A291759 -id:A291759 - cp's OEIS frontend

A304748 Restricted growth sequence transform of A291759(n), formed from 2-digits in ternary representation of A048673(n).

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

Offset: 1

Antti Karttunen, May 29 2018

nonn

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

Cf. A048673, A289814, A291759.

Cf. also A304746.

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));
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; };
v304748 = rgs_transform(vector(65537,n,A291759(n)));
A304748(n) = v304748[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); };

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];

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

Original entry on oeis.org

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)).

A291760 Binary encoding of 2-digits in ternary representation of A254103(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 5, 0, 3, 0, 1, 4, 7, 2, 1, 4, 5, 0, 9, 0, 1, 4, 5, 0, 1, 2, 1, 8, 9, 2, 13, 0, 3, 6, 1, 4, 7, 0, 9, 0, 3, 4, 17, 4, 7, 8, 1, 8, 11, 2, 9, 12, 15, 2, 1, 4, 5, 6, 1, 20, 23, 2, 17, 4, 5, 6, 25, 0, 1, 10, 5, 8, 11, 0, 1, 8, 9, 12, 13, 0, 1, 2, 17, 0, 1, 4, 5, 8, 9, 0, 33, 8, 9, 12, 13, 16, 17, 0, 1, 16, 17, 2, 21, 0, 3, 6, 17

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A254103, A286632, A286633, A289814, A291759, A291763.

a(n) = A289814(A254103(n)).

A291763 Binary encoding of 2-digits in ternary representation of A245612(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 3, 0, 1, 8, 1, 6, 5, 4, 1, 2, 1, 10, 23, 16, 1, 2, 13, 10, 9, 2, 7, 0, 1, 0, 3, 2, 1, 70, 21, 24, 45, 4, 33, 52, 1, 36, 3, 20, 25, 10, 21, 0, 17, 18, 5, 0, 13, 16, 3, 12, 1, 8, 1, 4, 5, 2, 5, 0, 1, 32, 139, 74, 41, 208, 49, 0, 89, 108, 11, 130, 65, 18, 103, 4, 1, 8, 73, 4, 5, 112, 41, 16, 49, 72, 19, 38, 41, 20, 1, 8, 33, 0, 35, 86, 9, 38

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A245612, A285712, A285714, A285715, A285716, A289814, A291759, A291760.

a(n) = A289814(A245612(n)).

A292252 Number of trailing 2-digits in ternary representation of A048673(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A007814, A007949, A048673, A291759, A292251 (even bisection subtracted by one).

Cf. also A292242, A292262.

If[First@ # == 2, Length@ #, 0] &@ Last@ Split@ IntegerDigits[#, 3] & /@ Table[(Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n, {n, 120}] (* Michael De Vlieger, Sep 12 2017 *)

(define (A292252 n) (A007949 (+ 1 (A048673 n))))
(define (A292252 n) (A007814 (+ 1 (A291759 n))))
(define (A292252 n) (if (odd? n) 0 (+ 1 (A292251 (/ n 2)))))

a(n) = A007949(1+A048673(n)).
a(n) = A007814(1+A291759(n)).
a(2n) = 1 + A292251(n/2), a(2n+1) = 0.

A292246 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+2 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 4, 1, 2, 14, 5, 12, 6, 7, 8, 2, 1, 0, 0, 9, 26, 22, 3, 20, 6, 5, 16, 10, 29, 10, 4, 11, 30, 2, 25, 60, 56, 13, 28, 54, 15, 48, 24, 17, 44, 8, 5, 12, 38, 3, 30, 26, 1, 24, 20, 1, 18, 6, 19, 62, 14, 53, 4, 14, 45, 0, 42, 7, 124, 118, 41, 50, 58, 13, 116, 106, 11, 40, 104, 33, 32, 98, 21, 92, 6, 59, 88, 18, 21, 82, 76, 9, 34, 36, 23, 74

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 2, the starting value is of the form 3k+2, after which follows A253889(3) = 1, the end point of iteration, which is not, thus a(2) = 1*(2^0) = 1.
For n = 4, the starting value is not of the form 3k+2, while A253889(4) = 2 is, thus a(4) = 0*(2^0) + 1*(2^1) = 2.

Cf. A064216, A253889, A254045, A289814, A292243, A292244, A292245.

Cf. also A291759.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Map[FromDigits[#, 2] &[IntegerDigits[#, 3] /. d_ /; d > 0 :> d - 1] &, Array[a, 96]] (* Michael De Vlieger, Sep 16 2017 *)

a(1) = 0; for n > 1, a(n) = 2*a(A253889(n)) + floor((n mod 3)/2).
a(n) = A289814(A292243(n)).
A000120(a(n)) = A254045(n).
a(n) AND A292244(n) = a(n) AND A292245(n) = 0, where AND is a bitwise-AND (A004198).

cp's OEIS Frontend

A304748 Restricted growth sequence transform of A291759(n), formed from 2-digits in ternary representation of A048673(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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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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A304759 Binary encoding of 1-digits in ternary representation of A048673(n).

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A291760 Binary encoding of 2-digits in ternary representation of A254103(n).

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A291763 Binary encoding of 2-digits in ternary representation of A245612(n).

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A292252 Number of trailing 2-digits in ternary representation of A048673(n).

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A292246 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+2 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

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