A305815 -id:A305815 - cp's OEIS frontend

A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

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

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A278222, A286622, A305794.

Cf. also A304105, A305793, A305815.

\\ Needs also code from A286622:
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; };
A305794(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A286622(d)-1))); (m); };
v305795 = rgs_transform(vector(up_to, n, A305794(n)));
A305795(n) = v305795[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A007814(i) = A007814(j).
  a(i) = a(j) => A093653(i) = A093653(j).
  a(i) = a(j) => A154402(i) = A154402(j).
  a(i) = a(j) => A305436(i) = A305436(j).

A305813 Restricted growth sequence transform of A305812, a filter sequence constructed from the GF(2)[X]-factorization signatures of the proper divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A278233, A305788, A305812, A305815.

Cf. also A305793.

\\ Needs also code from A305788:
up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
A305812(n) = if(1==n,0, my(m=1); fordiv(n,d,if((d>1)&&(dA305788(d)-1))); (m));
v305813 = rgs_transform(vector(up_to, n, A305812(n)));
A305813(n) = v305813[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A294881(i) = A294881(j).
  a(i) = a(j) => A294882(i) = A294882(j).

A305814 a(n) = Product_{d|n, d>1} prime(A305788(d)-1).

Original entry on oeis.org

1, 2, 2, 6, 3, 20, 2, 42, 10, 66, 2, 660, 2, 20, 42, 546, 13, 1700, 2, 3762, 12, 20, 5, 106260, 6, 20, 110, 660, 5, 106260, 2, 15834, 20, 806, 30, 2075700, 2, 20, 44, 1079694, 2, 6600, 5, 660, 4830, 170, 2, 42822780, 10, 660, 754, 660, 5, 691900, 12, 106260, 44, 170, 2, 2731625820, 2, 20, 660, 680862, 114, 17000, 2, 113646, 30, 56100, 5

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A008578, A278233, A305788, A305812, A305815 (rgs-transform).

A305814(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A305788(d)-1))); (m); }; \\ Needs also code from A305788.

a(n) = Product_{d|n} A008578(A305788(d)).

A305904 Filter sequence for all such sequences S, for which S(A091206(k)) = constant for all k >= 3, where A091206 gives primes whose binary representation encodes a polynomial irreducible over GF(2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 7, 11, 7, 12, 13, 14, 15, 16, 7, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 7, 28, 29, 30, 31, 32, 7, 33, 34, 35, 7, 36, 37, 38, 39, 40, 7, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 7, 52, 7, 53, 54, 55, 56, 57, 7, 58, 59, 60, 61, 62, 7, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 7

Offset: 1

Antti Karttunen, Jun 16 2018

nonn

Restricted growth sequence transform of the ordered pair [A305900(n), A305903(n)].

For all i, j: a(i) = a(j) => A305815(i) = A305815(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A091206, A305815, A305816, A305817, A305900, A305903.

up_to = 100000;
A305816(n) = (isprime(n)&&polisirreducible(Pol(binary(n))*Mod(1,2)));
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
v305817 = partialsums(A305816, up_to);
A305817(n) = v305817[n];
A305904(n) = if(n<7,n,if(A305816(n),7,3+n-A305817(n)));

For n < 7, a(n) = n; for >= 7, a(n) = 7 if A305816(n) = 1 [when n is in A091206[3..] = 7, 11, 13, 19, 31, 37, 41, ...], and 3+n-A305817(n) otherwise.

cp's OEIS Frontend

A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

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A305813 Restricted growth sequence transform of A305812, a filter sequence constructed from the GF(2)[X]-factorization signatures of the proper divisors of n.

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A305814 a(n) = Product_{d|n, d>1} prime(A305788(d)-1).

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A305904 Filter sequence for all such sequences S, for which S(A091206(k)) = constant for all k >= 3, where A091206 gives primes whose binary representation encodes a polynomial irreducible over GF(2).

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