A278233 -id:A278233 - cp's OEIS frontend

A284011 a(n) = least natural number with the same prime signature Stern polynomial B(n,x) has when it is factored over Z.

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

Offset: 1

Antti Karttunen, Mar 20 2017

nonn

			B_9(x) = x^2 + 2x + 1, which factorizes as (x + 1)^2, thus a(9) = 2^2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A046523, A125184, A186891 (positions of terms <= 2), A260443, A277013, A284010, A284012.
Cf. also A278233, A278243.
Differs from A046523 for the first time at n=25, where a(25) = 2, while A046523(25) = 4.

\\ After Charles R Greathouse IV's code in A046523 and A186891:
ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)));
A284011(n) = { my(p=0, f=vecsort(factor(ps(n))[, 2], ,4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }
for(n=1, 16384, write("b284011.txt", n, " ", A284011(n)));

a(1) = 1 (by convention), and for n > 1, a(n) = A284010(A260443(n)).

A304751 Filter sequence: Restricted growth sequence transform of function 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 Q.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 6, 4, 4, 2, 9, 2, 4, 6, 6, 2, 8, 2, 10, 4, 4, 4, 11, 2, 4, 4, 9, 2, 8, 2, 6, 8, 4, 2, 12, 4, 4, 4, 6, 2, 11, 2, 9, 4, 4, 2, 11, 2, 4, 8, 13, 4, 8, 2, 6, 2, 8, 2, 14, 2, 4, 8, 6, 2, 8, 2, 12, 2, 4, 2, 11, 4, 4, 2, 9, 2, 15, 2, 6, 4, 4, 4, 16, 2, 8, 6, 6, 2, 8, 2, 9, 8

Offset: 1

Antti Karttunen, Jun 08 2018

nonn

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

Cf. A206719, A206074 (gives the positions of 2's), A257000.

Cf. also A046523 (A101296), A278233, A284010, A284011.

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; };
Aux304751(n) = { my(p=0, f=vecsort((factor(Pol(binary(n)))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }
v304751 = rgs_transform(vector(up_to,n,Aux304751(n)));
A304751(n) = v304751[n];

For all i, j: a(i) = a(j) => A206719(i) = A206719(j).

For all i, j: a(i) = a(j) => A257000(i) = A257000(j).

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).

A286601 a(n) = A278222(A193231(n)).

Original entry on oeis.org

1, 2, 4, 2, 6, 2, 4, 8, 16, 8, 4, 12, 6, 12, 6, 2, 6, 2, 6, 12, 6, 30, 24, 12, 16, 32, 24, 8, 36, 12, 4, 12, 36, 12, 4, 12, 36, 72, 60, 12, 16, 48, 64, 32, 24, 8, 24, 72, 6, 12, 6, 2, 24, 12, 6, 30, 60, 12, 24, 48, 6, 30, 60, 30, 210, 30, 60, 120, 6, 30, 60, 30, 60, 180, 60, 12, 96, 48, 24, 120, 6, 30, 24, 12, 6, 2, 6, 12, 60, 30, 6, 30

Offset: 0

Antti Karttunen, Jun 04 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences related to binary expansion of n

Cf. A193231, A234022, A278222, A278231, A278233, A286602 (rgs-version of this sequence).

(define (A286601 n) (A278222 (A193231 n)))

a(n) = A278222(A193231(n)).

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).

A278231 Least number with the same prime signature as the n-th number in Blue-code: a(n) = A046523(A193231(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences computed from exponents in factorization of n

Cf. A046523, A193231, A278233.

(define (A278231 n) (A046523 (A193231 n)))

a(n) = A046523(A193231(n)).

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

Original entry on oeis.org

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)).

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

A284011 a(n) = least natural number with the same prime signature Stern polynomial B(n,x) has when it is factored over Z.

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A304751 Filter sequence: Restricted growth sequence transform of function 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 Q.

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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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A286601 a(n) = A278222(A193231(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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A278231 Least number with the same prime signature as the n-th number in Blue-code: a(n) = A046523(A193231(n)).

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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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