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A172438 Numbers k such that tau(k^2+1) - tau(k^2) = 1 where the function tau(k) is the number of positive divisors of k.

%I A172438 #34 Aug 16 2025 14:59:22
%S A172438 1,3,5,11,19,27,29,59,61,71,79,101,125,131,139,181,199,242,243,271,
%T A172438 333,349,379,387,409,423,449,461,477,521,569,571,603,631,641,661,739,
%U A172438 747,751,772,788,821,881,929,991,1017,1031,1039,1051,1058,1069,1075,1083
%N A172438 Numbers k such that tau(k^2+1) - tau(k^2) = 1 where the function tau(k) is the number of positive divisors of k.
%C A172438 Square roots of perfect squares in A055927. [_Juri-Stepan Gerasimov_, Apr 06 2011]
%D A172438 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A172438 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D A172438 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
%H A172438 Marius A. Burtea, <a href="/A172438/b172438.txt">Table of n, a(n) for n = 1..5587</a>
%H A172438 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A172438 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/s/s42andrews.html">Some debts I owe</a>, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
%H A172438 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113.
%H A172438 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper8/page1.htm">On The Number Of Divisors Of A Number</a>
%e A172438 k=1, tau(2) - tau(1) = 2 - 1 = 1.
%e A172438 k=3, tau(10) - tau(9) = 4 - 3 = 1.
%e A172438 k=5, tau(26) - tau(25) = 4 - 3 = 1.
%e A172438 k=387, tau(149770)- tau(149769) = 16 - 15 = 1.
%p A172438 with(numtheory): for n from 1 to 100000 do; if tau(n^2+1)-tau(n^2)= 1 then print(n); else fi ; od;
%t A172438 dsQ[n_]:=Module[{n2=n^2},DivisorSigma[0,n2+1]-DivisorSigma[0,n2]==1]; Select[Range[1200],dsQ] (* _Harvey P. Dale_, May 05 2011 *)
%t A172438 Select[Sqrt[#]&/@Flatten[Position[Partition[DivisorSigma[0,Range[1200000]],2,1],_?(#[[2]]-#[[1]]==1&),1,Heads->False]],IntegerQ] (* _Harvey P. Dale_, Apr 09 2022 *)
%o A172438 (Magma) [m:m in [1..1100]| #Divisors(m^2+1) - #Divisors(m^2) eq 1]; // _Marius A. Burtea_, Jul 12 2019
%Y A172438 Cf. A055927, A000005, A002183, A002182 (records), A005237, A006558.
%Y A172438 Cf. A048691, A193432.
%K A172438 nonn
%O A172438 1,2
%A A172438 _Michel Lagneau_, Feb 02 2010