A300840 - OEIS

A300840

Fermi-Dirac factorization prime shift towards smaller terms: a(n) = A052330(floor(A052331(n)/2)).

1, 1, 2, 3, 4, 2, 5, 3, 7, 4, 9, 6, 11, 5, 8, 13, 16, 7, 17, 12, 10, 9, 19, 6, 23, 11, 14, 15, 25, 8, 29, 13, 18, 16, 20, 21, 31, 17, 22, 12, 37, 10, 41, 27, 28, 19, 43, 26, 47, 23, 32, 33, 49, 14, 36, 15, 34, 25, 53, 24, 59, 29, 35, 39, 44, 18, 61, 48, 38, 20, 67, 21, 71, 31, 46, 51, 45, 22, 73, 52, 79, 37, 81, 30, 64, 41, 50, 27

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OFFSET

1,3

COMMENTS

With n having a unique factorization as fdp(i) * fdp(j) * ... * fdp(k), with i, j, ..., k all distinct, a(n) = fdp(i-1) * fdp(j-1) * ... * fdp(k-1), where fdp(0) = 1 and fdp(n) = A050376(n) for n >= 1.

Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - Andrew Howroyd, Aug 02 2018

LINKS

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

FORMULA

a(n) = A052330(floor(A052331(n)/2)).

For all n >= 1, a(A300841(n)) = n.

a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Nov 30 2019

MATHEMATICA

fdPrimeQ[n_] := Module[{f = FactorInteger[n], e}, Length[f] == 1 && (2^IntegerExponent[(e = f[[1, 2]]), 2] == e)];

prevFDPrime[n_] := Module[{k = n - 1}, While[! fdPrimeQ[k], k--]; k];

fd[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Table[If[b[[j]] > 0, p^(2^(m - j)), Nothing], {j, 1, m}]];

a[n_] := Times @@ prevFDPrime /@ Flatten[fd @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

PROG

(PARI)

up_to_e = 8192;

v050376 = vector(up_to_e);

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_e, 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); };

A300840(n) = A052330(A052331(n)>>1);

CROSSREFS

A left inverse of A300841.

Cf. A050376, A052330, A052331, A059897.

Cf. also A064989.

Sequence in context: A161759 A260643 A366295 * A243849 A286547 A157000

Adjacent sequences: A300837 A300838 A300839 * A300841 A300842 A300843

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Apr 13 2018

STATUS

approved