A304759 -id:A304759 - cp's OEIS frontend

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

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];

A304758 Restricted growth sequence transform of A304759(n), formed from 1-digits in ternary representation of A048673(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 2, 2, 5, 6, 7, 1, 3, 7, 7, 2, 8, 9, 10, 5, 11, 12, 2, 7, 5, 1, 2, 7, 11, 9, 13, 1, 14, 13, 13, 8, 15, 3, 13, 16, 17, 4, 18, 2, 3, 19, 10, 13, 8, 1, 2, 11, 3, 11, 2, 12, 17, 4, 13, 20, 21, 12, 2, 21, 22, 2, 23, 10, 21, 1, 24, 16, 25, 26, 2, 27, 21, 2, 24, 10, 28, 29, 7, 13, 30, 6, 7, 2, 5, 20, 31, 32, 15, 17, 2, 18, 14, 13, 2, 1, 3, 12, 18, 2, 5

Offset: 1

Antti Karttunen, May 30 2018

nonn

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

Cf. A048673, A289813, A304759.

Cf. also A304740, A304748.

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)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(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; };
v304758 = rgs_transform(vector(65537,n,A304759(n)));
A304758(n) = v304758[n];

A351031 a(n) = Product_{d|n, dA019565(A304759(d)).

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 6, 6, 12, 2, 18, 2, 2, 36, 90, 2, 180, 2, 180, 6, 4, 2, 810, 12, 10, 180, 30, 2, 180, 2, 9450, 12, 20, 12, 56700, 2, 30, 30, 56700, 2, 420, 2, 12, 1080, 10, 2, 1275750, 2, 120, 60, 30, 2, 31500, 24, 9450, 90, 20, 2, 238140, 2, 4, 2520, 10914750, 60, 84, 2, 420, 30, 31500, 2, 2946982500, 2, 6

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Index entries for sequences computed from indices in prime factorization

Cf. A019565, A048673, A289813, A304759, A351030, A351032, A351033 (rgs-transform).

Cf. also A293221.

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); } \\ From A289813
A304759(n) = A289813(A048673(n));
A351031(n) = { my(m=1); fordiv(n,d,if(dA019565(A304759(d)))); (m); };

A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

Original entry on oeis.org

1, 2, 1, 3, 4, 5, 2, 6, 7, 3, 3, 8, 1, 9, 2, 10, 11, 6, 4, 12, 11, 13, 3, 14, 9, 15, 3, 16, 12, 17, 3, 18, 3, 3, 7, 19, 3, 9, 19, 19, 6, 3, 5, 8, 12, 20, 1, 21, 8, 22, 12, 23, 11, 24, 12, 25, 6, 8, 26, 27, 12, 13, 12, 28, 13, 29, 4, 12, 9, 20, 26, 30, 31, 22, 10, 16, 5, 14, 6, 32, 33, 8, 3, 12, 10, 23, 15, 14, 19, 8

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

Restricted growth sequence transform of the ordered pair [A340381(n), A340382(n)], or equally, of the function f(n) = A290093(A048673(n)).

For all i, j: a(i) = a(j) => A286586(i) = A286586(j) => A286585(i) = A286585(j).

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

Cf. A048673, A278222, A290093, A291759, A304759, A340381, A340382, A340684.

Cf. also A290094, A305303.

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;
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));
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; };
Aux340383(n) = [A278222(A291759(n)),A278222(A304759(n))];
v340383 = rgs_transform(vector(up_to,n,Aux340383(n)));
A340383(n) = v340383[n];

A340684 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A291759(n), A278222(A304759(n))], for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

Restricted growth sequence transform of the ordered pair [A291759(n), A278222(A304759(n))].

For all i, j: a(i) = a(j) => A340383(i) = A340383(j).

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

Cf. A048673, A278222, A290093, A291759, A304759, A340383.

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;
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));
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; };
Aux340684(n) = [A291759(n),A278222(A304759(n))];
v340684 = rgs_transform(vector(up_to,n,Aux340684(n)));
A340684(n) = v340684[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));
```

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];

A305295 Binary encoding of 1-digits in ternary representation of A245612(n).

Original entry on oeis.org

1, 0, 2, 2, 6, 7, 0, 3, 14, 4, 12, 1, 2, 0, 4, 0, 30, 37, 0, 5, 26, 28, 0, 1, 6, 17, 8, 14, 10, 9, 4, 1, 62, 16, 72, 103, 2, 90, 8, 0, 54, 25, 60, 33, 2, 32, 0, 19, 14, 40, 32, 40, 18, 11, 24, 0, 22, 18, 16, 9, 10, 8, 0, 4, 126, 333, 36, 305, 146, 4, 204, 331, 6, 147, 176, 44, 18, 225, 8, 121, 110, 214, 48, 203, 122, 6, 64, 78, 6, 1

Offset: 0

Antti Karttunen, May 31 2018

Cf. A245612, A289813, A305296 (rgs-transform).
Cf. also A292260, A291763.
Cf. also A304759, A304760.

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;
A254049(n) = A048673((2*n)-1);
A245612(n) = if(n<2,1+n,if(!(n%2),(3*A245612(n/2))-1,A254049(A245612((n-1)/2))));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A305295(n) = A289813(A245612(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).

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.

cp's OEIS Frontend

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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A304758 Restricted growth sequence transform of A304759(n), formed from 1-digits in ternary representation of A048673(n).

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A351031 a(n) = Product_{d|n, dA019565(A304759(d)).

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A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

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A340684 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A291759(n), A278222(A304759(n))], for all i, j >= 1.

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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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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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A305295 Binary encoding of 1-digits in ternary representation of A245612(n).

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

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Formula

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