A293232 -id:A293232 - 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");

A293215 Restricted growth sequence transform of A293214, where A293214(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 7, 35, 36, 37, 2, 38, 28, 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, 73

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..16384

Cf. A001065, A019565, A048675, A293214, A293217 (a variant), A293223, A293224, A293226.

Differs from related A293232 for the first time at n=55, where a(55) = 28, while A293232(55) = 39.

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
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };
write_to_bfile(1,rgs_transform(vector(16384,n,A293214(n))),"b293215.txt");

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");

A293231 a(n) = Product_{d|n, dA019565(A193231(d)).

Original entry on oeis.org

1, 2, 2, 12, 2, 36, 2, 120, 6, 60, 2, 5400, 2, 360, 30, 25200, 2, 56700, 2, 21000, 180, 840, 2, 23814000, 10, 504, 630, 50400, 2, 661500, 2, 554400, 420, 132, 300, 392931000, 2, 792, 252, 242550000, 2, 24948000, 2, 2772000, 22050, 1980, 2, 605113740000, 60, 4851000, 66, 3880800, 2, 720373500, 700, 4889808000, 396, 2772, 2, 588305025000, 2, 1848

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. A019565, A193231, A290090, A293214, A293232 (rgs-version of this sequence).

Cf. also A001317, A045544, A053576.

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
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ This function from Franklin T. Adams-Watters
A293231(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A193231(d)))); m; };

a(n) = Product_{d|n, dA019565(A193231(d)).
For all n >= 1, A007814(a(n)) = A290090(n).
For n = 0..5, a(A001317((2^n)-1)) = A002110((2^n)-1).

A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

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

For all i, j: a(i) = a(j) => A300836(i) = A300836(j).

			For cases n=10 and 49, we see that 10 has proper divisors 1, 2 and 5 and these have Zeckendorf-representations (A014417) 1, 10 and 1000, while 49 has proper divisors 1 and 7 and these have Zeckendorf-representations 1 and 1010. When these Zeckendorf-representations are summed (columnwise without carries), result in both cases is 1011, thus a(10) = a(49).

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

Cf. A003714, A014417, A019565, A300834, A300836.

Cf. also A293215, A293217, A293223, A293224, A293232, A300833 for similar filtering sequences.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };
write_to_bfile(1,rgs_transform(vector(up_to,n,A300834(n))),"b300835.txt");

A318835 Restricted growth sequence transform of A318834, product_{d|n, dA019565(A000010(d)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

For all i, j: a(i) = a(j) => A051953(i) = A051953(j).

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

Cf. A000010, A019565, A318831, A318834,

Cf. also A293215, A293217, A293223, A293224, A293232, A300833, A300835.

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(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };
v318835 = rgs_transform(vector(up_to,n,A318834(n)));
A318835(n) = v318835[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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A293215 Restricted growth sequence transform of A293214, where A293214(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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A293231 a(n) = Product_{d|n, dA019565(A193231(d)).

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A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

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A318835 Restricted growth sequence transform of A318834, product_{d|n, dA019565(A000010(d)).

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