A045768 -id:A045768 - cp's OEIS frontend

A117348 Near-multiperfects with primes and powers of 2 excluded, abs(sigma(m) mod m) <= log(m).

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

Offset: 1

Walter Nissen, Mar 09 2006

nonn

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.

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Nissen, Near Multiperfects.
Eric Weisstein's World of Mathematics, Multiperfect Number

Cf. A045768, A045769, A045770, A077374, A087167, A087485.
Cf. A088007, A088008, A088010, A088011, A088012.
Cf. A117346, A117347, A117349, A117350.

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

Offset corrected by Amiram Eldar, Mar 05 2020

A230606 Numbers n such that sigma(n) = k*(n+1) for some integer k.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 104, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Jaroslav Krizek, Nov 29 2013

nonn

Numbers n such that A108775(n) = floor(sigma(n) / n) = sigma(n) mod n = A054024(n).

Union primes (A000040) and composite numbers A045768 (k = 1 for primes p, k = 2 for composite numbers).

			20 is in sequence because sigma(20) = 42 = 2*21.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000203(sigma(n)), A054024 (sigma(n) mod n), A108775.

Cf. A045768 (sigma(n) == 2 (mod n)).

Select[Range[300],Divisible[DivisorSigma[1,#],#+1]&] (* Harvey P. Dale, May 28 2019 *)

Example clarified by Harvey P. Dale, May 28 2019

A181601 Numbers m with divisor 32 | m and abundance sigma(m)-2*m = 32.

Original entry on oeis.org

992, 28544, 122624, 507392, 537248, 698528, 791264, 1081568, 1279136, 2279072, 5029184, 307801856, 623799776, 712023296, 11196261056, 14809750016, 34355412992, 59640734144, 340536203264, 637707589184, 1091487733184, 1473169206272, 1709840369984, 2526522709184

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

A subsequence of A175989. - R. J. Mathar, Nov 04 2010

Amiram Eldar, Table of n, a(n) for n = 1..45 (calculated using the b-file at A175989)

Cf. A000203, A005101, A005820, A045768, A097498, A118372, A153501, A175989, A181595, A181596, A181597, A181598, A181599.

Select[32Range[1000000],DivisorSigma[1,#]-2#==32&] (* Harvey P. Dale, Aug 16 2011 *)

Definition rephrased, a(5)-a(11) appended - R. J. Mathar, Nov 04 2010

a(12)-a(24) from Donovan Johnson, Dec 08 2011

A088820 Numbers k with abundance radius of 8, i.e., abs(sigma(k)-2*k) = 8.

Original entry on oeis.org

22, 56, 130, 184, 368, 836, 1012, 2272, 11096, 17816, 18904, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 128768, 254012, 388076, 391612, 527872, 1090912, 2087936, 2291936, 13174976, 17619844, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216

Offset: 1

Labos Elemer, Oct 20 2003

nonn

Original definition: Abundance-radius=8, that is Abs[sigma[n]-2n]=8 (either +8 or -8). A045770 from 3rd term complemented by -8 cases.

			22 is in the sequence since sigma(22) = 1 + 2 + 11 + 22 = 36 = 2*22 - 8.
56 is in the sequence since sigma(56) = 1 + 2 + 4 + 7 + 8 + 14 + 28 + 56 = 120 = 2*56 + 8. - _Michael B. Porter_, Jul 20 2016

Amiram Eldar, Table of n, a(n) for n = 1..66 (terms below 10^18, from the b-files at A088833 and A125247)

Disjoint union of A088833 (abundance 8) and A125247 (deficiency 8).
Cf. A045770, A045768, A055708, A077374, A088007, A088008, A088010, A088011, A088012, A088818, A088819.
Cf. A000203 (sigma), A033880 (abundance), A005100 (deficient numbers).

[n: n in [1..2*10^7] | Abs(DivisorSigma(1, n) - 2*n) eq 8]; // Vincenzo Librandi, Jul 20 2016

Select[Range[1, 10^6], Abs[DivisorSigma[1, #] - 2 #] == 8 &] (* Vincenzo Librandi, Jul 20 2016 *)

is(n)=abs(sigma(n)-2*n)==8 \\ Use, e.g., select(is,[1..10^5]*2). - M. F. Hasler, Jul 19 2016

More terms from David Wasserman, Aug 18 2005
Edited by M. F. Hasler, Jul 19 2016
a(33)-a(34) from Amiram Eldar, Mar 11 2025

A181599 Numbers m with divisor 16 | m and abundance sigma(m)-2*m = 16.

Original entry on oeis.org

1504, 30592, 4526272, 8353792, 361702144, 1081850752, 1845991216, 2146926592, 21818579968, 34357510144, 228354264064, 549746900992, 2169800814592, 8796057370624, 24038405705152, 80952364306432, 140737345748992, 2737658648639872, 23810602502029312, 36979953305070592

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

Cf. A097498, A141547, A181595, A181596, A181597, A181598, A118372, A045768, A000396, A005101, A153501, A005820.

A008598 INTERSECT A141547. - R. J. Mathar, Nov 04 2010

Definition rephrased - R. J. Mathar, Nov 04 2010
a(9)-a(13) from Donovan Johnson, Dec 08 2011
a(14)-a(20) from the b-file at A141547 added by Amiram Eldar, Aug 03 2024

A063785 Numbers n such that sigma(n) = 2n + omega(n), where omega(n) is the number of distinct prime divisors of n.

Original entry on oeis.org

20, 104, 464, 1952, 4030, 5830, 130304, 522752, 1848964, 8382464, 134193152

Offset: 1

Jason Earls, Aug 17 2001

It is easily proved that if 2^m-3 is prime then 2^(m-1)*(2^m-3) is in the sequence. - Farideh Firoozbakht, Feb 12 2008

Cf. A045768.

for(n=1,10^8, if(sigma(n)==2*n+omega(n),print(n)))

cp's OEIS Frontend

A117348 Near-multiperfects with primes and powers of 2 excluded, abs(sigma(m) mod m) <= log(m).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A230606 Numbers n such that sigma(n) = k*(n+1) for some integer k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A181601 Numbers m with divisor 32 | m and abundance sigma(m)-2*m = 32.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A088820 Numbers k with abundance radius of 8, i.e., abs(sigma(k)-2*k) = 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A181599 Numbers m with divisor 16 | m and abundance sigma(m)-2*m = 16.

Views

Author

Keywords

Crossrefs

Formula

Extensions

A063785 Numbers n such that sigma(n) = 2n + omega(n), where omega(n) is the number of distinct prime divisors of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI