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A276630
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a(n) = number of prime signature permutations which can prohibit the appearance of terms in A026477 that are members of the same signature set (see explanation in "Comments" and "Examples").
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1
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0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 3, 2, 2, 0, 1, 2, 2, 2, 2
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OFFSET
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1,26
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COMMENTS
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Explanation:
Prime signatures are multisets of prime exponents in descending order, representing the prime factors of k. For example, k = 200 has prime signature {3,2} because 2^3*5^2 = 200. Prime signature sets contain all numbers that have the same signature; so {72, 108, 200, 392, 500, 675, 968, 1125, 1323..} are members of signature set {3,2}. All signature sets are infinite except {} == {0} = 1 (because p^0 = 1).
A026477 begins a(1)=1, a(2)=2 and a(3)=3; a(n) is smallest positive integer > a(n-1) not a product of any 3 distinct prior terms. Consequently, either all or no terms appear in A026477 that are members of a given three-exponent signature set Z. Call "IN" those Z's whose members are terms which appear, and call "OUT" those Z's whose members are terms which do not appear.
Determining which Z's are IN or OUT involves taking three IN's and summing various permutations of exponents, such that no two are identical in both order and placement (because all terms in A026477 are unique). If no permutation exists which generates a specific Z, then that Z is IN, otherwise OUT. Evaluating Z in graded reverse lexicographic order (see A080577), the first few IN are: {}, {1}, {2}, {4}, {1,1,1,1}, {3,1,1}, {3,3}, {2,2,2,1}, {1,1,1,1,1,1,1}. All other Z whose signature sums < 8 are OUT.
Only one permutation of exponents summing to Z is required for Z to be OUT. However, more than one permutation may exist; a(n) therefore represents the number of three IN's whose sums permute to Z, where n reflects the position of Z in the above order of partitions. Thus a(n)=0 are IN, all others are OUT. See "Examples" for additional discussion.
It appears that a(n) tends to increase as n increases.
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LINKS
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EXAMPLE
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NOTE: Partitions are in graded reverse lexicographic order.
a(8) = 0; the 8th partition is {4}, therefore members of {4} (i.e., primes p^4) appear in A026477.
a(14) = 1; the 14th partition is {4,1}. The one group of three previously-appearing members of signature sets in A026477 whose sums permute to {4,1} is {4} + {1} + {}, so members of {4,1} do not appear in A026477. Note that while {2} (p^2) also appears and {2} + {2} + {1} also permutes to {4,1}, it does not pertain here because {2} + {2} = {4} iff p^2*p^2 = p^4, violating the condition that all terms are products of distinct prior terms. So a(14) = 1, rather than 2.
a(59) = 3; the 59th partition is {3,3,1,1}. The three applicable signature sets which permute to {3,3,1,1} are A: {1,1,1,1} + {2} + {2}; B: {3,1,1} + {2} + {1} and C: {3,3} + {1} + {1}; so members of {3,3,1,1} do not appear in A026477. Note that A and C pertain here with repeated signature sets ({2} for A and {1} for C) because, unlike a(14), their placement in the permutation is different. So for instance, {1,1,1,1} in A = primes p*q*r*s, so one {2} may = p^2 while the other {2} may = q^2.
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CROSSREFS
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Cf. A276631 (signature sets whose members appear in A026477; i.e, partitions corresponding with a(n)=0 in this sequence).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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