A340376 -id:A340376 - cp's OEIS frontend

A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

1, 0, 2, 2, 3, 0, 0, 6, 7, 4, 1, 2, 4, 4, 0, 14, 5, 12, 6, 10, 9, 0, 4, 6, 1, 0, 4, 10, 5, 8, 1, 30, 8, 8, 14, 26, 2, 8, 13, 22, 3, 16, 0, 2, 17, 12, 8, 14, 1, 0, 10, 2, 10, 0, 9, 22, 3, 8, 11, 18, 9, 0, 18, 62, 0, 20, 12, 18, 1, 24, 13, 54, 15, 0, 28, 18, 0, 24, 12, 46, 37, 4, 8, 34, 7, 4, 0, 6, 11, 32, 23, 26, 22, 0

Offset: 1

Antti Karttunen, May 30 2018

Compare the logarithmic scatterplot to those of A291759, A292250 and A304760.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences related to binary expansion of n

Cf. A048673, A289813, A304758 (rgs-transform), A340381.
Cf. also A291759, A292250, A304760, A305295.
Cf. A340376 (positions of zeros), A340378 (binary weight).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));

a(n) = A289813(A048673(n)).

A340377 Numbers k such that there are no 2-digits in the ternary expansion of A048673(k).

Original entry on oeis.org

1, 3, 5, 9, 13, 17, 19, 21, 35, 47, 53, 59, 67, 71, 73, 91, 93, 95, 121, 123, 129, 143, 145, 157, 163, 173, 175, 179, 207, 211, 229, 233, 239, 255, 267, 291, 297, 299, 321, 327, 351, 355, 371, 381, 405, 413, 437, 451, 477, 479, 485, 487, 499, 503, 505, 523, 527, 541, 547, 549, 557, 595, 643, 645, 647, 661, 691, 701

Offset: 1

Antti Karttunen, Jan 15 2021

nonn

All terms are odd, because A048673(2n) = 3*A048673(n) - 1, which forces the least significant digit in the ternary expansion of A048673(2n) to be "2".

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of zeros in A291759 and in A340379. Positions of ones in A340382.

Cf. A048673, A340376.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
isA340377(n) = (0==A291759(n));

A340378 Number of 1-digits in the ternary representation of A048673(n).

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 0, 2, 3, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 2, 2, 0, 1, 2, 1, 0, 1, 2, 2, 1, 1, 4, 1, 1, 3, 3, 1, 1, 3, 3, 2, 1, 0, 1, 2, 2, 1, 3, 1, 0, 2, 1, 2, 0, 2, 3, 2, 1, 3, 2, 2, 0, 2, 5, 0, 2, 2, 2, 1, 2, 3, 4, 4, 0, 3, 2, 0, 2, 2, 4, 3, 1, 1, 2, 3, 1, 0, 2, 3, 1, 4, 3, 3, 0, 1, 4, 1, 0, 1, 1, 2, 1, 0, 2, 3

Offset: 1

Antti Karttunen, Jan 15 2021

nonn

Binary weight of A304759(n).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A048673, A062756, A286585, A304759, A340379.

Cf. A340376 (positions of zeros).

PARI
```
A340378(n) = hammingweight(A304759(n));
```

a(n) = A062756(A048673(n)) = A000120(A304759(n)).

a(n) = A286585(n) - 2*A340379(n).

A340381 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A304759(i)) = A278222(A304759(j)), for all i, j >= 1.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 2, 3, 4, 1, 1, 1, 1, 1, 2, 4, 5, 3, 3, 5, 5, 2, 1, 3, 1, 2, 1, 5, 5, 1, 1, 6, 1, 1, 4, 7, 1, 1, 7, 7, 3, 1, 2, 1, 5, 3, 1, 4, 1, 2, 5, 1, 5, 2, 5, 7, 3, 1, 7, 5, 5, 2, 5, 8, 2, 5, 3, 5, 1, 3, 7, 9, 6, 2, 4, 5, 2, 3, 3, 10, 11, 1, 1, 5, 4, 1, 2, 3, 7, 1, 10, 7, 7, 2, 1, 6, 1, 2, 1, 1, 5, 1, 2, 3, 7

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

For all i, j: A304758(i) = A304758(j) => a(i) = a(j) => A340378(i) = A340378(j).

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

Cf. A278222, A304758, A304759, A340378, A340382, A340383.
Cf. A340376 (positions of 2's).
Cf. also A305301.

up_to = 65537;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
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));
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; };
v340381 = rgs_transform(vector(up_to,n,A278222(A304759(n))));
A340381(n) = v340381[n];

A341345 a(n) = A048673(n) mod 3.

Original entry on oeis.org

1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 0, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 2, 0, 2, 0, 2, 1, 2, 1, 2, 1, 2, 0, 2, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 0, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1

Offset: 1

Antti Karttunen, Feb 09 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A010872, A048673, A292247, A292248, A292251, A292252.
Cf. A007395 (even bisection), A341346 (odd bisection), A341347.
Cf. also A291759, A292243, A292244, A292245, A292246, A292250, A304759, A340376, A340377.
Cf. also A292603.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341345(n) = (((A003961(n)+1)/2)%3);

a(n) = A010872(A048673(n)).
a(n) = 0 iff A292247(n) is odd.
a(n) = 0 iff A292250(n) is odd, or equally, iff both A291759(n) and A304759(n) are even.
a(n) = 0 iff A292251(n) > 0.
a(n) = 1 iff A292248(n) is odd.
a(n) = 1 iff A304759(n) is odd, or equally, iff both A291759(n) and A292250(n) are even.
a(2n) = 2.

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A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

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A340377 Numbers k such that there are no 2-digits in the ternary expansion of A048673(k).

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A340378 Number of 1-digits in the ternary representation of A048673(n).

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A340381 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A304759(i)) = A278222(A304759(j)), for all i, j >= 1.

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A341345 a(n) = A048673(n) mod 3.

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