A289814 -id:A289814 - cp's OEIS frontend

A305433 Restricted growth sequence transform of ordered pair [A278222(A305295(n)), A278222(A291763(n))], constructed from runlengths of 1-digits and 2-digits in base-3 representation of A245612(n).

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

Offset: 0

Antti Karttunen, Jun 01 2018

nonn

Restricted growth sequence transform of A290093(A245612(n)).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A245612, A278222, A290093, A290094, A291763, A305295, A305431, A305432, A305434.

Cf. also A305303.

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
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289813
A305295(n) = A289813(A245612(n));
A291763(n) = A289814(A245612(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; };
Aux305433(n) = [A278222(A305295(n)), A278222(A291763(n))];
v305433 = rgs_transform(vector(65538,n,Aux305433(n-1)));
A305433(n) = v305433[1+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));

A343229 A binary encoding of the digits "-1" in balanced ternary representation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 15, 14, 14, 13, 12, 12, 13, 12, 12, 11, 10, 10, 9, 8, 8, 9, 8, 8, 11, 10, 10, 9, 8, 8, 9, 8, 8, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0

Offset: 0

Rémy Sigrist, Apr 08 2021

The ones in the binary representation of a(n) correspond to the digits "-1" in the balanced ternary representation of n.

We can extend this sequence to negative indices: a(-n) = A343228(n) for any n >= 0.

			The first terms, alongside the balanced ternary representation of n (with "T" instead of digits "-1") and the binary representation of a(n), are:
  n   a(n)  ter(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     1      1T          1
   3     0      10          0
   4     0      11          0
   5     3     1TT         11
   6     2     1T0         10
   7     2     1T1         10
   8     1     10T          1
   9     0     100          0
  10     0     101          0
  11     1     11T          1
  12     0     110          0
  13     0     111          0
  14     7    1TTT        111
  15     6    1TT0        110

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Wikipedia, Balanced ternary

Cf. A059095, A060373, A140267, A289814, A289831, A343228, A343230, A343231, A005836 (indices of 0's).

a(n) = { my (v=0, b=1, t); while (n, t=centerlift(Mod(n, 3)); if (t==-1, v+=b); n=(n-t)\3; b*=2); v }

a(n) = A289831(A060373(n)).

A305298 Restricted growth sequence transform of A291763, formed from 2-digits in ternary representation of A291763(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 31 2018

nonn

For all i, j: a(i) = a(j) => A292262(i) = A292262(j).

For all i, j: a(i) = a(j) => A305432(i) = A305432(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A245612, A289814, A291763, A292262, A305432.
Cf. also A305296, A305297.
Cf. also A304746, 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;
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))));
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291763(n) = A289814(A245612(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; };
v305298 = rgs_transform(vector(65538,n,A291763(n-1)));
A305298(n) = v305298[1+n];

A305432 Restricted growth sequence transform of A278222(A291763(n)), constructed from runlengths of 2-digits in base-3 representation of A245612(n).

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 3, 4, 2, 2, 2, 2, 4, 5, 2, 2, 2, 6, 4, 4, 2, 7, 1, 2, 1, 3, 2, 2, 6, 8, 3, 9, 2, 4, 6, 2, 4, 3, 4, 6, 4, 8, 1, 4, 4, 4, 1, 6, 2, 3, 3, 2, 2, 2, 2, 4, 2, 4, 1, 2, 2, 9, 8, 8, 6, 6, 1, 9, 10, 6, 4, 4, 4, 11, 2, 2, 2, 8, 2, 4, 7, 8, 2, 6, 4, 6, 6, 8, 4, 2, 2, 4, 1, 6, 9, 4, 6, 12, 2, 6, 7

Offset: 0

Antti Karttunen, Jun 01 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A245612, A278222, A291763, A305298, A305431, A305433.

Cf. also A305302.

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))));
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291763(n) = A289814(A245612(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; };
v305432 = rgs_transform(vector(65538,n,A278222(A291763(n-1))));
A305432(n) = v305432[1+n];

A340382 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A291759(i)) = A278222(A291759(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 16 2021

nonn

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

Cf. A278222, A291759, A304748, A340381,
Cf. A340377 (positions of ones).
Cf. also A305302.

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));
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; };
v340382 = rgs_transform(vector(up_to,n,A278222(A291759(n))));
A340382(n) = v340382[n];

A351032 a(n) = Product_{d|n, dA019565(A291759(d)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 24, 1, 6, 1, 8, 1, 12, 1, 8, 3, 6, 1, 480, 1, 2, 1, 72, 1, 120, 1, 16, 3, 2, 3, 480, 1, 2, 1, 32, 1, 216, 1, 120, 5, 6, 1, 13440, 3, 60, 1, 120, 1, 168, 3, 1440, 1, 6, 1, 144000, 1, 10, 3, 32, 1, 1080, 1, 8, 3, 72, 1, 26880, 1, 10, 75, 24, 9, 1080, 1, 128, 7, 10, 1, 86400, 1, 30, 3

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Index entries for sequences computed from indices in prime factorization

Cf. A019565, A048673, A289814, A291759, A351030, A351031, A351034 (rgs-transform).

Cf. also A293222.

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); };

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];

A351094 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351092(i) = A351092(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 2, 1, 1, 3, 4, 1, 4, 1, 5, 2, 6, 2, 7, 1, 8, 1, 1, 3, 2, 4, 1, 1, 4, 4, 9, 1, 1, 5, 1, 2, 2, 6, 3, 2, 8, 7, 7, 1, 6, 8, 10, 1, 8, 1, 11, 3, 12, 2, 13, 4, 14, 1, 15, 1, 11, 4, 16, 4, 17, 9, 18, 1, 12, 1, 19, 5, 20, 1, 15, 2, 1, 2, 2, 6, 21, 3, 4, 2, 4, 8, 3, 7, 1, 7, 22, 1, 3, 6, 8, 8

Offset: 1

Antti Karttunen, Jan 31 2022

Restricted growth sequence transform of A351092.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A351090, A351092, A351093.

up_to = 20000;
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); };
A289814(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); }; \\ From A289814
A351092(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289814(d))); (m); };
v351094 = rgs_transform(vector(up_to, n, A351092(n)));
A351094(n) = v351094[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

A305433 Restricted growth sequence transform of ordered pair [A278222(A305295(n)), A278222(A291763(n))], constructed from runlengths of 1-digits and 2-digits in base-3 representation of A245612(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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A343229 A binary encoding of the digits "-1" in balanced ternary representation of n.

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A305298 Restricted growth sequence transform of A291763, formed from 2-digits in ternary representation of A291763(n).

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A305432 Restricted growth sequence transform of A278222(A291763(n)), constructed from runlengths of 2-digits in base-3 representation of A245612(n).

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A340382 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A291759(i)) = A278222(A291759(j)), for all i, j >= 1.

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A351032 a(n) = Product_{d|n, dA019565(A291759(d)).

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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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A351094 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351092(i) = A351092(j), for all i, j >= 1.

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