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A052331 Inverse of A052330; A binary encoding of Fermi-Dirac factorization of n, shown in decimal. 34
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

Index entries for sequences that are permutations of the natural numbers

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. 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 November 16 22:43 EST 2018. Contains 317275 sequences. (Running on oeis4.)