A302023 Permutation of natural numbers mapping ordinary factorization to "Fermi-Dirac factorization": a(1) = 1, a(2n) = 2*A300841(a(n)), a(A003961(n)) = A300841(a(n)).

1, 2, 3, 6, 4, 8, 5, 24, 12, 10, 7, 30, 9, 14, 15, 120, 11, 40, 13, 42, 21, 18, 16, 168, 20, 22, 60, 54, 17, 56, 19, 840, 27, 26, 28, 210, 23, 32, 33, 216, 25, 72, 29, 66, 84, 34, 31, 1080, 35, 70, 39, 78, 37, 280, 36, 264, 48, 38, 41, 270, 43, 46, 108, 7560, 44, 88, 47, 96, 51, 90, 49, 1512, 53, 50, 105, 102, 45, 104, 59, 1320

Offset: 1

Author

Antti Karttunen, Apr 15 2018

Keywords

• nonn

Comments

See comments and additional formulas in A302024.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A302024 (inverse).

Cf. A050376, A052330, A064989, A156552, A300841.

Cf. also A091202, A302025.

Programs

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));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
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); };
A300841(n) = A052330(2*A052331(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A302023(n) = if(1==n,n,if(!(n%2),2*A300841(A302023(n/2)),A300841(A302023(A064989(n)))));

Formula

a(1) = 1; a(2n) = 2*A300841(a(n)), a(2n+1) = A300841(a(A064989(2n+1))). [corrected Jun 10 2018]

a(n) = A052330(A156552(n)).

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