A290091 -id:A290091 - cp's OEIS frontend

A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 3, 2, 3, 10, 5, 2, 5, 7, 3, 21, 5, 10, 36, 14, 5, 27, 12, 2, 5, 16, 5, 14, 23, 7, 12, 29, 3, 21, 5, 21, 78, 27, 5, 27, 12, 10, 78, 14, 36, 136, 44, 14, 90, 25, 5, 27, 23, 27, 90, 61, 12, 42, 38, 2, 5, 16, 5, 14, 23, 16, 23, 67, 5, 27, 23, 14, 44, 40, 23, 61, 80, 7, 12, 67, 12, 25, 80, 29, 38, 121, 3, 21, 5, 21, 78, 27, 5, 27, 12, 21, 465, 27, 78, 300, 90, 27

Offset: 0

Antti Karttunen, Jul 25 2017

nonn

For all i, j: a(i) = a(j) => A006047(i) = A006047(j) => A053735(i) = A053735(j).

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A000027, A278222, A289813, A289814, A290091, A290092.

Cf. A006047, A053735, A290079 (some of the matched sequences).

(define (A290093 n) (* (/ 1 2) (+ (expt (+ (A290091 n) (A290092 n)) 2) (- (A290091 n)) (- (* 3 (A290092 n))) 2)))

a(n) = (1/2)*(2 + ((A290091(n)+A290092(n))^2) - A290091(n) - 3*A290092(n)).

A293221 a(n) = Product_{d|n, dA019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of proper divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 12, 6, 6, 2, 36, 2, 4, 18, 12, 2, 30, 2, 360, 12, 10, 2, 540, 6, 60, 30, 360, 2, 900, 2, 120, 30, 10, 12, 2700, 2, 4, 180, 360, 2, 540, 2, 360, 450, 6, 2, 5400, 4, 120, 30, 360, 2, 210, 30, 5040, 12, 14, 2, 1701000, 2, 84, 180, 2520, 180, 1260, 2, 840, 18, 12600, 2, 94500, 2, 140, 180, 840, 20, 18900, 2, 756000, 210, 210, 2, 23814000, 30

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..6561

Cf. A019565, A289813, A293214, A293222, A293223 (restricted growth sequence transform), A293226.

Cf. also A290091.

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); } \\ From Remy Sigrist
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };

a(n) = Product_{d|n, dA019565(A289813(d)).

For all n >= 0, a(3^n) = A002110(n).

A293223 Restricted growth sequence transform of A293221, a product formed from the 1-digits present in the ternary expansion of proper divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

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

Cf. A293221.

Cf. also A290091, A293224, A293225, A293226, 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); } \\ From Remy Sigrist
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
write_to_bfile(1,rgs_transform(vector(19683,n,A293221(n))),"b293223.txt");

A290094 Restricted growth sequence transform of A290093.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..59049

For all i, j: a(i) = a(j) <=> A290093(n) = A290093(n), thus this matches to all the same base-3 (ternary) related sequences as A290093: A006047, A053735, A062756, A081603, A117942, A206424, A227428, A290091, A290092, A290079, and many others.

A290092 Filter based on 2-digits of base-3 expansion: a(n) = A278222(A289814(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A278222, A289814, A290091, A290093.

(define (A290092 n) (A278222 (A289814 n)))

a(n) = A278222(A289814(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

A290093 Compound filter (for base-3 digit runlengths): a(n) = P(A290091(n), A290092(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A293221 a(n) = Product_{d|n, dA019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of proper divisors of n.

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A293223 Restricted growth sequence transform of A293221, a product formed from the 1-digits present in the ternary expansion of proper divisors of n.

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PARI

A290094 Restricted growth sequence transform of A290093.

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A290092 Filter based on 2-digits of base-3 expansion: a(n) = A278222(A289814(n)).

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

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