A248663 -id:A248663 - cp's OEIS frontend

A283975 Odd bisection of A264977.

1, 3, 1, 7, 5, 7, 1, 15, 13, 7, 5, 11, 13, 15, 1, 31, 29, 7, 13, 3, 1, 11, 5, 19, 21, 15, 13, 19, 29, 31, 1, 63, 61, 7, 29, 19, 25, 3, 13, 11, 9, 11, 1, 23, 25, 19, 5, 35, 37, 15, 21, 11, 9, 19, 13, 43, 37, 31, 29, 35, 61, 63, 1, 127, 125, 7, 61, 51, 41, 19, 29, 59, 49, 3, 25, 31, 17, 11, 13, 27, 25, 11, 9, 31, 21, 23, 1, 47, 33, 19, 25, 63, 41, 35, 5, 67, 69

Offset: 0

Antti Karttunen, Mar 25 2017

Antti Karttunen, Table of n, a(n) for n = 0..8190
Index entries for sequences related to binary expansion of n

Cf. A000120, A248663, A264977, A277324.

Cf. also A284266.

A264977[n_]:= If[n<2, n, If[EvenQ[n], 2 A264977[n/2], BitXor[A264977[(n - 1)/2], A264977[(n + 1)/2]]]]; Table[A264977[2n + 1], {n, 0, 150}] (* Indranil Ghosh, Mar 28 2017 *)

a(n) = if(n<2, n, if(n%2, bitxor(a((n - 1)/2), a((n + 1)/2)), 2*a(n/2))); for(n=0, 150, print1(a(2*n + 1),", ")) \\ Indranil Ghosh, Mar 28 2017

(define (A283975 n) (A264977 (+ n n 1)))

a(n) = A264977((2*n)+1).
a(n) = A248663(A277324(n)).
A000120(a(n)) = A284266(n).

A335424 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335423(i)) = A046523(A335423(j)) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 2, 1, 4, 2, 2, 2, 4, 3, 1, 2, 2, 2, 2, 4, 4, 2, 3, 1, 4, 2, 2, 2, 5, 2, 2, 4, 4, 3, 1, 2, 4, 4, 4, 2, 6, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 3, 4, 4, 4, 4, 2, 3, 2, 4, 2, 1, 4, 6, 2, 2, 4, 6, 2, 2, 2, 4, 2, 2, 3, 6, 2, 2, 1, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 3, 2, 2, 2, 1, 2, 6, 2, 4, 5

Offset: 1

Antti Karttunen, Jun 13 2020

nonn

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A162642(i) = A162642(j).

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

Cf. A046523, A162642, A248663, A335423, A335425.

Cf. also A286622, A305800.

up_to = 65537;
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; };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A248663(n) = A048675(core(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A335423(n) = A005940(1+A248663(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v335424 = rgs_transform(vector(up_to,n,A046523(A335423(n))));
A335424(n) = v335424[n];

A335425 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000188(i) = A000188(j) and A335424(i) = A335424(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 5, 2, 7, 4, 8, 2, 9, 2, 5, 7, 7, 2, 10, 11, 7, 9, 5, 2, 12, 2, 13, 7, 7, 4, 14, 2, 7, 7, 15, 2, 16, 2, 5, 9, 7, 2, 13, 17, 18, 7, 5, 2, 19, 7, 15, 7, 7, 2, 10, 2, 7, 9, 20, 7, 16, 2, 5, 7, 16, 2, 21, 2, 7, 18, 5, 4, 16, 2, 13, 22, 7, 2, 15, 7, 7, 7, 15, 2, 23, 7, 5, 7, 7, 7, 24, 2, 25, 9, 26, 2, 16, 2, 15, 12

Offset: 1

Antti Karttunen, Jun 13 2020

nonn

Restricted growth sequence transform of the ordered pair [A000188(n), A046523(A335423(n))].

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A000188, A001222, A046523, A162642, A248663, A305800, A335423, A335424.

up_to = 65537;
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; };
A000188(n) = core(n, 1)[2]; \\ From A000188
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A248663(n) = A048675(core(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A335423(n) = A005940(1+A248663(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux335425(n) = [A000188(n),A046523(A335423(n))];
v335425 = rgs_transform(vector(up_to,n,Aux335425(n)));
A335425(n) = v335425[n];

A335426 a(1) = 0; thereafter a(2^(2^k)) = 0 for k > 0, and for other even numbers n, a(n) = 1+a(n/2), and for odd numbers n, a(n) = 2*a(A064989(n)).

Original entry on oeis.org

0, 1, 2, 0, 4, 3, 8, 1, 0, 5, 16, 4, 32, 9, 6, 0, 64, 1, 128, 6, 10, 17, 256, 5, 0, 33, 2, 10, 512, 7, 1024, 1, 18, 65, 12, 2, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 0, 1, 66, 34, 32768, 3, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 2, 36, 19, 262144, 66, 258, 13, 524288, 3, 1048576, 2049, 2, 130, 24, 35, 2097152, 8, 0, 4097

Offset: 1

Antti Karttunen, Jun 15 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A001146, A003961, A048675, A064989, A209229, A225546, A248663, A335427, A335428.

Cf. A082522 (gives indices of zeros after a(1)=0).

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)};
A209229(n) = (n && !bitand(n,n-1));
A335426(n) = if(n<=2,n-1, if(!(n%2),if(A209229(n) && A209229(valuation(n,2)),0,1+A335426(n/2)), 2*A335426(A064989(n))));
\\ Alternatively:
A335426(n) = if(1==n,0, my(e=isprimepower(n)); if(e>1 && A209229(e), 0, if(!(n%2),1+A335426(n/2), 2*A335426(A064989(n)))));

a(1) = 0, and then after, a(2^(2^k)) = 0 for k > 0, and for other even numbers n, a(n) = 1+a(n/2), and for odd numbers n, a(n) = 2*a(A064989(n)).
a(n) = A335427(A225546(n)).
a(A003961(n)) = 2 * a(n).

cp's OEIS Frontend

A283975 Odd bisection of A264977.

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A335424 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335423(i)) = A046523(A335423(j)) for all i, j >= 1.

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A335425 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000188(i) = A000188(j) and A335424(i) = A335424(j) for all i, j >= 1.

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A335426 a(1) = 0; thereafter a(2^(2^k)) = 0 for k > 0, and for other even numbers n, a(n) = 1+a(n/2), and for odd numbers n, a(n) = 2*a(A064989(n)).

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