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A276553 Numbers n such that n^2 and (n + 1)^2 have the same number of divisors. 3
2, 14, 15, 21, 33, 34, 38, 44, 57, 75, 81, 85, 86, 93, 94, 98, 116, 118, 122, 133, 135, 141, 142, 145, 147, 158, 171, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 253, 272, 285, 296, 298, 301, 302, 326, 332, 334, 375, 381, 387, 393, 394, 405, 429, 434, 445 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Except for a(1), all the terms are composite.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

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.

But 16 is not in the sequence, since 17 is prime and 17^2 consequently only has three divisors.

MAPLE

N:= 1000: # to get all terms <= N

T:= map(t -> numtheory:-tau(t^2), [$1..N+1]):

select(t -> T[t]=T[t+1], [$1..N]); # Robert Israel, Apr 10 2017

MATHEMATICA

Select[Range[1000], DivisorSigma[0, #^2] == DivisorSigma[0, (# + 1)^2] &]

PROG

(PARI) k=[]; for(n=1, 1000, a=numdiv(n^2); b=numdiv((n+1)^2); if(a==b, k=concat(k, n))); k

(Python)

from sympy.ntheory import divisor_count

print([n for n in range(1, 501) if divisor_count(n**2) == divisor_count((n + 1)**2)]) # Indranil Ghosh, Apr 10 2017

(Scheme, with Antti Karttunen's IntSeq-library) (define A276553 (ZERO-POS 1 1 A284570)) ;; Antti Karttunen, Apr 15 2017

CROSSREFS

Cf. A000290, A048691, A062832, A276542, A284378.

Cf. A052213 (a subsequence).

Positions of zeros in A284570.

Sequence in context: A072163 A084674 A009774 * A032933 A194230 A109993

Adjacent sequences:  A276550 A276551 A276552 * A276554 A276555 A276556

KEYWORD

nonn

AUTHOR

K. D. Bajpai, Apr 10 2017

STATUS

approved

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Last modified September 26 13:34 EDT 2021. Contains 347668 sequences. (Running on oeis4.)