A302791 -id:A302791 - cp's OEIS frontend

A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2A300841(n)) = 2a(n), a(A300841(n)) = A003961(a(n)).

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

Offset: 1

Antti Karttunen, Apr 15 2018

nonn

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".

Cf. A302023 (inverse).

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

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));

a(n) = A005940(1+A052331(n)).
a(A050376(n)) = A000040(n).
A001221(a(n)) = A302790(n).
A001222(a(n)) = A064547(n).

A304535 Restricted growth sequence transform of A278222(A304533(n)).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 4, 2, 4, 5, 2, 4, 6, 4, 7, 3, 6, 4, 8, 4, 7, 9, 6, 10, 4, 6, 8, 5, 11, 6, 8, 2, 4, 6, 4, 6, 4, 5, 4, 5, 8, 2, 4, 6, 4, 6, 4, 6, 3, 6, 11, 3, 6, 12, 6, 9, 7, 9, 7, 10, 5, 11, 13, 11, 14, 6, 12, 15, 11, 13, 12, 16, 4, 5, 8, 5, 8, 5, 8, 6, 17, 5, 18, 5, 11, 19, 18, 20, 11, 19, 5, 8, 16, 8, 17, 2, 4, 5, 4, 5, 4, 5, 4, 5, 18, 4

Offset: 0

Antti Karttunen, May 15 2018

nonn

For all i, j: a(i) = a(j) => A304536(i) = A304536(j).

Antti Karttunen, Table of n, a(n) for n = 0..74430

Cf. A278222, A302791, A304531, A304533, A304536.

A302787 a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 5, 1, 9, 1, 3, 1, 17, 5, 1, 1, 33, 1, 3, 9, 65, 1, 7, 1, 129, 17, 5, 1, 11, 1, 257, 33, 513, 3, 9, 1, 1025, 65, 13, 1, 19, 1, 17, 5, 2049, 1, 129, 1, 4097, 257, 33, 1, 35, 9, 21, 513, 8193, 1, 7, 1, 16385, 3, 65, 17, 67, 1, 129, 1025, 25, 1, 37, 1, 32769, 2049, 257, 5, 131, 1, 33, 1, 65537, 1, 11, 65, 131073, 4097, 69, 1, 41, 9

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

After n=1, differs from A240535 (which gives the same terms, but with mirrored binary expansion) for the first time at n=30, where a(30) = 11, while A240535(30) = 13. Note how 11 = "1011" and 13 = "1101" in binary.

For all i, j: a(i) = a(j) => A302791(i) = A302791(j).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000265, A050376, A052331, A240535, A302786, A302788, A302789, A302790, A302791.

Cf. also A246277.

up_to = 8192;
v050376 = vector(up_to);
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); };
A000265(n) = (n/2^valuation(n, 2));
A302787(n) = if(1==n,0,A000265(A052331(n)));

a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).
For n > 1, a(n) = A030101(A240535(n)).
For n >= 1, A069010(a(n)) = A302790(n).

A305979 Filter sequence for a(Fermi-Dirac primes) = constant sequences.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 8, 2, 2, 9, 2, 10, 11, 12, 2, 13, 2, 14, 15, 16, 2, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 2, 40, 41, 42, 43, 44, 2, 45, 46, 47, 2, 48, 2, 49, 50, 51, 52, 53, 2, 54, 2, 55, 2, 56, 57, 58, 59, 60, 2, 61, 62, 63, 64, 65, 66, 67, 2, 68, 69, 70, 2

Offset: 1

Antti Karttunen, Jul 02 2018

nonn

For all i, j:
  a(i) = a(j) => A305976(i) = A305976(j).
  a(i) = a(j) => A302791(i) = A302791(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A050376, A302777, A302778, A302791, A305975, A305976.

up_to = 100000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
A209229(n) = (n && !bitand(n, n-1));
A302777(n) = A209229(isprimepower(n));
v302778 = partialsums(A302777,up_to);
A302778(n) = v302778[n];
A305979(n) = if(1==n,n,if(1==A302777(n),2,1+n-A302778(n)));

a(1) = 1; for n > 1, if A302777(n) is 1 [when n is in A050376], a(n) = 2, otherwise a(n) = 1 + n - A302778(n) = running count from 3 onward.

A304097 Restricted growth sequence transform of A278222(A302853(n)).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 2, 4, 5, 4, 6, 7, 2, 4, 5, 4, 5, 4, 6, 3, 5, 8, 5, 9, 4, 6, 7, 5, 10, 11, 10, 12, 2, 4, 5, 4, 5, 4, 5, 4, 11, 13, 11, 10, 14, 4, 11, 13, 11, 13, 11, 10, 5, 13, 15, 13, 16, 11, 10, 14, 13, 12, 17, 12, 2, 4, 11, 4, 5, 4, 5, 3, 5, 10, 6, 7, 9, 3, 5, 13, 5, 8, 5, 10, 6, 10, 14, 7, 16, 11, 13, 14, 13, 18, 2, 4, 5, 4, 5, 4, 5, 4, 11, 13, 11

Offset: 0

Antti Karttunen, May 17 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..60822

Cf. A278222, A302791, A282291, A302853, A304098.

Compare also the scatter-plot to that of A304535.

\\ Needs also code from A282291 and A302853:
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
write_to_bfile(0,rgs_transform(vector(60823,n,A278222(A302853(n-1)))),"b304097.txt");

A304744 Restricted growth sequence transform of A046523(A052330(n)).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 7, 2, 3, 3, 8, 6, 7, 9, 10, 2, 3, 3, 8, 6, 7, 9, 10, 3, 8, 8, 11, 9, 10, 12, 13, 4, 6, 5, 7, 14, 15, 15, 16, 6, 9, 7, 10, 17, 18, 18, 19, 6, 9, 7, 10, 17, 18, 18, 19, 9, 12, 10, 13, 20, 21, 21, 22, 2, 3, 3, 8, 6, 7, 9, 10, 3, 8, 8, 11, 9, 10, 12, 13, 3, 8, 8, 11, 9, 10, 12, 13, 8, 11, 11, 23, 12, 13, 24, 25, 6, 9, 7, 10, 17, 18, 18, 19

Offset: 0

Antti Karttunen, May 27 2018

nonn

For all i, j: a(i) = a(j) => A000120(i) = A000120(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A046523, A052330.

Cf. also A286622 (A278222), A302791, A304535, A304745.

up_to_e = 17; \\ Good for computing up to n = (2^up_to_e)-1
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));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v304744 = rgs_transform(vector(65538,n,A046523(A052330(n-1))));
A304744(n) = v304744[1+n];

cp's OEIS Frontend

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)).

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A304535 Restricted growth sequence transform of A278222(A304533(n)).

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A302787 a(1) = 0; for n > 1, a(n) = A000265(A052331(n)).

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A305979 Filter sequence for a(Fermi-Dirac primes) = constant sequences.

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A304097 Restricted growth sequence transform of A278222(A302853(n)).

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A304744 Restricted growth sequence transform of A046523(A052330(n)).

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A302024 Permutation of natural numbers mapping "Fermi-Dirac factorization" to ordinary factorization: a(1) = 1, a(2A300841(n)) = 2a(n), a(A300841(n)) = A003961(a(n)).