A045770 -id:A045770 - cp's OEIS frontend

A117350 Near-multiperfects with primes, powers of 2, 6 * prime and 2^n * prime excluded, abs(sigma(n) mod n) <= log(n).

70, 110, 120, 650, 672, 884, 1155, 4030, 5830, 8925, 11096, 17816, 18632, 18904, 30240, 32445, 32760, 45356, 70564, 77744, 85936, 91388, 100804, 116624, 244036, 254012, 388076, 391612, 430272, 442365, 523776, 1090912, 1848964, 2178540

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 in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.
The 2-perfect numbers are excluded because they are 2^n * prime.

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

Donovan Johnson, Table of n, a(n) for n = 1..114 (terms <= 10^12)
Eric Weisstein's World of Mathematics, Multiperfect Number.

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117346 through A117349.

Offset corrected by Donovan Johnson, Oct 01 2012

A076495 Smallest x such that sigma(x) mod x = n, or 0 if no such x exists.

Original entry on oeis.org

2, 20, 4, 9, 0, 25, 8, 10, 15, 14, 21, 24, 27, 22, 16, 26, 39, 208, 36, 34, 51, 38, 57, 112, 95, 46, 69, 48, 115, 841, 32, 58, 45, 62, 93, 660, 155, 1369, 162, 44, 63, 1681, 50, 82, 123, 52, 129, 60, 75, 94, 72, 352, 235, 90, 329, 84, 99, 68, 265, 96, 371, 118, 64, 76

Offset: 1

Labos Elemer, Oct 21 2002

nonn

At present, the 0 entry for n=5 is only a conjecture.
For n <= 1000, a(5) and a(898) are the only terms not found using x <= 10^11. - Donovan Johnson, Sep 20 2012
10^11 < a(898) <= 140729946996736. - Donovan Johnson, Sep 28 2013
a(898) > 10^13 and the same bound holds for a(5), if it exists. - Giovanni Resta, Apr 02 2014
a(5) > 1.5*10^14, if it exists. - Jud McCranie, Jun 02 2019

			n=1: a(1) = smallest prime = 2.
n=3: a(3) = 4 since sigma(4) mod 4 = 7 mod 4 = 3.
n=5: Very difficult case (see Comments section).

Donovan Johnson, Table of n, a(n) for n = 1..1000
Carl Pomerance, On the congruences σ(n) ≡ a (mod n) and n ≡ a (mod φ(n)), Acta Arithmetica 26:3 (1974-1975), pp. 265-272. (See theorem 4.)

Cf. A045768, A045769, A045770, A054024.

f[x_] := s=Mod[DivisorSigma[1, n], n]; t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000}]; t

a(n)=my(k);while(sigma(k++)%k!=n,);k \\ Charles R Greathouse IV, Dec 28 2013

A117347 Near-multiperfects with primes excluded, abs(sigma(m) mod m) <= log(m).

Original entry on oeis.org

4, 6, 8, 10, 16, 20, 28, 32, 64, 70, 88, 104, 110, 120, 128, 136, 152, 256, 464, 496, 512, 592, 650, 672, 884, 1024, 1155, 1888, 1952, 2048, 2144, 4030, 4096, 5830, 8128, 8192, 8384, 8925, 11096, 16384, 17816, 18632, 18904, 30240, 32128, 32445, 32760, 32768

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) (where sigma is the sum-of-divisors function) really "near" a multiple of n, for n = 9? Or n = 18?

			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, A088009, A088010, A088011, A088012.
Cf. A117346, A117348, A117349, A117350.

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

Offset corrected by Amiram Eldar, Mar 05 2020

A084306 Numbers x such that sigma(x) mod x = 12 and x is not divisible by 6. Singular solutions mentioned in A076496.

Original entry on oeis.org

121, 304, 127744, 33501184, 8589082624

Offset: 1

Labos Elemer, Jun 11 2003

nonn

If n = P*q, where P is a multiple perfect number and q is prime so that gcd(P,q) = 1, then sigma(n) = kn(q+1). Consequently sigma(n) = knq + kn sigma(n) mod n = kn. Such values of n are regular solutions to this and analogous cases. Here, not these but the additional eccentric solutions are collected. Cf. A076496.
a(6) > 10^11. - Donovan Johnson, Sep 20 2012
If p = 2^k - 13 > 3 is a prime number, then 2^(k-1)*p is a term. This happens for k = 5, 9, 13, 17, 57, 105, 137, 3217, ... (A096818). - Giovanni Resta, Apr 01 2014

			n = 33501184 = 4096*8179; sigma(n) = 2n + 12 = 67002380.

Cf. A076496, A076495, A000396, A045768-A045770, A007691, A096818.

Do[s=Mod[DivisorSigma[1, n], n]; If[IntegerQ[n/100000], Print[{n}]]; If[Equal[s, 12]&&!Equal[Mod[n, 6], 0], Print[n]], {n, 1, 100000000}]

a(5) from Donovan Johnson, Sep 20 2012

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A117350 Near-multiperfects with primes, powers of 2, 6 * prime and 2^n * prime excluded, abs(sigma(n) mod n) <= log(n).

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A076495 Smallest x such that sigma(x) mod x = n, or 0 if no such x exists.

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A117347 Near-multiperfects with primes excluded, abs(sigma(m) mod m) <= log(m).

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A084306 Numbers x such that sigma(x) mod x = 12 and x is not divisible by 6. Singular solutions mentioned in A076496.

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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

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

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