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 A052331 Inverse of A052330; A binary encoding of Fermi-Dirac factorization of n, shown in decimal. 42
 0, 1, 2, 4, 8, 3, 16, 5, 32, 9, 64, 6, 128, 17, 10, 256, 512, 33, 1024, 12, 18, 65, 2048, 7, 4096, 129, 34, 20, 8192, 11, 16384, 257, 66, 513, 24, 36, 32768, 1025, 130, 13, 65536, 19, 131072, 68, 40, 2049, 262144, 258, 524288, 4097, 514, 132, 1048576, 35 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 LINKS Antti Karttunen, Table of n, a(n) for n = 1..4096 FORMULA a(1)=0; a(n*A050376(k)) = a(n) + 2^k for a(n) < 2^k, k=0, 1, ... - Thomas Ordowski, Mar 23 2005 From Antti Karttunen, Apr 13 2018: (Start) a(1) = 0; for n > 1, a(n) = A000079(A302785(n)-1) + a(A302776(n)). For n > 1, a(n) = A000079(A302786(n)-1) * A302787(n). a(n) = A064358(n)-1. A000120(a(n)) = A064547(n). A069010(a(n)) = A302790(n). (End) EXAMPLE 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 PROG (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>1} \\ M. F. Hasler, Apr 08 2015 (PARI) up_to_e = 8192; v050376 = vector(up_to_e); ispow2(n) = (n && !bitand(n, n-1)); i = 0; for(n=1, oo, if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e, break)); 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 CROSSREFS Cf. A050376, A052330, A064547, A302024, A302029, A302776, A302778, A302785, A302786, A302787, A302790, A302784. Cf. A182979 (same sequence shown in binary). One less than A064358. Cf. also A156552. Sequence in context: A277272 A109588 A254788 * A242365 A119436 A277695 Adjacent sequences:  A052328 A052329 A052330 * A052332 A052333 A052334 KEYWORD nonn AUTHOR Christian G. Bower, Dec 15 1999 STATUS approved

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Last modified January 16 17:26 EST 2021. Contains 340206 sequences. (Running on oeis4.)