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 A300841 Fermi-Dirac factorization prime shift towards larger terms: a(n) = A052330(2*A052331(n)). 12
 1, 3, 4, 5, 7, 12, 9, 15, 11, 21, 13, 20, 16, 27, 28, 17, 19, 33, 23, 35, 36, 39, 25, 60, 29, 48, 44, 45, 31, 84, 37, 51, 52, 57, 63, 55, 41, 69, 64, 105, 43, 108, 47, 65, 77, 75, 49, 68, 53, 87, 76, 80, 59, 132, 91, 135, 92, 93, 61, 140, 67, 111, 99, 85, 112, 156, 71, 95, 100, 189, 73, 165, 79, 123, 116, 115, 117, 192, 81 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS With n having a unique factorization as A050376(i) * A050376(j) * ... * A050376(k), with i, j, ..., k all distinct, a(n) = A050376(1+i) * A050376(1+j) * ... * A050376(1+k). Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - Andrew Howroyd, Aug 02 2018 LINKS Antti Karttunen, Table of n, a(n) for n = 1..32768 FORMULA a(n) = A052330(2*A052331(n)). For all n >= 1, a(A050376(n)) = A050376(1+n). For all n >= 1, A300840(a(n)) = n. a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Nov 23 2019 EXAMPLE For n = 6 = A050376(1)*A050376(2), a(6) = A050376(2)*A050376(3) = 3*4 = 12. For n = 12 = A050376(2)*A050376(3), a(12) = A050376(3)*A050376(4) = 4*5 = 20. PROG (PARI) up_to_e = 8192; v050376 = vector(up_to_e); A050376(n) = v050376[n]; ispow2(n) = (n && !bitand(n, n-1)); i = 0; for(n=1, oo, if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e, break)); A052330(n) = { my(p=1, i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); }; A052331(n) = { my(s=0, e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); }; A300841(n) = A052330(2*A052331(n)); CROSSREFS Cf. A050376, A052330, A052331, A059897, A300840 (a left inverse). Cf. also A003961. Range of values is A003159. Sequence in context: A285224 A322991 A120635 * A023713 A032890 A092859 Adjacent sequences:  A300838 A300839 A300840 * A300842 A300843 A300844 KEYWORD nonn,mult AUTHOR Antti Karttunen, Apr 12 2018 STATUS approved

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Last modified December 1 22:24 EST 2020. Contains 338858 sequences. (Running on oeis4.)