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 A344534 For any number n with binary expansion Sum_{k = 1..m} 2^e_k (where 0 <= e_1 < ... < e_m), a(n) = Product_{k = 1..m} prime(1+A002262(e_k))^2^A025581(e_k) (where prime(k) denotes the k-th prime number). 3
 1, 2, 4, 8, 3, 6, 12, 24, 16, 32, 64, 128, 48, 96, 192, 384, 9, 18, 36, 72, 27, 54, 108, 216, 144, 288, 576, 1152, 432, 864, 1728, 3456, 5, 10, 20, 40, 15, 30, 60, 120, 80, 160, 320, 640, 240, 480, 960, 1920, 45, 90, 180, 360, 135, 270, 540, 1080, 720, 1440 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The ones in the binary expansion of n encode the Fermi-Dirac factors of a(n). The following table gives the rank of the bit corresponding to the Fermi-Dirac factor p^2^k:     ...       7| 9       5| 5 8       3| 2 4 7       2| 0 1 3 6     ---+--------     p/k| 0 1 2 3 ... This sequence is a bijection from the nonnegative integers to the positive integers with inverse A344536. This sequence establishes a bijection from A261195 to A225547. This sequence and A344535 each map between two useful choices for encoding sets of elements drawn from a 2-dimensional array. To give a very specific example, each mapping is an isomorphism between two alternative integer representations of the polynomial ring GF2[x,y]. The relevant set is {x^i*y^j : i, j >= 0}. The mappings between the two representations of the ring's addition operation are from XOR (A003987) to A059897(.,.) and for the multiplication operation, they are from A329331(.,.) to A329329(.,.). - Peter Munn, May 31 2021 LINKS Rémy Sigrist, Table of n, a(n) for n = 0..8192 Encyclopedia of Mathematics, Isomorphism FORMULA a(n) = A344535(A344531(n)). a(n) = A344535(n) iff n belongs to A261195. A064547(a(n)) = A000120(n). a(A036442(n)) = prime(n). a(A006125(n+1)) = 2^2^n for any n >= 0. a(m + n) = a(m) * a(n) when m AND n = 0 (where AND denotes the bitwise AND operator). From Peter Munn, Jun 06 2021: (Start) a(n) = A225546(A344535(n)). a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987. a(A329331(n, k)) = A329329(a(n), a(k)). (End) EXAMPLE For n = 42: - 42 = 2^5 + 2^3 + 2^1, - so we have the following Fermi-Dirac factors p^2^k:       5| X       3|       2|   X X     ---+------     p/k| 0 1 2 - a(42) = 2^2^1 * 2^2^2 * 5^2^0 = 320. PROG (PARI) A002262(n)=n-binomial(round(sqrt(2+2*n)), 2) A025581(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1) a(n) = { my (v=1, e); while (n, n-=2^e=valuation(n, 2); v* = prime(1 + A002262(e))^2^A025581(e)); v } CROSSREFS Cf. A000120, A002262, A006125, A025581, A036442, A064547, A225546, A225547, A261195, A329050, A344531, A344535, A344536. Comparable mappings that also use Fermi-Dirac factors: A052330, A059900. Maps binary operations A003987 to A059897, A329331 to A329329. Sequence in context: A247555 A340730 A101942 * A050170 A087089 A197382 Adjacent sequences:  A344531 A344532 A344533 * A344535 A344536 A344537 KEYWORD nonn,base AUTHOR Rémy Sigrist, May 22 2021 STATUS approved

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Last modified September 16 05:49 EDT 2021. Contains 347469 sequences. (Running on oeis4.)