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.
1, 3, 5, 11, 19, 27, 29, 59, 61, 71, 79, 101, 125, 131, 139, 181, 199, 242, 243, 271, 333, 349, 379, 387, 409, 423, 449, 461, 477, 521, 569, 571, 603, 631, 641, 661, 739, 747, 751, 772, 788, 821, 881, 929, 991, 1017, 1031, 1039, 1051, 1058, 1069, 1075, 1083
Offset: 1
Keywords
Examples
k=1, tau(2) - tau(1) = 2 - 1 = 1. k=3, tau(10) - tau(9) = 4 - 3 = 1. k=5, tau(26) - tau(25) = 4 - 3 = 1. k=387, tau(149770)- tau(149769) = 16 - 15 = 1.
References
- 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.
- T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
- D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
Links
- Marius A. Burtea, Table of n, a(n) for n = 1..5587
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
- P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
- S. Ramanujan, On The Number Of Divisors Of A Number
Programs
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Magma
[m:m in [1..1100]| #Divisors(m^2+1) - #Divisors(m^2) eq 1]; // Marius A. Burtea, Jul 12 2019
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Maple
with(numtheory): for n from 1 to 100000 do; if tau(n^2+1)-tau(n^2)= 1 then print(n); else fi ; od;
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Mathematica
dsQ[n_]:=Module[{n2=n^2},DivisorSigma[0,n2+1]-DivisorSigma[0,n2]==1]; Select[Range[1200],dsQ] (* Harvey P. Dale, May 05 2011 *) Select[Sqrt[#]&/@Flatten[Position[Partition[DivisorSigma[0,Range[1200000]],2,1],?(#[[2]]-#[[1]]==1&),1,Heads->False]],IntegerQ] (* _Harvey P. Dale, Apr 09 2022 *)
Comments