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A225547
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Fixed points of A225546.
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11
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1, 2, 9, 12, 18, 24, 80, 108, 160, 216, 625, 720, 960, 1250, 1440, 1792, 1920, 2025, 3584, 4050, 5625, 7500, 8640, 11250, 15000, 16128, 17280, 18225, 21504, 24300, 32256, 36450, 43008, 48600, 50000, 67500, 100000, 135000, 143360, 162000, 193536, 218700, 286720, 321489, 324000, 387072, 437400, 450000, 600000
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OFFSET
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1,2
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COMMENTS
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Every number in this sequence is the product of a unique subset of A225548.
From Peter Munn, Feb 11 2020: (Start)
The terms are the numbers whose Fermi-Dirac factors (see A050376) occur symmetrically about the main diagonal of A329050.
Closed under the commutative binary operation A059897(.,.). As numbers are self-inverse under A059897, the sequence thereby forms a subgroup of the positive integers under A059897.
(End)
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LINKS
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Paul Tek, Table of n, a(n) for n = 1..10000
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EXAMPLE
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The Fermi-Dirac factorization of 160 is 2 * 5 * 16. The factors 2, 5 and 16 are A329050(0,0), A329050(2,0) and A329050(0,2), having symmetry about the main diagonal of A329050. So 160 is in the sequence.
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PROG
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(PARI) A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
ff(fa) = {for (i=1, #fa~, my(p=fa[i, 1]); fa[i, 1] = A019565(fa[i, 2]); fa[i, 2] = 2^(primepi(p)-1); ); fa; } \\ A225546
pos(k, fs) = for (i=1, #fs, if (fs[i] == k, return(i)); );
normalize(f) = {my(list = List()); for (k=1, #f~, my(fk = factor(f[k, 1])); for (j=1, #fk~, listput(list, fk[j, 1])); ); my(fs = Set(list)); my(m = matrix(#fs, 2)); for (i=1, #m~, m[i, 1] = fs[i]; for (k=1, #f~, m[i, 2] += valuation(f[k, 1], fs[i])*f[k, 2]; ); ); m; }
isok(n) = my(fa=factor(n), fb=ff(fa)); normalize(fb) == fa; \\ Michel Marcus, Aug 05 2022
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CROSSREFS
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Cf. A050376, A225546, A329050.
Closed under A059895, A059896, A059897, A306697, A329329.
Subsequences: A191554, A191555, A225548.
Cf. fixed points of the comparable A122111 involution: A088902.
Sequence in context: A050855 A031070 A350630 * A325755 A360453 A324570
Adjacent sequences: A225544 A225545 A225546 * A225548 A225549 A225550
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KEYWORD
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nonn
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AUTHOR
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Paul Tek, May 10 2013
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STATUS
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approved
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