A290094 -id:A290094 - cp's OEIS frontend

A293226 Restricted growth sequence transform of A293225, a filter combining two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 4, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 12, 36, 2, 37, 38, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 67, 68, 69, 70, 71, 2, 72

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A001065, A293221, A293222, A293223, A293224, A293225.

Cf. also A290093, A290094, A293215, A293217, A293232.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
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); };
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
Anot_submitted(n) = (1/2)*(2 + ((A293222(n) + A293221(n))^2) - A293222(n) - 3*A293221(n)); \\ Eq.class-wise equal to A293225.
write_to_bfile(1,rgs_transform(vector(19683,n,Anot_submitted(n))),"b293226.txt");

A293225 Compound filter: a(n) = P(A293224(n), A293223(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 2, 5, 2, 8, 2, 12, 4, 13, 2, 32, 2, 40, 30, 33, 2, 59, 2, 58, 42, 69, 2, 143, 8, 80, 29, 83, 2, 178, 2, 197, 38, 96, 25, 239, 2, 100, 121, 163, 2, 221, 2, 202, 194, 103, 2, 448, 61, 365, 59, 245, 2, 333, 48, 576, 187, 256, 2, 720, 2, 278, 546, 718, 138, 606, 2, 503, 114, 1009, 2, 1101, 2, 437, 651, 678, 532, 831, 2, 1400, 172, 213, 2, 1508, 71, 500, 597

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A000027, A019565, A293221, A293222, A293223, A293224, A293226 (rgs-version of this filter).

Cf. also A289813, A289814, A290093, A290094.

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(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
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); };
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
v293223 = rgs_transform(vector(19683,n,A293221(n)));
A293223(n) = v293223[n];
v293224 = rgs_transform(vector(19683,n,A293222(n)));
A293224(n) = v293224[n];
A293225(n) = (1/2)*(2 + ((A293224(n) + A293223(n))^2) - A293224(n) - 3*A293223(n));

(define (A293225 n) (* 1/2 (+ (expt (+ (A293224 n) (A293223 n)) 2) (- (A293224 n)) (- (* 3 (A293223 n))) 2)))

a(n) = (1/2)*(2 + ((A293224(n) + A293223(n))^2) - A293224(n) - 3*A293223(n)).

A305303 Restricted growth sequence transform of ordered pair [A278222(A304760(n)), A278222(A291760(n))], constructed from runlengths of 1-digits and 2-digits in base-3 representation of A254103(n).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 3, 7, 2, 4, 4, 8, 9, 10, 11, 12, 7, 13, 4, 14, 3, 15, 5, 16, 5, 7, 4, 17, 2, 4, 14, 18, 4, 8, 14, 19, 20, 21, 6, 22, 22, 23, 11, 8, 24, 10, 4, 25, 22, 23, 4, 22, 7, 26, 4, 27, 21, 28, 7, 14, 4, 8, 10, 29, 16, 30, 14, 17, 4, 8, 31, 32, 9, 12, 8, 27, 12, 19, 24, 33, 14, 17, 10, 34, 2, 4, 22, 16, 11, 14, 14, 17, 3, 15, 11, 35, 14, 17, 31, 34

Offset: 0

Antti Karttunen, May 30 2018

Restricted growth sequence transform of A290093(A254103(n)).

For all i, j: a(i) = a(j) => A286633(i) = A286633(j) => A286632(i) = A286632(j).

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

Cf. A254103, A289813, A289814, A291760, A304740, A304746, A304760.

Cf. also A286632, A286633, A290093, A290094, A305301, A305302.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\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
A304760(n) = A289813(A254103(n));
A291760(n) = A289814(A254103(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; };
Aux305303(n) = [A278222(A304760(n)), A278222(A291760(n))];
v305303 = rgs_transform(vector(65538,n,Aux305303(n-1)));
A305303(n) = v305303[1+n];

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).

Original entry on oeis.org

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];

A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 20, 15, 14, 23, 24, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 31, 38, 39, 40, 41, 42, 43, 44, 45, 34, 47, 48, 43, 50, 51, 52, 53, 54, 29, 56, 33, 38, 59, 60, 47, 62, 45, 32, 59, 42, 41, 68, 69, 50, 71, 72, 35, 62, 51, 44, 71, 78, 53, 80, 81

Offset: 0

Antti Karttunen, Jan 12 2020

For all i, j:
  a(i) = a(j) => A290094(i) = A290094(j).
For all i, j > 0:
  a(i) = a(j) => A007949(i) = A007949(j).

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A007949, A030102, A263273, A290094.

Cf. also A330740, A331166.

A030102(n) = { my(r=[n%3]); while(0A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
A331173(n) = min(n, A263273(n));

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];

A291771 Filter based on runlengths of 0-digits in base-3 expansion of n: a(n) = A278222(A291770(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 8, 4, 4, 6, 2, 2, 6, 2, 2, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 8, 4, 4, 6, 2, 2, 6, 2, 2, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 16, 8, 8, 12, 4, 4, 12, 4, 4, 12, 6, 6, 6, 2, 2, 6, 2, 2, 12, 6, 6, 6, 2, 2, 6

Offset: 1

Antti Karttunen, Sep 11 2017

Antti Karttunen, Table of n, a(n) for n = 1..59049

Cf. A007089, A077267, A278222, A291770, A290091, A290092, A290093, A290094.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Array[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]]] &@ f[FromDigits[IntegerDigits[#, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2], 1, 1] &, 96] (* Michael De Vlieger, Sep 11 2017 *)

a(n) = A278222(A291770(n)).

cp's OEIS Frontend

A293226 Restricted growth sequence transform of A293225, a filter combining two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A293225 Compound filter: a(n) = P(A293224(n), A293223(n)), where P(n,k) is sequence A000027 used as a pairing function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Scheme

Formula

A305303 Restricted growth sequence transform of ordered pair [A278222(A304760(n)), A278222(A291760(n))], constructed from runlengths of 1-digits and 2-digits in base-3 representation of A254103(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A291771 Filter based on runlengths of 0-digits in base-3 expansion of n: a(n) = A278222(A291770(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula