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A052331 Inverse of A052330; A binary encoding of Fermi-Dirac factorization of n, shown in decimal. 44
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
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<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
(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. A182979 (same sequence shown in binary).
One less than A064358.
Cf. also A156552.
Sequence in context: A254788 A347669 A352812 * A242365 A119436 A277695
KEYWORD
nonn
AUTHOR
Christian G. Bower, Dec 15 1999
STATUS
approved

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Last modified March 28 04:05 EDT 2024. Contains 371235 sequences. (Running on oeis4.)