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A283931 Numbers n such that tau(2^n) = tau(2^n + 1). 1
1, 63, 511 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

tau(n) is the number of divisors of n (A000005).

There are no other terms <= 880.

Numbers n such that A046798(n) = n + 1.

Numbers n such that A000005(A000079(n)) = A000005(A000051(n)).

Corresponding values of tau(2^n): 2, 64, 512, ...

Corresponding pairs of numbers (2^n, 2^n + 1): (2, 3); (9223372036854775808, 9223372036854775809); ...

From Jon E. Schoenfield, Mar 26 2017 [expanded Mar 30 2017]: (Start)

All terms in the sequence are odd. (If n were an even number n = 2k, then tau(2^n) = n + 1 = 2k + 1 would be odd, but tau(2^n + 1) = tau(2^(2k) + 1) = tau(4^k + 1) would be even, since 4^k + 1 can never be a square.)

2^881 + 1 may be difficult to factor completely, but it is divisible by each of the two primes 22907 and 58147, yet it is divisible by neither 22907^2 nor 58147^2, so each of those two primes will appear in the factorization of 2^881 + 1 with multiplicity 1, and thus tau(2^881 + 1) is some multiple of 4, whereas tau(2^881) = 882 = 2*441 is not, so 881 is not a term. Using partial factorizations of 2^n + 1 for values of n > 881, it is not difficult to show that there are no more terms < 1087.

Each of the known terms a(1), a(2), and a(3) is of the form 2^j - 1 (so tau(2^n) = 2^j); the corresponding values of j are 1, 6, and 9. Are there additional terms of this form? The sequence does not include 2^10 - 1 = 1023; even without completely factoring 2^1023 + 1, it can be seen that 3 must appear in that factorization with a multiplicity of 2, so tau(2^1023 + 1) is divisible by 3, and thus cannot be 1024. Since 2^2047 + 1 is 3 * 179 * 2796203 * 53484017 * 62020897 * 18584774046020617 times a 576-digit composite (coprime to each of those 6 prime factors listed), 2047 will be a term iff that 576-digit composite has exactly 2048/2^6 = 32 divisors.

(End)

LINKS

Table of n, a(n) for n=1..3.

EXAMPLE

For n = 1, tau(2) = tau(3) = 2.

PROG

(MAGMA) [n: n in [0..500] | NumberOfDivisors(2^n) eq NumberOfDivisors(2^n + 1)]

CROSSREFS

Cf. A000005, A000051, A000079, A046798, A283930.

Sequence in context: A075939 A075935 A331011 * A038858 A307527 A217264

Adjacent sequences:  A283928 A283929 A283930 * A283932 A283933 A283934

KEYWORD

nonn,more,bref

AUTHOR

Jaroslav Krizek, Mar 18 2017

STATUS

approved

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Last modified December 7 12:17 EST 2021. Contains 349581 sequences. (Running on oeis4.)