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A236841 Numbers that occur as results of downward remultiplication (N -> GF(2)[X]) of some number; A234741 sorted and duplicates removed. 10
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The range of A234741: numbers n which encode by their binary representation a polynomial in GF(2)[X] whose multiset of irreducible polynomial factors P, Q, ..., W (where n = P x Q x ... x W, where P, Q, ..., W are irreducible polynomials encoded by A014580, and are not necessarily distinct, and x stands for carryless multiplication of such polynomials: A048720) can be grouped to at least one such multiset (P x Q, W), (P x W, Q), (P, Q x W), (P x Q x W), etc., in such a way that all its members are primes.
Above condition implies that none of the terms of A091214 occur here.
LINKS
FORMULA
Use the characteristic function A236861(n) to determine whether n is a term of this sequence or not. Specifically, all primes occur in this sequence. A composite number n occurs only if there exists at least one such pair of k, m < n that n = A048720(k,m) and k and m both occur here. This implies that none of the terms of A091214 are present.
EXAMPLE
17 is a term because it factors as 3 x 3 x 3 x 3 in GF(2)[X], and these can be grouped as (3x3x3x3), (3x3 * 3x3), (3 * 3 * 3x3) and (3 * 3 * 3 * 3) that is, as 17, (5 * 5), (3 * 3 * 5) and (3 * 3 * 3 * 3) which give the four different k, 17, 25, 45 and 81, for which A234741(k) = 17. (Note that A236833(17) = 4. In the grouping (3 * 3x3x3) = (3 * 15) 15 is not a prime, so it is discarded,)
25 is not a term because it is an irreducible in GF(2)[X], but not a prime in N.
43 = 3 x 25 is a term because 43 itself is a prime in N.
125 = 3 x 3 x 25 is a term, because both 3 and (3 x 25) = 43 are primes in N. Their product 3*43 = 129 gives one such k that A234741(k) = 125.
1951 = 25 x 87 is a member, as although both 25 and 87 are in A091214, 1951 is itself a prime in N.
PROG
(Scheme, two different implementations, using Antti Karttunen's IntSeq-library)
(define A236841 (NONZERO-POS 1 0 A236833))
(define A236841 (NONZERO-POS 1 0 A236861))
CROSSREFS
Positions of nonzero terms in A236833.
Complement of A236834.
Characteristic function: A236861.
A subsequence: A236839.
Sequence in context: A243903 A296867 A228144 * A236850 A263028 A044921
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 31 2014
STATUS
approved

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Last modified June 19 18:58 EDT 2024. Contains 373507 sequences. (Running on oeis4.)