A302792 a(1) = 1; for n>1, a(n) = n/(smallest Fermi-Dirac factor of n).

1, 1, 1, 1, 1, 3, 1, 4, 1, 5, 1, 4, 1, 7, 5, 1, 1, 9, 1, 5, 7, 11, 1, 12, 1, 13, 9, 7, 1, 15, 1, 16, 11, 17, 7, 9, 1, 19, 13, 20, 1, 21, 1, 11, 9, 23, 1, 16, 1, 25, 17, 13, 1, 27, 11, 28, 19, 29, 1, 20, 1, 31, 9, 16, 13, 33, 1, 17, 23, 35, 1, 36, 1, 37, 25, 19, 11, 39, 1, 16, 1, 41, 1, 28, 17, 43, 29, 44, 1, 45, 13, 23, 31, 47, 19, 48, 1

Offset

1,6

Comments

The positive integers that are absent from this sequence are A036554, integers that have 2 as a Fermi-Dirac factor. - Peter Munn, Apr 23 2018

a(n) is the largest aliquot infinitary divisor of n, for n > 1 (cf. A077609). - Amiram Eldar, Nov 19 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = n / A223490(n).

a(n) = A300841(A302023(A032742(A302024(n)))).

Mathematica

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := n / Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 27 2020 *)

Prog

(PARI)
up_to = 65537;
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); };
A001511(n) = 1+valuation(n, 2);
A223490(n) = if(1==n, n, A050376(A001511(A052331(n))));
A302792(n) = (n/A223490(n));
(PARI) a(n) = {if(n==1, 1, my(f = factor(n)); for(i=1, #f~, f[i, 1] = f[i, 1]^(1<<valuation(f[i, 2], 2))); n/vecmin(f[, 1])); } \\ Amiram Eldar, Nov 19 2022

Crossrefs

Cf. A001511, A036554, A050376, A052331, A077609, A223490, A302023, A302024, A302776, A300841.

Cf. A084400 (gives the positions of 1's).

Cf. also A032742.

Sequence in context: A324542 A349042 A375820 * A179820 A364098 A363521

Adjacent sequences: A302789 A302790 A302791 * A302793 A302794 A302795

Keyword

nonn

Author

Antti Karttunen, Apr 13 2018

Status

approved