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A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors.

%I A276553 #38 May 08 2021 23:05:18
%S A276553 2,14,15,21,33,34,38,44,57,75,81,85,86,93,94,98,116,118,122,133,135,
%T A276553 141,142,145,147,158,171,177,201,202,205,213,214,217,218,230,244,253,
%U A276553 272,285,296,298,301,302,326,332,334,375,381,387,393,394,405,429,434,445
%N A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors.
%C A276553 Except for a(1), all the terms are composite.
%H A276553 Antti Karttunen, <a href="/A276553/b276553.txt">Table of n, a(n) for n = 1..10000</a>
%e A276553 We see that 14^2 = 196, the divisors of which are 1, 2, 4, 7, 14, 28, 49, 98, 196, and there are nine of them. And we see that 15^2 = 225, the divisors of which are 1, 3, 5, 9, 15, 25, 45, 75, 225, and there are nine of them. Both 14^2 and 15^2 have the same number of divisors, hence 14 is in the sequence.
%e A276553 And we see that 16^2 = 256, the divisors of which are the powers of 2 from 2^0 to 2^8, that's nine divisors. Both 15^2 and 16^2 have the same number of divisors, hence 15 is also in the sequence.
%e A276553 But 16 is not in the sequence, since 17 is prime and 17^2 consequently only has three divisors.
%p A276553 N:= 1000: # to get all terms <= N
%p A276553 T:= map(t -> numtheory:-tau(t^2), [$1..N+1]):
%p A276553 select(t -> T[t]=T[t+1], [$1..N]); # _Robert Israel_, Apr 10 2017
%t A276553 Select[Range[1000], DivisorSigma[0, #^2] == DivisorSigma[0, (# + 1)^2] &]
%o A276553 (PARI) k=[]; for(n=1, 1000, a=numdiv(n^2); b=numdiv((n+1)^2); if(a==b, k=concat(k, n))); k
%o A276553 (Python)
%o A276553 from sympy.ntheory import divisor_count
%o A276553 print([n for n in range(1, 501) if divisor_count(n**2) == divisor_count((n + 1)**2)]) # _Indranil Ghosh_, Apr 10 2017
%o A276553 (Scheme, with Antti Karttunen's IntSeq-library) (define A276553 (ZERO-POS 1 1 A284570)) ;; _Antti Karttunen_, Apr 15 2017
%Y A276553 Cf. A000290, A048691, A062832, A276542, A284378.
%Y A276553 Cf. A052213 (a subsequence).
%Y A276553 Positions of zeros in A284570.
%K A276553 nonn
%O A276553 1,1
%A A276553 _K. D. Bajpai_, Apr 10 2017