A294027 - OEIS

%I #21 Aug 23 2018 02:13:13

%S 945,8505,15015,19305,21735,33345,34125,21961263753,39891817245

%N Odd bi-unitary abundant numbers with a record small gap to the next term odd bi-unitary abundant number.

%C The corresponding gaps are 7560, 1890, 1050, 330, 210, 150, 30, 12, 6.

%C The upper ends are 8505, 10395, 16065, 19635, 21945, 33495, 34155, 21961263765, 39891817251.

%C The bi-unitary version of A294025.

%e Odd bi-unitary abundant numbers are 945, 8505, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, ...

%e Their differences are 7560, 1890, 1890, 2730, 1050, 1890, 1350, 330, 2100, 210, ...

%e The records of small differences are 7560, 1890, 1050, 330, 210, ...

%e And the corresponding terms are 945, 8505, 15015, 19305, 21735, ...

%t f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bOddAbundantQ[n_] := OddQ[n] && bsigma[n] > 2 n; s = Select[Range[1000000], bOddAbundantQ]; a = {}; dmin = 50000; Do[d = s[[j + 1]] - s[[j]]; If[d < dmin, AppendTo[a, s[[j]]]; dmin = d], {j, 1, Length[s] - 1}]; a

%o (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

%o gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

%o biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));

%o biusig(n) = vecsum(biudivs(n));

%o isok(n) = (n % 2) && (biusig(n) > 2*n);

%o lista(nn) = {last = 0; gap = oo; forstep(n=1, nn, 2, if (isok(n), if (last, if (n - last < gap, print1(last, ", "); gap = n - last)); last = n;););} \\ _Michel Marcus_, Dec 15 2017

%Y Cf. A293186, A294025.

%K nonn,fini,more

%O 1,1

%A _Amiram Eldar_, Oct 22 2017

%E a(1) added by _Amiram Eldar_, Aug 22 2018.

%E a(8)-a(9) from _Giovanni Resta_, Aug 22 2018