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

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

1,2

Comments

Square roots of perfect squares in A055927. [Juri-Stepan Gerasimov, Apr 06 2011]

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

Example

n=1, tau(2) - tau(1) = 2 - 1 = 1.
n=3, tau(10) - tau(9) = 4 - 3 = 1.
n=5, tau(26) - tau(25) = 4 - 3 = 1.
n=387, tau(149770)- tau(149769) = 16 - 15 = 1.

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;

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 *)

Prog

(Magma) [m:m in [1..1100]| #Divisors(m^2+1) - #Divisors(m^2) eq 1]; // Marius A. Burtea, Jul 12 2019

Crossrefs

Cf. A055927, A000005, A002183, A002182 (records), A005237, A006558.

Cf. A048691, A193432.

Sequence in context: A168161 A217792 A284036 * A023233 A048161 A284034

Adjacent sequences: A172435 A172436 A172437 * A172439 A172440 A172441

Keyword

nonn

Author

Michel Lagneau, Feb 02 2010

Status

approved