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%I #44 Apr 27 2018 17:09:40
%S 0,1,2,4,8,3,16,5,32,9,64,6,128,17,10,256,512,33,1024,12,18,65,2048,7,
%T 4096,129,34,20,8192,11,16384,257,66,513,24,36,32768,1025,130,13,
%U 65536,19,131072,68,40,2049,262144,258,524288,4097,514,132,1048576,35
%N Inverse of A052330; A binary encoding of Fermi-Dirac factorization of n, shown in decimal.
%C Every number can be represented uniquely as a product of numbers of the form p^(2^k), sequence A050376. This sequence is a binary representation of this factorization, with a(p^(2^k)) = 2^(i-1), where i is the index (A302778) of p^(2^k) in A050376. Additive with a(p^e) = sum a(p^(2^e_k)) where e = sum(2^e_k) is the binary representation of e and a(p^(2^k)) is as described above. - _Franklin T. Adams-Watters_, Oct 25 2005 - Index offset corrected by _Antti Karttunen_, Apr 17 2018
%H Antti Karttunen, <a href="/A052331/b052331.txt">Table of n, a(n) for n = 1..4096</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(1)=0; a(n*A050376(k)) = a(n) + 2^k for a(n) < 2^k, k=0, 1, ... - _Thomas Ordowski_, Mar 23 2005
%F From _Antti Karttunen_, Apr 13 2018: (Start)
%F a(1) = 0; for n > 1, a(n) = A000079(A302785(n)-1) + a(A302776(n)).
%F For n > 1, a(n) = A000079(A302786(n)-1) * A302787(n).
%F a(n) = A064358(n)-1.
%F A000120(a(n)) = A064547(n).
%F A069010(a(n)) = A302790(n).
%F (End)
%e n = 84 has Fermi-Dirac factorization A050376(5) * A050376(3) * A050376(2) = 7*4*3. Thus a(84) = 2^(5-1) + 2^(3-1) + 2^(2-1) = 16 + 4 + 2 = 22 ("10110" in binary = A182979(84)). - _Antti Karttunen_, Apr 17 2018
%o (PARI) A052331=a(n)={for(i=1,#n=factor(n)~,n[2,i]>1||next; m=binary(n[2,i]); n=concat(n,Mat(vector(#m-1,j,[n[1,i]^2^(#m-j),m[j]]~)));n[2,i]%=2); n||return(0); m=vecsort(n[1,]); forprime(p=1,m[#m],my(j=0);while(p^2^j<m[#m],setsearch(m,p^2^j)||n=concat(n,[p^2^j,0]~);j++));vector(#n=vecsort(n),i,2^i)*n[2,]~>>1} \\ _M. F. Hasler_, Apr 08 2015
%o (PARI)
%o up_to_e = 8192;
%o v050376 = vector(up_to_e);
%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 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); }; \\ _Antti Karttunen_, Apr 12 2018
%Y Cf. A050376, A052330, A064547, A302024, A302029, A302776, A302778, A302785, A302786, A302787, A302790, A302784.
%Y Cf. A182979 (same sequence shown in binary).
%Y One less than A064358.
%Y Cf. also A156552.
%K nonn
%O 1,3
%A _Christian G. Bower_, Dec 15 1999
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