A305788 -id:A305788 - cp's OEIS frontend

A305812 a(1) = 0; for n > 1, a(n) = Product_{d|n, 1 < d < n} prime(A305788(d)-1).

0, 1, 1, 2, 1, 4, 1, 6, 2, 6, 1, 60, 1, 4, 6, 42, 1, 100, 1, 198, 4, 4, 1, 4620, 3, 4, 10, 60, 1, 4620, 1, 546, 4, 26, 6, 56100, 1, 4, 4, 26334, 1, 600, 1, 60, 210, 10, 1, 1381380, 2, 132, 26, 60, 1, 18700, 6, 4620, 4, 10, 1, 66625020, 1, 4, 60, 15834, 6, 1000, 1, 2418, 10, 3300, 1, 334187700, 1, 4, 84, 60, 4, 2200, 1, 14036022, 110, 4, 1

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, A305813 (rgs-transform), A305814.

Cf. also A305792, A304102.

A305812(n) = if(1==n,0, my(m=1); fordiv(n,d,if((d>1)&&(dA305788(d)-1))); (m)); \\ Needs also code from A305788.

a(1) = 0; for n > 1, a(n) = Product_{d|n, dA008578(A305788(d)).

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)).

A278233 Filter-sequence for GF(2)[X]-factorization: sequence that gives the least natural number with the same prime signature that (0, 1)-polynomial encoded in the binary expansion of n has when it is factored over GF(2).

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 2, 8, 6, 12, 2, 12, 2, 6, 8, 16, 16, 30, 2, 36, 4, 6, 6, 24, 2, 6, 12, 12, 6, 24, 2, 32, 6, 48, 6, 60, 2, 6, 12, 72, 2, 12, 6, 12, 24, 30, 2, 48, 6, 6, 32, 12, 6, 60, 2, 24, 12, 30, 2, 72, 2, 6, 12, 64, 36, 30, 2, 144, 4, 30, 6, 120, 2, 6, 24, 12, 6, 60, 6, 144, 4, 6, 30, 36, 64, 30, 2, 24, 6, 120, 2, 60, 6, 6, 12, 96, 2, 30, 12, 12, 30, 96, 2

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

a(n) = the least number with the same prime signature as A091203(n).
This sequence works as an A046523-analog in the polynomial ring GF(2)[X] and can be used as a filter which matches with (and thus detects) any sequence in the database where a(n) depends only on the exponents of irreducible factors when the polynomial corresponding to n (via base-2 encoding) is factored over GF(2). These sequences are listed in the Crossrefs section, "Sequences that partition N into ...".
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

			3 is "11" in binary, encodes polynomial x + 1, and 7 is "111" in binary, encodes polynomial x^2 + x + 1, both which are irreducible over GF(2). We can multiply their codes with carryless multiplication A048720 as A048720(3,7) = 9, A048720(9,3) = 27, A048720(9,7) = 63. Now a(27) = a(63) because the exponents occurring in both codes 27 and 63 are one 1 and two 2's, and their order is not significant when computing prime signature. Moreover a(27) = a(63) = 12 because that is the least number with a prime signature (1,2) in the more familiar domain of natural numbers.
a(25) = 2, because 25 is "11001" in binary, encoding polynomial x^4 + x^3 + 1, which is irreducible in the ring GF(2)[X], i.e., 25 is in A014580, whose initial term is 2.

Cf. A014580 (gives the positions of 2's), A048720, A057889, A091203, A091205, A193231, A235042, A278231, A278238, A278239.
Similar filtering sequences: A046523, A278222, A278226, A278236, A278243.
Sequences that partition N into same or coarser equivalence classes: A091220, A091221, A091222, A106493, A106494.
Cf. also A304529, A304751, A305788 (rgs-transform), A305789.

A278233(n) = { my(p=0, f=vecsort((factor(Pol(binary(n))*Mod(1, 2))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ Antti Karttunen, Jun 10 2018

(define (A278233 n) (A046523 (A091203 n)))

a(n) = A046523(A091203(n)) = A046523(A091205(n)) = A046523(A235042(n)). [Because of the "sorting" essentially performed by A046523, any map from GF(2)[X] to Z can be used, as long as it is fully (cross-)multiplicative and preserves also the exponents intact.]
Other identities. For all n >= 1:
a(A014580(n)) = 2.
a(n) = a(A057889(n)) = a(A193231(n)).
a(A000695(n)) = A278238(n).
a(A277699(n)) = A278239(n).

A305789 Filter-sequence combining prime signature of n (A046523) and similar signature for GF(2)[X]-factorization (A278233).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 6, 7, 8, 2, 9, 2, 5, 10, 11, 12, 13, 2, 14, 15, 5, 16, 17, 18, 5, 19, 9, 16, 20, 2, 21, 5, 22, 5, 23, 2, 5, 8, 24, 2, 25, 16, 9, 26, 27, 2, 28, 7, 29, 30, 9, 16, 31, 32, 17, 8, 27, 2, 33, 2, 5, 9, 34, 35, 36, 2, 37, 15, 36, 16, 38, 2, 5, 26, 9, 5, 39, 16, 40, 41, 5, 42, 43, 44, 27, 32, 17, 16, 45, 32, 46, 5, 5, 8, 47, 2, 13, 9, 48, 42

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

Restricted growth sequence transform of ordered pair [A046523(n), A278233(n)].

For all i, j: a(i) = a(j) => A305802(i) = A305802(j).

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

Cf. A046523, A101296, A278233, A305788, A305802.

Cf. also A305790.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278233(n) = { my(p=0, f=vecsort((factor(Pol(binary(n))*Mod(1, 2))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
Aux305789(n) = [A046523(n), A278233(n)];
v305789 = rgs_transform(vector(up_to,n,Aux305789(n)));
A305789(n) = v305789[n];

A331166 a(n) = min(n, A057889(n)), where A057889 is bijective base-2 reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 19, 22, 27, 28, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 37, 42, 43, 44, 45, 46, 47, 48, 35, 38, 51, 44, 43, 54, 55, 56, 39, 46, 55, 60, 47, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 69, 74, 83, 84, 85, 86, 87, 88, 77, 90, 91, 92, 93, 94, 95, 96, 67, 70

Offset: 0

Antti Karttunen, Jan 12 2020

There is a large number of sequences b, related to binary expansion of n (A007088), for which it holds that b(n) = b(a(n)) for all n >= 0 (or n >= 1). For example, we have:
For all i, j:
  a(i) = a(j) => A002487(i) = A002487(j),
  a(i) = a(j) => A005811(i) = A005811(j),
  a(i) = a(j) => A286622(i) = A286622(j) => A000120(i) = A000120(j).
For all i, j > 0:
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A280505(i) = A280505(j),
  a(i) = a(j) => A305788(i) = A305788(j) => A091222(i) = A091222(j).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000120, A002487, A005811, A007088, A007814, A030101, A057889, A091222, A280505, A286622, A305788.

Cf. also A330740, A331167, A331173.

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A331166(n) = min(n, A057889(n));

a(n) = min(n, A057889(n)).

A305815 Restricted growth sequence transform of A305814, a filter sequence constructed from the GF(2)[X]-factorization signatures of the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 6, 7, 8, 2, 9, 2, 5, 6, 10, 11, 12, 2, 13, 14, 5, 15, 16, 3, 5, 17, 9, 15, 16, 2, 18, 5, 19, 20, 21, 2, 5, 22, 23, 2, 24, 15, 9, 25, 26, 2, 27, 7, 9, 28, 9, 15, 29, 14, 16, 22, 26, 2, 30, 2, 5, 9, 31, 32, 33, 2, 34, 20, 35, 15, 36, 2, 5, 37, 9, 5, 38, 15, 39, 40, 5, 41, 42, 43, 26, 5, 16, 15, 44, 45, 46, 5, 5, 8, 47, 2, 12, 48, 49, 41, 50

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, A305813, A305814.

Cf. also A305795.

\\ 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; };
A305814(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A305788(d)-1))); (m); };
v305815 = rgs_transform(vector(up_to, n, A305814(n)));
A305815(n) = v305815[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A294883(i) = A294883(j).
  a(i) = a(j) => A294884(i) = A294884(j).

A304529 a(1) = 0, a(2n) = n, a(2n+1) = a(A305422(2n+1)).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 4, 3, 5, 1, 6, 1, 7, 4, 8, 8, 9, 1, 10, 2, 11, 11, 12, 1, 13, 6, 14, 7, 15, 1, 16, 25, 17, 7, 18, 1, 19, 14, 20, 1, 21, 19, 22, 12, 23, 1, 24, 3, 25, 16, 26, 13, 27, 1, 28, 22, 29, 1, 30, 1, 31, 5, 32, 10, 33, 1, 34, 2, 35, 59, 36, 1, 37, 44, 38, 55, 39, 13, 40, 2, 41, 9, 42, 32, 43, 1, 44, 47, 45, 1, 46, 19, 47, 26, 48, 1, 49, 50, 50

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

This is GF(2)[X] analog of A246277.
For all i, j: a(i) = a(j) => A278233(i) = A278233(j).
For all i, j: a(i) = a(j) => A305788(i) = A305788(j).

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

Cf. A305422, A305425.
Cf. A014580 (positions of 1's), A278233, A305788.
Cf. also A246277.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A304529(n) = if(1==n,0,while(n%2, n = A305422(n)); n/2);

a(1) = 0, a(2n) = n, a(2n+1) = a(A305422(2n+1)).

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).

A305903 Filter sequence for all such sequences b, for which b(A014580(k)) = constant for all k >= 3.

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, 7, 22, 23, 24, 25, 26, 7, 27, 28, 29, 30, 31, 7, 32, 33, 34, 7, 35, 36, 37, 38, 39, 7, 40, 41, 42, 43, 44, 45, 46, 7, 47, 48, 49, 7, 50, 7, 51, 52, 53, 54, 55, 7, 56, 57, 58, 59, 60, 7, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 7, 74, 75, 76, 7, 77, 78, 79, 80, 81, 7

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A305900(A091203(n)).
This is GF(2)[X] analog of A305900.
For all i, j:
  a(i) = a(j) => A304529(i) = A304529(j) => A305788(i) = A305788(j).
  a(i) = a(j) => A268389(i) = A268389(j).

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

Cf. A091203, A268389, A278233, A304529, A305788, A305900.

up_to = 1000;
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
prepare_v091226(up_to) = { my(v = vector(up_to), c=0); for(i=1,up_to,c += A091225(i); v[i] = c); (v); }
v091226 = prepare_v091226(up_to);
A091226(n) = if(!n,n,v091226[n]);
A305903(n) = if(n<7,n,if(A091225(n),7,3+n-A091226(n)));

For n < 7, a(n) = n, for >= 7, a(n) = 7 when n is in A014580[3..] (= 7, 11, 13, 19, 25, 31, ...), and a(n) = 3+n-A091226(n) when n is in A091242[4..] (= 8, 9, 10, 12, 14, 15, ...).

cp's OEIS Frontend

A305812 a(1) = 0; for n > 1, a(n) = Product_{d|n, 1 < d < n} prime(A305788(d)-1).

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

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A278233 Filter-sequence for GF(2)[X]-factorization: sequence that gives the least natural number with the same prime signature that (0, 1)-polynomial encoded in the binary expansion of n has when it is factored over GF(2).

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A305789 Filter-sequence combining prime signature of n (A046523) and similar signature for GF(2)[X]-factorization (A278233).

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A331166 a(n) = min(n, A057889(n)), where A057889 is bijective base-2 reverse.

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A305815 Restricted growth sequence transform of A305814, a filter sequence constructed from the GF(2)[X]-factorization signatures of the divisors of n.

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A304529 a(1) = 0, a(2n) = n, a(2n+1) = a(A305422(2n+1)).

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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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A305903 Filter sequence for all such sequences b, for which b(A014580(k)) = constant for all k >= 3.

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