A225547 Fixed points of A225546.

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

Offset

1,2

Comments

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)

Links

Paul Tek, Table of n, a(n) for n = 1..10000

Example

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.

Prog

(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

Crossrefs

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

Keyword

nonn

Author

Paul Tek, May 10 2013

Status

approved