A291766 -id:A291766 - cp's OEIS frontend

A291755 Compound filter (multiplicative order of 2 mod 2n+1 & eulerphi(2n+1)): a(n) = P(A002326(n), A037225(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 5, 25, 31, 61, 181, 265, 59, 261, 613, 142, 507, 761, 613, 1513, 566, 416, 607, 2521, 607, 1731, 1499, 607, 2301, 1912, 749, 5305, 1731, 1396, 6613, 7081, 826, 1723, 8581, 2102, 5391, 3169, 1731, 3946, 6709, 5725, 13285, 2493, 3431, 4764, 3415, 2356, 5707, 10201, 3946, 19801, 11527

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000010, A000027, A002326, A037225, A291766 (rgs-version of this filter).
Cf. also A292249, A292268.
Cf. A037226, A053006.

A002326[n_] := MultiplicativeOrder[2, 2n+1];
a[n_] := (1/2)*(2 + ((A002326[n] + EulerPhi[2n+1])^2) - A002326[n] - 3* EulerPhi[2n+1]);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 23 2024 *)

A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
A291755(n) = (1/2)*(2 + ((A002326(n)+eulerphi(n+n+1))^2) - A002326(n) - 3*eulerphi(n+n+1));

(define (A291755 n) (* 1/2 (+ (expt (+ (A002326 n) (A000010 (+ 1 n n))) 2) (- (A002326 n)) (- (* 3 (A000010 (+ 1 n n)))) 2)))

a(n) = (1/2)*(2 + ((A002326(n) + A000010(2n+1))^2) - A002326(n) - 3*A000010(2n+1)).

A291769 Restricted growth sequence transform of A292249; filter combining multiplicative order of 2 mod 2n+1 & prime signature of 2n+1 (A002326 & A278223).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 18, 32, 33, 34, 35, 36, 37, 38, 39, 40, 25, 41, 12, 18, 17, 42, 43, 44, 45, 46, 47, 48, 19, 42, 15, 49, 22, 50, 51, 27, 52, 53, 54, 55, 28, 56, 57, 58, 59, 60, 41, 61, 62, 63, 64, 27, 26, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 60, 42

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

Also restricted growth sequence transform of the odd bisection of A286573.

Antti Karttunen, Table of n, a(n) for n = 0..32768

Cf. A002326, A046523, A278223, A286573, A291766, A292249.

Cf. A291766, A292267 for related filters.

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])); }
A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A292249(n) = (1/2)*(2 + ((A002326(n)+A046523(n+n+1))^2) - A002326(n) - 3*A046523(n+n+1));
write_to_bfile(0,rgs_transform(vector(32769,n,A292249(n-1))),"b291769_upto32768.txt");

A292267 Restricted growth sequence transform of A292268; filter combining multiplicative order of 2 mod 2n+1 and the number of trailing 1's in binary expansion of 2n+1.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768

Cf. A002326, A007814, A292268.

Cf. A291766, A291769 for related filters.

allocatemem(2^30);
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])); }
A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
A007814(n) = valuation(n,2);
A292268(n) = (1/2)*(2 + ((A002326(n)+A007814(2*(1+n)))^2) - A002326(n) - 3*A007814(2*(1+n)));
write_to_bfile(0,rgs_transform(vector(32769,n,A292268(n-1))),"b292267_upto32768.txt");

cp's OEIS Frontend

A291755 Compound filter (multiplicative order of 2 mod 2n+1 & eulerphi(2n+1)): a(n) = P(A002326(n), A037225(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A291769 Restricted growth sequence transform of A292249; filter combining multiplicative order of 2 mod 2n+1 & prime signature of 2n+1 (A002326 & A278223).

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A292267 Restricted growth sequence transform of A292268; filter combining multiplicative order of 2 mod 2n+1 and the number of trailing 1's in binary expansion of 2n+1.

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