A061193 Number of digits in n-th even perfect number (A000396). (Formerly N0213)

1, 2, 3, 4, 8, 10, 12, 19, 37, 54, 65, 77, 314, 366, 770, 1327, 1373, 1937, 2561, 2663, 5834, 5985, 6751, 12003, 13066, 13973, 26790, 51924, 66530, 79502, 130100, 455663, 517430, 757263, 841842, 1791864, 1819050, 4197919, 8107892, 12640858, 14471465, 15632458, 18304103, 19616714, 22370543

Offset

1,2

Comments

The next known values following a(48) are 44677235, 46498850, and 49724095, but these may not be the next terms. [Updated by M. F. Hasler, Nov 28 2017, Ivan Panchenko, Apr 07 2018, Apr 17 2018, Amiram Eldar, Oct 16 2024]

References

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.

Martin Gardner, Mathematical Magic Show, Alfred A. Knopf, 1977, p. 165.

Paul Hoffman, Archimedes' Revenge, Penguin, 1988, p. 11.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Donald D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co., 1995, p. 80.

Links

Ivan Panchenko, Table of n, a(n) for n = 1..48 [Updated by Ivan Panchenko, Apr 17 2018, Amiram Eldar, Oct 16 2024]

A. Byerley, Relationship Between Mersenne Primes and Perfect Numbers.

Chris K. Caldwell, Table of Known Mersenne Primes.

J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]

J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]

J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]

H. J. Smith, Mersenne Primes. [broken link]

Formula

a(n) = ceiling((2*A000043(n)-1)*A007524), with A000043 = Mersenne prime exponents, A007524 = log_10(2). - M. F. Hasler, Nov 28 2017

Mathematica

Table[n=MersennePrimeExponent@k; IntegerLength[2^(n-1)(2^n-1)], {k, 45}] (* Giorgos Kalogeropoulos, Sep 03 2020 *)
Array[IntegerLength@*PerfectNumber, 18] (* Giorgos Kalogeropoulos, Sep 03 2020 *)

Prog

(PARI) apply( p->(2*p-1)*log(2)\log(10)+1, A000043) \\ where A000043 is the vector of the known Mersenne primes. - M. F. Hasler, Nov 28 2017

Crossrefs

Cf. A000043, A000396, A007524, A055642.

Sequence in context: A356641 A296161 A121612 * A346971 A180941 A190896

Adjacent sequences: A061190 A061191 A061192 * A061194 A061195 A061196

Keyword

nonn,base

Author

N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Lekraj Beedassy, May 30 2001.

Extensions

More terms from Harry J. Smith, Apr 16 2003

Entry revised by N. J. A. Sloane, Jun 10 2012

a(39) through a(45) from M. F. Hasler, Nov 28 2017

Status

approved