A291759 -id:A291759 - cp's OEIS frontend

A292250 Binary encoding of 0-digits in ternary representation of A048673(n).

0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 3, 0, 2, 2, 1, 4, 6, 2, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 5, 6, 1, 4, 1, 4, 0, 8, 0, 14, 1, 4, 12, 0, 7, 0, 4, 2, 1, 0, 5, 8, 2, 0, 4, 2, 4, 0, 4, 6, 5, 0, 5, 8, 3, 12, 2, 4, 2, 8, 0, 4, 1, 8, 3, 2, 1, 16, 16, 2, 3, 28, 0, 0, 1, 8, 0, 26, 8, 0, 9, 0, 15, 0, 1, 10, 14, 4, 0, 4, 7, 0, 4, 12, 6, 16, 5, 6, 8, 0, 2, 10, 9, 4

Offset: 1

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A048673, A291759, A291770, A292251, A292252.

Cf. also A292240, A292260.

Map[FromDigits[IntegerDigits[#, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2] &, Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 116}]] (* Michael De Vlieger, Sep 12 2017 *)

a(n) = A291770(A048673(n)).

A351030 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351031(i) = A351031(j) and A351032(i) = A351032(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 23, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 46, 36, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of the ordered pair [A351031(n), A351032(n)], or equally, of the ordered pair [A351033(n), A351034(n)].

For all i, j: a(i) = a(j) => A349910(i) = A349910(j).

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

Cf. A019565, A048673, A289813, A289814, A291759, A304759, A351031, A351032 A351033, A351034.

Cf. also A293226, A349910.

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; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1+factorback(f))/2; };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A291759(n) = A289814(A048673(n));
A304759(n) = A289813(A048673(n));
A351031(n) = { my(m=1); fordiv(n,d,if(dA019565(A304759(d)))); (m); };
A351032(n) = { my(m=1); fordiv(n,d,if(dA019565(A291759(d)))); (m); };
Aux351030(n) = [A351031(n),A351032(n)];
v351030 = rgs_transform(vector(up_to, n, Aux351030(n)));
A351030(n) = v351030[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));

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.

A351034 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351032(i) = A351032(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 5, 1, 6, 1, 7, 1, 6, 8, 5, 1, 9, 1, 2, 1, 10, 1, 11, 1, 12, 8, 2, 8, 9, 1, 2, 1, 13, 1, 14, 1, 11, 15, 5, 1, 16, 8, 17, 1, 11, 1, 18, 8, 19, 1, 5, 1, 20, 1, 21, 8, 13, 1, 22, 1, 6, 8, 10, 1, 23, 1, 21, 24, 4, 25, 22, 1, 26, 27, 21, 1, 28, 1, 29, 8, 30, 1, 31, 8, 10, 15, 2, 1

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of A351032.

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

Cf. A019565, A048673, A289814, A291759, A351030, A351032, A351033.

Cf. also A293224.

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; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1+factorback(f))/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));
A351032(n) = { my(m=1); fordiv(n,d,if(dA019565(A291759(d)))); (m); };
v351034 = rgs_transform(vector(up_to, n, A351032(n)));
A351034(n) = v351034[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

A292250 Binary encoding of 0-digits in ternary representation of A048673(n).

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A351030 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351031(i) = A351031(j) and A351032(i) = A351032(j), for all i, j >= 1.

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

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A351034 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351032(i) = A351032(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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