A117347 -id:A117347

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A117346

Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

+10
5

1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 16, 17, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 70, 71, 73, 79, 83, 88, 89, 97, 101, 103, 104, 107, 109, 110, 113, 120, 127, 128, 131, 136, 137, 139, 149, 151, 152, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Sigma is the sum_of_divisors function.

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B2.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Multiperfect Number.

EXAMPLE

70 is in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

MATHEMATICA

asmlQ[n_]:=Module[{p=Mod[DivisorSigma[1, n], n]}, If[p>n/2, p=n-p]; p<=Log[n]];

Select[Range[200], asmlQ] (* Harvey P. Dale, Dec 25 2013 *)

CROSSREFS

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117347 through A117350.

KEYWORD

nonn

AUTHOR

Walter Nissen, Mar 09 2006

EXTENSIONS

First term prepended by Harvey P. Dale, Dec 25 2013

STATUS

approved

A117349

Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

+10
5

6, 10, 20, 28, 70, 88, 104, 110, 120, 136, 152, 464, 496, 592, 650, 672, 884, 1155, 1888, 1952, 2144, 4030, 5830, 8128, 8384, 8925, 11096, 17816, 18632, 18904, 30240, 32128, 32445, 32760, 32896, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 116624

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B2.

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..180 (terms <= 10^12)

Eric Weisstein's World of Mathematics, Multiperfect Number

FORMULA

sigma(n) = k*n + r, abs(r) <= log(n).

EXAMPLE

70 is a term because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

CROSSREFS

Cf. A045768, A045769, A045770, A077374, A087167, A087485.

Cf. A088007, A088008, A088009, A088010, A088011, A088012.

Cf. A117346, A117347, A117348, A117350.

KEYWORD

nonn

AUTHOR

Walter Nissen, Mar 09 2006

EXTENSIONS

Offset corrected by Donovan Johnson, Oct 01 2012

STATUS

approved

A117348

Near-multiperfects with primes and powers of 2 excluded, abs(sigma(m) mod m) <= log(m).

+10
3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near". E.g., is sigma(n) really "near" a multiple of n, for n = 9? Or n = 18? Sigma is the sum_of_divisors function.

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B2.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Walter Nissen, Near Multiperfects.

Eric Weisstein's World of Mathematics, Multiperfect Number

FORMULA

sigma(n) = k * n + r, abs(r) <= log(n).

EXAMPLE

70 is a term because sigma(70) = 144 = 2 * 70 + 4, while 4 < log (70) ~= 4.248.

CROSSREFS

Cf. A045768, A045769, A045770, A077374, A087167, A087485.