,53,87,76,80,59,132,91,135,92,93,61,140,67,111,99,85,112,156,71,95,100,189,73,165,79,123,116,115,117,192,81
%N Fermi-Dirac factorization prime shift towards larger terms: a(n) = A052330(2*A052331(n)).
%C With n having a unique factorization as A050376(i) * A050376(j) * ... * A050376(k), with i, j, ..., k all distinct, a(n) = A050376(1+i) * A050376(1+j) * ... * A050376(1+k).
%C Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - _Andrew Howroyd_, Aug 02 2018
%H Antti Karttunen, <a href="/A300841/b300841.txt">Table of n, a(n) for n = 1..32768</a>
%F a(n) = A052330(2*A052331(n)).
%F For all n >= 1, a(A050376(n)) = A050376(1+n).
%F For all n >= 1, A300840(a(n)) = n.
%F a(A059897(n,k)) = A059897(a(n), a(k)). - _Peter Munn_, Nov 23 2019
%e For n = 6 = A050376(1)*A050376(2), a(6) = A050376(2)*A050376(3) = 3*4 = 12.
%e For n = 12 = A050376(2)*A050376(3), a(12) = A050376(3)*A050376(4) = 4*5 = 20.
%t fdPrimeQ[n_] := Module[{f = FactorInteger[n], e}, Length[f] == 1 && (2^IntegerExponent[(e = f[[1, 2]]), 2] == e)];
%t nextFDPrime[n_] := Module[{k = n + 1}, While[! fdPrimeQ[k], k++]; k];
%t fd[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Table[If[b[[j]] > 0, p^(2^(m - j)), Nothing], {j, 1, m}]];
%t a[n_] := Times @@ nextFDPrime /@ Flatten[fd @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Sep 07 2023 *)
%o (PARI)
%o up_to_e = 8192;
%o v050376 = vector(up_to_e);
%o A050376(n) = v050376[n];
%o ispow2(n) = (n && !bitand(n,n-1));
%o i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
%o A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
%o A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
%o A300841(n) = A052330(2*A052331(n));
%Y Cf. A050376, A052330, A052331, A059897, A300840 (a left inverse).
%Y Cf. also A003961.
%Y Range of values is A003159.
%K nonn,mult
%O 1,2
%A _Antti Karttunen_, Apr 12 2018
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