This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129657 #17 Feb 16 2025 08:33:05
%S A129657 1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,
%T A129657 28,29,31,32,33,34,35,36,37,38,39,41,43,44,45,46,47,48,49,50,51,52,53,
%U A129657 55,57,58,59,61,62,63,64,65,67,68,69,71,73,74,75,76,77,79,80,81,82,83,84
%N A129657 Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.
%C A129657 For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~1.144n. It follows that the density of the infinitary deficient numbers is 1/1.144, which is about 0.874.
%H A129657 Amiram Eldar, <a href="/A129657/b129657.txt">Table of n, a(n) for n = 1..10000</a>
%H A129657 Graeme L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an Integer's Infinitary Divisors</a>, Mathematics of Computation, Vol. 54, No. 189, (1990), pp. 395-411.
%H A129657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfinitaryDivisor.html">Infinitary Divisor</a>.
%e A129657 The sixth integer that exceeds its proper infinitary divisor sum is 7. Hence a(6)=7.
%t A129657 ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],_?(And@@#&),{1}]] ]] ] Null;properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]-k;InfinitaryDeficientNumberQ[k_]:=If[properinfinitarydivisorsum[k]<k,True,False];Select[Range[100],InfinitaryDeficientNumberQ[ # ] &] (* end of program *)
%t A129657 fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[100], isigma[#] < 2 # &] (* _Amiram Eldar_, Jun 09 2019 *)
%Y A129657 Cf. A126168, A049417, A127666, A129656, A007357.
%K A129657 easy,nonn
%O A129657 1,2
%A A129657 _Ant King_, Apr 29 2007