A091255 -id:A091255 - cp's OEIS frontend

A280505 The palindromic kernel of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A091255(n,A057889(n)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 1, 14, 15, 16, 17, 18, 1, 20, 21, 2, 3, 24, 1, 2, 27, 28, 3, 30, 31, 32, 33, 34, 7, 36, 1, 2, 5, 40, 1, 42, 3, 4, 45, 6, 1, 48, 7, 2, 51, 4, 3, 54, 1, 56, 5, 6, 1, 60, 1, 62, 63, 64, 65, 66, 1, 68, 1, 14, 3, 72, 73, 2, 15, 4, 3, 10, 7, 80, 1, 2, 9, 84, 85, 6, 1, 8, 3, 90, 1, 12, 93, 2, 5, 96, 1, 14, 99, 4, 9, 102, 1, 8, 15, 6

Offset: 1

Antti Karttunen, Jan 09 2017

a(n) = the maximal GF(2)[X]-divisor of n which in base 2 is either a palindrome or becomes a palindrome if trailing 0's are omitted.
More precisely: a(n) = the unique term m of A057890 for which A280500(n,m) > 0 and A091222(m) >= A091222(k) for all such terms k of A057890 for which A280500(n,k) > 0.
All terms are in A057890 and each term of A057890 occurs an infinite number of times.

Cf. A048720, A057889, A057890, A091222, A091255, A280501, A280503, A280506.

(define (A280505 n) (A091255bi n (A057889 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n,A057889(n)).
Other identities. For all n >= 1:
a(A057889(n)) = a(n).
A048720(a(n), A280506(n)) = n.

A325634 a(n) = A091255(n, sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 1, 1, 1, 6, 1, 4, 1, 2, 3, 1, 3, 3, 1, 2, 1, 2, 3, 12, 1, 2, 5, 28, 3, 6, 1, 1, 3, 10, 1, 1, 1, 2, 1, 10, 1, 2, 1, 4, 5, 6, 1, 4, 1, 1, 3, 2, 3, 10, 1, 8, 5, 6, 1, 4, 1, 2, 1, 1, 21, 6, 1, 6, 1, 14, 3, 9, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 7, 28, 5, 6, 1, 4, 3, 2, 1, 4, 1, 2, 5, 12, 1, 1, 5, 1, 3, 10, 1, 2, 3

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A009194, A091255, A169813, A325632, A325633, A325635, A325639 (fixed points, n such that a(n) = n).

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325634(n) = A091255sq(n, sigma(n));

a(n) = A091255(n, A000203(n)).

a(n) = A091255(n, A169813(n)) = A091255(A000203(n), A169813(n)).

A325635 a(n) = A091255(2n, sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 6, 12, 2, 1, 1, 6, 2, 4, 2, 4, 6, 1, 6, 3, 2, 2, 2, 4, 6, 12, 1, 2, 10, 56, 6, 12, 2, 1, 6, 10, 2, 1, 2, 4, 2, 10, 2, 4, 2, 4, 10, 12, 2, 4, 1, 1, 6, 2, 6, 20, 2, 8, 10, 6, 2, 8, 2, 4, 2, 1, 42, 12, 2, 6, 2, 28, 6, 9, 2, 2, 2, 4, 6, 4, 2, 2, 1, 2, 14, 56, 10, 12, 2, 4, 6, 2, 2, 8, 2, 4, 10, 12, 2, 1, 10, 1, 6, 20, 2, 2, 6

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A091255, A318467, A325634, A325636, A325638 (n such that a(n) = 2n).

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325635(n) = A091255sq(n+n, sigma(n));

a(n) = A091255(2n, A000203(n)).

a(n) = A091255(2n, A318467(n)) = A091255(A000203(n), A318467(n)).

A280501 "Blue kernel" of n: a(n) = A091255(n, A193231(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 7, 1, 7, 6, 1, 6, 1, 7, 1, 1, 1, 18, 19, 20, 21, 1, 1, 6, 1, 1, 7, 7, 1, 6, 1, 1, 1, 6, 7, 18, 1, 19, 1, 20, 1, 21, 1, 1, 7, 6, 1, 6, 7, 1, 1, 1, 19, 18, 1, 7, 1, 6, 1, 20, 1, 1, 21, 1, 21, 6, 1, 20, 1, 7, 1, 18, 1, 1, 1, 19, 1, 6, 7, 20, 1, 1, 7, 21, 1, 6, 1, 1, 1, 18, 1, 6, 7, 1, 19, 6, 1, 7, 1, 1, 7, 6, 1, 1, 1, 106, 107, 108, 109, 1, 1, 7

Offset: 1

Antti Karttunen, Jan 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences operating on polynomials in ring GF(2)[X]

Cf. A048720, A091255, A193231, A280502, A280505, A331167.

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A280501(n) = A091255sq(n, A193231(n)); \\ Antti Karttunen, Jan 12 2020

(define (A280501 n) (A091255bi n (A193231 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n, A193231(n)).
Other identities. For all n >= 1:
a(A193231(n)) = a(n).
A048720(a(n), A280502(n)) = n.

A325639 Numbers n for which A091255(n, sigma(n)) = n.

Original entry on oeis.org

1, 6, 28, 120, 312, 428, 456, 496, 504, 672, 760, 6552, 8128, 30240, 31452, 32760, 429240, 523776, 2178540, 5009850, 7505976, 23569920, 33550336, 45532800, 142990848, 186076800, 379975680

Offset: 1

Antti Karttunen, May 21 2019

Numbers n for which A000203(n) = A048720(n, k) for some k. The value of k for the initial terms is: 1, 2, 2, 7, 3, 3, 6, 2, 5, 3, 3, 6, 2, 4, 6, 4, 6, 7, 4, 3, 6, 4, 2, 4, 4, 7, 7, ...
Conjecture: all terms after the initial one are even. If this is true, then there are no odd perfect numbers.
A007691(11) = 2178540 is the first term of A007691 which is not present in this sequence.

Index entries for sequences where any odd perfect numbers must occur

Fixed points of A325632 and A325634.
Cf. A000203, A007691, A048720, A091255.
Cf. A000396, A325638 (subsequences).

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325634(n) = A091255sq(n, sigma(n));
isA325639(n) = (A325634(n)==n);

A325632 a(n) = gcd(n, A325634(n)) = gcd(n, A091255(n, sigma(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 4, 5, 2, 1, 4, 1, 1, 3, 2, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 6, 1, 2, 1, 14, 1, 9, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 28, 5, 2, 1, 4, 1, 2, 1, 4, 1, 2, 5, 12, 1, 1, 1, 1, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A091255, A325634, A325639 (fixed points).
Differs from A009194 for the first time at n=42, where a(42) = 2, while A009194(42) = 6.
Differs from A325633 and A325640 for the first time at n=45, where a(45) = 5, while A325633(45) = 1 and A325640(45) = 3.

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325634(n) = A091255sq(n, sigma(n));
A325632(n) = gcd(n,A325634(n));

a(n) = gcd(n, A325634(n)) = gcd(n, A091255(n, A000203(n))).

A280503 a(n) = A091255(n,A056539(n)).

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 9, 10, 1, 12, 1, 2, 15, 2, 17, 2, 1, 2, 21, 2, 3, 4, 1, 2, 27, 4, 3, 2, 31, 2, 33, 6, 7, 18, 1, 38, 5, 6, 1, 42, 3, 2, 45, 6, 1, 12, 7, 2, 51, 52, 3, 18, 1, 56, 5, 6, 1, 12, 1, 2, 63, 2, 65, 2, 1, 2, 1, 2, 3, 2, 73, 2, 15, 2, 3, 2, 7, 2, 1, 2, 9, 2, 85, 2, 1, 2, 3, 2, 1, 2, 93, 2, 5, 4, 1, 2, 99, 4, 9, 2, 1, 4, 15, 2, 107, 4, 1, 2, 3, 8

Offset: 1

Antti Karttunen, Jan 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A048720, A056539, A091255, A280502, A280505.

(define (A280503 n) (A091255bi n (A056539 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n,A056539(n)).
Other identities. For all n >= 1:
a(A056539(n)) = a(n).
A048720(a(n), A280504(n)) = n.

A325633 a(n) = gcd(A009194(n), A325634(n)) = gcd(A009194(n), A091255(n, sigma(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 3, 2, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 28, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 5, 12, 1, 1, 1, 1, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A091255, A325634.
Differs from A009194 for the first time at n=42, where a(42) = 2, while A009194(42) = 6.
Differs from A325632 and A325640 for the first time at n=45, where a(45) = 1, while A325632(45) = 5 and A325640(45) = 3.

A009194(n) = gcd(n,sigma(n));
A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325634(n) = A091255sq(n, sigma(n));
A325633(n) = gcd(A009194(n),A325634(n));

a(n) = gcd(A009194(n), A325634(n)) = gcd(A009194(n), A091255(n, A000203(n))).

A369293 Lexicographically earliest sequence of distinct positive integers such that a(1) = 1, a(2) = 2, and for any n > 1, A091255(a(n), a(n+1)) <> 1.

Original entry on oeis.org

1, 2, 4, 6, 3, 5, 9, 7, 14, 8, 10, 12, 15, 17, 18, 16, 20, 22, 11, 29, 23, 13, 26, 24, 27, 21, 28, 30, 32, 34, 33, 31, 62, 36, 35, 42, 38, 19, 53, 39, 40, 43, 25, 50, 44, 46, 45, 48, 51, 54, 49, 56, 52, 57, 58, 60, 63, 65, 66, 64, 68, 70, 72, 71, 61, 122, 74

Offset: 1

Rémy Sigrist, Jan 18 2024

In other words, the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive terms (beyond the initial term) are not coprime.
This sequence is a variant of the EKG sequence (A064413).
Is this a permutation of the positive integers?

			The first terms, alongside A091255(a(n), a(n+1)), are:
  n   a(n)  A091255(a(n), a(n+1))
  --  ----  ---------------------
   1     1                      1
   2     2                      2
   3     4                      2
   4     6                      3
   5     3                      3
   6     5                      3
   7     9                      7
   8     7                      7
   9    14                      2
  10     8                      2

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 100000 terms
Rémy Sigrist, PARI program
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences related to EKG sequence

See A369281 and A369294 for similar sequences.

Cf. A064413, A091255, A369302.

PARI
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A325640 a(n) = A091255(n, A009194(n)) = A091255(n, gcd(n, sigma(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 28, 1, 2, 1, 4, 1, 18, 1, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A007691 (fixed points), A009194, A091255, A325634.

Differs from A325632 and A325633 for the first time at n=45, where a(45) = 3, while A325632(45) = 5 and A325633(45) = 1.

A009194(n) = gcd(n,sigma(n));
A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325640(n) = A091255sq(n, A009194(n));

a(n) = A091255(n, A009194(n)) = A091255(n, gcd(n, sigma(n))).

cp's OEIS Frontend

A280505 The palindromic kernel of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A091255(n,A057889(n)).

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A325634 a(n) = A091255(n, sigma(n)).

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A325635 a(n) = A091255(2n, sigma(n)).

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A280501 "Blue kernel" of n: a(n) = A091255(n, A193231(n)).

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A325639 Numbers n for which A091255(n, sigma(n)) = n.

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A325632 a(n) = gcd(n, A325634(n)) = gcd(n, A091255(n, sigma(n))).

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A280503 a(n) = A091255(n,A056539(n)).

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A325633 a(n) = gcd(A009194(n), A325634(n)) = gcd(A009194(n), A091255(n, sigma(n))).

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A369293 Lexicographically earliest sequence of distinct positive integers such that a(1) = 1, a(2) = 2, and for any n > 1, A091255(a(n), a(n+1)) <> 1.

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