A295043 a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.

1, 0, 3, 7, 0, 31, 0, 127, 217, 381, 889, 0, 3937, 8191, 11811, 27559, 57337, 131071, 253921, 524287, 1040257, 1777447, 4063201, 7281799, 16646017, 32247967, 66584449, 116522119, 225735769, 516026527, 1073602561, 2147483647, 4294434817, 7515217927, 15032385529

Offset: 0

Author

Jaroslav Krizek, Nov 13 2017

Keywords

• nonn

Comments

If a(n) > 0, then it is a term of A046528 (numbers that are a product of distinct Mersenne primes).

Examples

			a(0) = 1 because 1 is the largest number k with sigma(k) = 1 = 2^0.
a(5) = 31 because 31 is the largest number k with sigma(k) = 32 = 2^5.
a(6) = 0 because there is no number k with sigma(k) = 64 = 2^6.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..250
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000043, A000203, A046528, A048947, A057637, A180221.

Cf. A247956 (the smallest number k instead of the largest).

Cf. A078426 (no solution to the equation sigma(x)=2^n).

A000668 (Mersenne primes) is a subsequence.

Programs

PARI

a(n) = {local(r, k); r=0; for(k=1, 2^n, if(sigma(k) == 2^n, r=k)); return(r)}; \\ Michael B. Porter, Nov 14 2017

PARI

a(n) = forstep(k=2^n, 1, -1, if (sigma(k)==2^n, return (k))); return (0) \\ Rémy Sigrist, Jan 08 2018

PARI

a(n) = invsigmaMax(1<Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

Formula

a(A078426(n)) = 0.

a(A180221(n)) > 0.

a(n) <= 2^n - 1 with equality when n is a Mersenne exponent (A000043). - Michael B. Porter, Nov 14 2017