A302846 -id:A302846 - cp's OEIS frontend

A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

0, 1, 3, 2, 15, 14, 12, 13, 5, 6, 4, 7, 10, 9, 11, 8, 21, 20, 22, 23, 16, 19, 17, 18, 26, 27, 25, 24, 31, 28, 30, 29, 63, 62, 60, 61, 48, 49, 51, 50, 58, 57, 59, 56, 53, 54, 52, 55, 42, 43, 41, 40, 47, 44, 46, 45, 37, 36, 38, 39, 32, 35, 33, 34, 255, 254, 252, 253, 240, 241, 243, 242, 250, 249, 251, 248, 245, 246, 244

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302846 (inverse).

Cf. A006068, A057300, A064707, A163355, A302843, A302782.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A064707(n) = A006068(A006068(n));
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d,f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r))));
A302845(n) = A163355(A064707(n));

a(n) = A163355(A064707(n)).

a(n) = A302843(A006068(n)).

A302844 Permutation of nonnegative integers: a(n) = A003188(A163356(n)).

Original entry on oeis.org

0, 1, 2, 3, 12, 15, 14, 13, 10, 9, 8, 11, 4, 5, 6, 7, 24, 27, 26, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 21, 20, 23, 40, 43, 42, 41, 46, 47, 44, 45, 34, 35, 32, 33, 38, 37, 36, 39, 56, 57, 58, 59, 52, 55, 54, 53, 50, 49, 48, 51, 60, 61, 62, 63, 192, 195, 194, 193, 198, 199, 196, 197, 202, 203, 200, 201, 206, 205

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

When A207901, which is a multiplicative walk permutation, is composed from the right with this permutation, the result is A302781, another multiplicative walk permutation.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302843 (inverse).

Cf. A003188, A006068, A057300, A163356, A302846, A302781.

A003188(n) = bitxor(n, n>>1);
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163356(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); (((((2+(i%2))^d)%5)-1)*f) + if(3==d,f-1-A163356(r),A057300(A163356(r)))); \\ Antti Karttunen, Apr 14 2018
A302844(n) = A003188(A163356(n));

a(n) = A003188(A163356(n)).

a(n) = A006068(A302846(n)).

A303773 Permutation of nonnegative integers constructed with a greedy algorithm producing either-subset-or-superset-mask type of walk in binary lattice (see comments for the exact definition).

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 5, 7, 15, 11, 10, 8, 9, 13, 12, 14, 30, 22, 20, 16, 17, 19, 18, 26, 24, 25, 27, 31, 23, 21, 29, 28, 60, 44, 40, 32, 33, 35, 34, 38, 36, 37, 39, 47, 41, 43, 42, 46, 62, 58, 50, 48, 49, 51, 55, 53, 52, 54, 118, 82, 66, 64, 65, 67, 71, 69, 68, 70, 78, 74, 72, 73, 75, 79, 77, 76, 92, 88, 80, 81, 83, 87, 85, 84, 86, 94, 90, 91, 89, 93, 95, 127

Offset: 0

Antti Karttunen, May 05 2018

a(0) = 0 and for n > 0, if there are one or more k_i that are not already present in the sequence among terms a(0) .. a(n-1), and for which bitor(k_i,a(n-1)) = a(n-1), then a(n) = that k_i which gives minimum value of A003961(k_i) amongst them; otherwise, when no such k_i exists, a(n) = the least number not already present that can be obtained by toggling a single 0-bit of a(n-1) to 1. This is done by trying to toggle successive vacant bits from the least significant end of the binary representation of a(n-1), until such a sum a(n-1) + 2^h (= a(n-1) bitxor 2^h) is found that is not already present in the sequence.

Shares with permutations like A003188, A006068, A163252, A300838, A302846, A303763, A303765, A303767 and A303775 the property that when moving from any a(n) to a(n+1) either a subset of 0-bits are toggled on (changed to 1's, in this case always only a single bit), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes may occur at the same step.

Cf. A303774 (inverse).
Cf. A303770.
Cf. A303763, A303765, A303767, A303775 for similar constructions.

up_to = (2^18)-1;
A006519(n) = (2^valuation(n, 2));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
v303773 = vector(up_to);
m303774 = Map();
w=1; for(n=1,up_to,s = Set([]); for(m=1,w, if((bitor(w,m)==w) && !mapisdefined(m303774,m), s = setunion(Set([A003961(m)]),s))); if(length(s)>0, w = A064989(vecmin(s)), b=A006519(1+w); while(bitand(w,b) || mapisdefined(m303774,w+b), b <<= 1); w += b); v303773[n] = w; mapput(m303774,w,n));
A303773(n) = if(!n,n,v303773[n]);
A303774(n) = if(!n,n,mapget(m303774,n));

For all n >= 0, A019565(a(n)) = A303770(n).

A302853 Suspected permutation of nonnegative integers: a(n) = A052331(A282291(1+n)).

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 12, 8, 9, 11, 10, 14, 30, 16, 17, 19, 18, 22, 20, 28, 24, 25, 27, 26, 31, 5, 7, 15, 13, 29, 21, 23, 87, 64, 65, 67, 66, 70, 68, 76, 72, 73, 75, 74, 78, 94, 80, 81, 83, 82, 86, 84, 92, 88, 89, 91, 90, 95, 69, 71, 79, 77, 93, 85, 117, 32, 33, 41, 40, 44, 36, 52, 48, 49, 57, 56, 60, 124, 96, 97, 105, 104, 108, 100, 116, 112, 113, 121, 120, 125

Offset: 0

Antti Karttunen, May 17 2018

nonn

Shares with sequences like A003188, A006068, A300838, A302846, A303765, A303767, A304083 and A304533 the property that when moving from any a(n) to a(n+1) either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes may occur at the same step.

Cf. A302854 (inverse).
Cf. A052331, A282291.
Cf. also A304533.

up_to_e = 2^15;
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));
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); };
A302853(n) = A052331(A282291(1+n)); \\ Needs also code from A282291.

a(n) = A052331(A282291(1+n)).

cp's OEIS Frontend

A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

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A302844 Permutation of nonnegative integers: a(n) = A003188(A163356(n)).

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A303773 Permutation of nonnegative integers constructed with a greedy algorithm producing either-subset-or-superset-mask type of walk in binary lattice (see comments for the exact definition).

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A302853 Suspected permutation of nonnegative integers: a(n) = A052331(A282291(1+n)).

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