A297172 -id:A297172 - cp's OEIS frontend

A297162 Restricted growth sequence transform of A297172, which is Möbius transform of A253564.

1, 2, 3, 2, 4, 2, 5, 6, 3, 3, 7, 3, 8, 4, 3, 9, 10, 6, 11, 4, 12, 5, 13, 14, 4, 7, 14, 5, 15, 3, 16, 17, 18, 8, 4, 9, 19, 10, 20, 21, 22, 4, 23, 7, 12, 11, 24, 25, 5, 9, 26, 8, 27, 9, 18, 28, 29, 13, 30, 31, 32, 15, 18, 33, 34, 5, 35, 10, 36, 31, 37, 17, 38, 16, 14, 11, 5, 7, 39, 40, 25, 19, 41, 42, 43, 22, 44, 45, 46, 14, 20, 13, 47, 23

Offset: 1

Antti Karttunen, Dec 27 2017

nonn

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

Cf. A253564, A297172.

Cf. also A297113, A297157, A297161.

up_to = 8192;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A253564(n) = A156552(A122111(n));
A297172(n) = sumdiv(n,d,moebius(n/d)*A253564(d));
write_to_bfile(1,rgs_transform(vector(up_to,n,A297172(n))),"b297162.txt");

A297112 Möbius transform of A156552.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 8, 4, 4, 4, 16, 4, 32, 8, 4, 8, 64, 4, 128, 8, 8, 16, 256, 8, 8, 32, 8, 16, 512, 4, 1024, 16, 16, 64, 8, 8, 2048, 128, 32, 16, 4096, 8, 8192, 32, 8, 256, 16384, 16, 16, 8, 64, 64, 32768, 8, 16, 32, 128, 512, 65536, 8, 131072, 1024, 16, 32, 32, 16, 262144, 128, 256, 8, 524288, 16, 1048576, 2048, 8

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A008683, A064989, A156552, A297113.

Cf. also A297106, A297156, A297171, A297172.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));

;; With memoization-macro definec.
(definec (A297112 n) (cond ((<= n 2) (- n 1)) ((odd? n) (* 2 (A297112 (A064989 n)))) ((= 2 (modulo n 4)) (A297112 (/ n 2))) (else (* 2 (A297112 (/ n 2)))))) ;; Antti Karttunen, Dec 27 2017

a(1) = 0, a(2) = 1, after which, a(2n+1) = 2*a(A064989(2n+1)), a(4n) = 2*a(2n), a(4n+2) = a(2n+1).
a(n) = Sum_{d|n} A008683(n/d)*A156552(d).
For n > 1, a(n) = A000079(A297113(n)-1).

A297171 Möbius transform of A243071.

Original entry on oeis.org

0, 1, 3, 1, 7, 2, 15, 2, 2, 6, 31, 5, 63, 14, 3, 4, 127, 2, 255, 13, 11, 30, 511, 10, 4, 62, 4, 29, 1023, 4, 2047, 8, 27, 126, 5, 4, 4095, 254, 59, 26, 8191, 12, 16383, 61, 10, 510, 32767, 20, 8, 4, 123, 125, 65535, 4, 21, 58, 251, 1022, 131071, 7, 262143, 2046, 26, 16, 53, 28, 524287, 253, 507, 6, 1048575, 8

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A008683, A064989, A243071, A297161 (rgs-transform of this sequence).

Cf. also A297112, A297156, A297172.

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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));

a(n) = Sum_{d|n} A008683(n/d)*A243071(d).

A297156 Möbius transform of A243354.

Original entry on oeis.org

0, 1, 3, 1, 7, 2, 15, 3, 1, 6, 31, 6, 63, 14, 2, 5, 127, 2, 255, 14, 10, 30, 511, 10, 1, 62, 7, 30, 1023, 4, 2047, 11, 26, 126, 2, 2, 4095, 254, 58, 26, 8191, 12, 16383, 62, 14, 510, 32767, 22, 1, 2, 122, 126, 65535, 6, 18, 58, 250, 1022, 131071, 4, 262143, 2046, 30, 21, 50, 28, 524287, 254, 506, 4, 1048575, 6

Offset: 1

Antti Karttunen, Dec 28 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A006068, A156552, A243354, A297157 (rgs-transform of this sequence).

Cf. also A297112, A297171, A297172.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ Essentially Joerg Arndt's Jul 19 2012 code.
A243354(n) = A006068(A156552(n));
A297156(n) = sumdiv(n,d,moebius(n/d)*A243354(d));

cp's OEIS Frontend

A297162 Restricted growth sequence transform of A297172, which is Möbius transform of A253564.

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A297112 Möbius transform of A156552.

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A297171 Möbius transform of A243071.

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A297156 Möbius transform of A243354.

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