A293225 -id:A293225 - 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");

A293450 Restricted growth sequence transform of (3*A293225(n) + A010872(n)), a filter combining (n mod 3) with 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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 06 2017

nonn

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

Cf. A002324, A010872, A293221, A293222, A293225, A293226.

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)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my (d=digits(n, 3)); from digits(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.
Anot2submitted(n) = ((3*Anot_submitted(n))+(n%3));
write_to_bfile(1,rgs_transform(vector(59049,n,Anot2submitted(n))),"b293450.txt");

For all i, j: a(i) = a(j) => A002324(i) = A002324(j).

A293214 a(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to binary expansion of n

Cf. A001065, A002110, A019565, A048675, A091954, A292257, A293215 (restricted growth sequence transform).

Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };

a(n) = Product_{d|n, dA019565(d).
a(n) = A300830(n) * A300831(n) * A300832(n). - Antti Karttunen, Mar 16 2018
Other identities.
For n >= 0, a(2^n) = A002110(n).
For n >= 1:
A048675(a(n)) = A001065(n).
A001222(a(n)) = A292257(n).
A007814(a(n)) = A091954(n).
A087207(a(n)) = A218403(n).
A248663(a(n)) = A227320(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");

A293224 Restricted growth sequence transform of A293222, a product formed from the 2-digits present in the ternary expansions of proper divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2017

nonn

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

Cf. A293222.

Cf. also A290092, A293223, A293215, A293217, A293232, A293225, A293226.

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
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Remy Sigrist
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
write_to_bfile(1,rgs_transform(vector(19683,n,A293222(n))),"b293224.txt");

A294933 Compound filter related to base-3 expansion of the exponents in prime factorization of n: a(n) = P(A294932(n), A294931(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 2, 3, 2, 7, 2, 4, 3, 7, 2, 5, 2, 7, 7, 16, 2, 5, 2, 5, 7, 7, 2, 16, 3, 7, 4, 5, 2, 29, 2, 8, 7, 7, 7, 10, 2, 7, 7, 16, 2, 29, 2, 5, 5, 7, 2, 67, 3, 5, 7, 5, 2, 16, 7, 16, 7, 7, 2, 12, 2, 7, 5, 6, 7, 29, 2, 5, 7, 29, 2, 8, 2, 7, 5, 5, 7, 29, 2, 67, 16, 7, 2, 12, 7, 7, 7, 16, 2, 12, 7, 5, 7, 7, 7, 23, 2, 5, 5, 10, 2, 29, 2, 16, 29, 7, 2, 8, 2, 29, 7, 67

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

For all i, j: a(i) = a(j) => A038148(i) = A038148(j).

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

Cf. A294931, A294932.
Cf. also A293225, A293226 and A293442 (analogous filter for base-2).
Cf. A010057, A038148, A240231.

a(n) = (1/2)*(2 + ((A294932(n) + A294931(n))^2) - A294932(n) - 3*A294931(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.

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A293450 Restricted growth sequence transform of (3*A293225(n) + A010872(n)), a filter combining (n mod 3) with 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.

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A293214 a(n) = Product_{d|n, dA019565(d).

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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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A293224 Restricted growth sequence transform of A293222, a product formed from the 2-digits present in the ternary expansions of proper divisors of n.

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A294933 Compound filter related to base-3 expansion of the exponents in prime factorization of n: a(n) = P(A294932(n), A294931(n)), where P(n,k) is sequence A000027 used as a pairing function.

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