A340378 -id:A340378 - 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)).

A340376 Numbers k such that there are no 1-digits in the ternary expansion of A048673(k).

Original entry on oeis.org

2, 6, 7, 15, 22, 26, 43, 50, 54, 62, 65, 74, 77, 87, 94, 98, 103, 138, 183, 190, 198, 214, 218, 221, 235, 278, 302, 343, 353, 406, 421, 426, 430, 439, 463, 465, 467, 475, 498, 506, 534, 574, 578, 610, 633, 646, 662, 666, 682, 734, 799, 843, 862, 869, 870, 882, 886, 910, 949, 967, 977, 987, 1013, 1014, 1087, 1121

Offset: 1

Antti Karttunen, Jan 15 2021

nonn

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 A304759 and in A340378. Positions of 2's in A340381.

Cf. A048673, A340377.

PARI
```
isA340376(n) = (0==A304759(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];

A340379 Number of 2-digits in the ternary representation of A048673(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 15 2021

nonn

Binary weight of A291759(n).

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

Cf. A000120, A048673, A081603, A286585, A291759, A292252.

Cf. A340377 (positions of zeros).

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));
A340379(n) = hammingweight(A291759(n));

a(n) = A081603(A048673(n)) = A000120(A291759(n)).
a(n) = (A286585(n) - A340378(n)) / 2.
For all n >= 1, a(n) >= A292252(n).

cp's OEIS Frontend

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

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A340376 Numbers k such that there are no 1-digits in the ternary expansion of A048673(k).

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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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A340379 Number of 2-digits in the ternary representation of A048673(n).

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