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

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

Author

N. J. A. Sloane

Keywords

• nonn
• base

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.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 244-245.
• 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]

Crossrefs

Cf. A000043, A000396, A007524, A055642.

Programs

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 *)

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

Formula

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

Extensions

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

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