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A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2*A300841(n)) = 2*a(n), a(A300841(n)) = A003961(a(n)). 9
1, 2, 3, 5, 7, 4, 11, 6, 13, 10, 17, 9, 19, 14, 15, 23, 29, 22, 31, 25, 21, 26, 37, 8, 41, 34, 33, 35, 43, 12, 47, 38, 39, 46, 49, 55, 53, 58, 51, 18, 59, 20, 61, 65, 77, 62, 67, 57, 71, 74, 69, 85, 73, 28, 91, 30, 87, 82, 79, 27, 83, 86, 121, 95, 119, 44, 89, 115, 93, 50, 97, 42, 101, 94, 111, 145, 143, 52, 103, 133, 107, 106, 109, 45, 161 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Because "Fermi-Dirac factorization" is fundamentally different from ordinary prime factorization (as no exponents larger than 1 are allowed) this pair of permutations mapping between them is not always very intuitive. For example, we have ("as expected") A302776(n) = A302023(A052126(A302024(n))), while on the other hand, we have A302792(n) = A300841(A302023(A032742(A302024(n)))), where an additional shift-operator A300841 is needed for "correction".

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..32768

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = A005940(1+A052331(n)).

a(A050376(n)) = A000040(n).

A001221(a(n)) = A302790(n).

A001222(a(n)) = A064547(n).

PROG

(PARI)

up_to = 32768;

v050376 = vector(up_to);

A050376(n) = v050376[n];

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, 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); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler

A302024(n) = A005940(1+A052331(n));

CROSSREFS

Cf. A302023 (inverse).

Cf. A050376, A052331, A003961, A005940, A300840, A300841, A302791.

Sequence in context: A284145 A284189 A125151 * A273665 A212646 A103866

Adjacent sequences:  A302021 A302022 A302023 * A302025 A302026 A302027

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 15 2018

STATUS

approved

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Last modified April 25 12:04 EDT 2019. Contains 322456 sequences. (Running on oeis4.)