A077374 -id:A077374 - cp's OEIS frontend

A117349 Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

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? Log is the natural logarithm. 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.

Donovan Johnson, Table of n, a(n) for n = 1..180 (terms <= 10^12)
Eric Weisstein's World of Mathematics, Multiperfect Number

Cf. A045768, A045769, A045770, A077374, A087167, A087485.
Cf. A088007, A088008, A088009, A088010, A088011, A088012.
Cf. A117346, A117347, A117348, A117350.

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

Offset corrected by Donovan Johnson, Oct 01 2012

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

Original entry on oeis.org

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

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

A077371 Fibonacci numbers whose internal digits form a Fibonacci number. Equivalently, Fibonacci numbers from which deleting the MSD and LSD leaves a Fibonacci number.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 233, 610, 987

Offset: 1

Amarnath Murthy, Nov 06 2002

Conjecture: The sequence is finite.
No more terms < 10^6. - Lars Blomberg, May 20 2015
From Manfred Scheucher, Jun 02 2015 (Start)
No more terms < 10^10000.
When considering binary representations, the sequence would be 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 144, and no further terms < 2^150000 (about 10^44095).
When considering k-ary representations with k=2..100, each of the sequences has some small terms in the beginning (as in the 10-ary case) and no further terms <10^1000.
The sequence seems to be finite for any base, not just for base 10.
Another observation: When considering k-ary representations with k=55,144,377,... (Fibonacci numbers with even index, A001906), the number of "initial terms" (terms <10^1000) increases very fast.
(End)

Manfred Scheucher, Sage Script

Cf. A077372, A077373, A077374, A077375.

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

A088006 Abundance values of numbers whose abundance is (+-1) times a prime.

Original entry on oeis.org

-2, -5, -2, 3, 2, -19, 19, -41, -7, -41, 17, 2, -109, -2, 7, -47, -271, 199, -127, 71, 2, -37, 499, 2, -71, 199, 353, -811, 1021, -929, 59, -457, -449, -163, -683, -157, 41, -751, 251, 2, -2161, -19, 401, 467, -61, 967, -631, -3659, -2777, 3391, -4421, 269, -2333, -1201, -4969, 4999, -1103, -2647, -1097, 269

Offset: 1

Labos Elemer, Oct 18 2003

sign

			For n = 4: A088005(4) = 18, sigma(18) = 18 + 9 + 6 + 3 + 2 + 1 = 39, 2*18 = 36, abundance = 39 - 36 = 3 = a(4).
For n = 6: A088005(6) = 25, sigma(25)_= 25 + 5 + 1 = 31, 2*25 = 50, abundance = 31 - 50 = -19 = a(6).

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

Cf. A000040, A000203, A033880, A077374, A087998, A088005.

ab[x_] := DivisorSigma[1, x]-2*x Do[If[PrimeQ[s=ab[n]], Print[s]], {n, 1, 10000}]
Select[Table[DivisorSigma[1,n]-2n,{n,7000}],PrimeQ[Abs[#]]&] (* Harvey P. Dale, Aug 21 2011 *)

list(lim) = {my(ab); for(k = 1, lim, ab = sigma(k) - 2*k; if(isprime(abs(ab)), print1(ab, ", ")));} \\ Amiram Eldar, Feb 16 2025

Solutions x to sigma(k) - 2k = x where abs(x) is a prime number.

a(n) = A033880(A088005(n)). - Amiram Eldar, Feb 16 2025

A301859 Abundant numbers whose abundance is a perfect number.

Original entry on oeis.org

48, 2002, 2632, 4540, 5170, 6952, 8925, 29056, 32445, 32980, 88330, 133042, 174856, 189472, 280228, 442365, 518368, 566752, 892552, 1266952, 2030368, 2052256, 2218450, 3959752, 4120672, 4558936, 5568448, 9071752, 15921112, 38551936, 65969536, 70114936, 88149352, 97364848

Offset: 1

Waldemar Puszkarz, Mar 27 2018

nonn

There are 34 terms up to 10^8. The abundance of odd terms (only 3 terms) is 6 (see also A087167). The abundance of even terms is 28, 496, 8128, and 33550336 (for 97364848). There exist deficient numbers whose abundance is a perfect number in absolute terms, e.g., 7, 29, 62.

			48 is a term as it is abundant and its abundance, sigma(48)-2*48 = 28, is the second perfect number.

Amiram Eldar, Table of n, a(n) for n = 1..68

Cf. A005101 (abundant numbers), A033880 (abundance), A000396 (perfect numbers), A087167, A088834, A088012, A077374 (sequences related to the odd terms of this sequence).

Select[Range[10^8], PerfectNumberQ[DivisorSigma[1,# ]-2#]&]

for(n=1,10^8, a=sigma(n)-2*n; a>0&&sigma(a)==2*a&&print1(n ","))

A087415 Odd numbers k such that abs(sigma(k)-2k) <= sqrt(k). Abundance-radius = abs(sigma(k)-2k) does not exceed square root of k and k is odd.

Original entry on oeis.org

1, 315, 945, 1155, 2205, 7425, 8415, 8925, 9405, 9555, 24885, 26145, 26325, 28035, 30555, 31815, 32445, 33705, 34335, 35595, 40005, 40365, 41265, 43155, 46035, 49875, 51765, 55335, 78975, 80535, 83265, 91455, 96915, 101475, 106425, 130815, 191565, 338415

Offset: 1

Labos Elemer, Oct 20 2003

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Intersection of A005408 and A088007.

Cf. A077374, A088008-A088012, A000396, A000079, A005100, A005101.

abu[x_] := Abs[DivisorSigma[1, x]-2*x]; Do[If[ !Greater[abu[n], Sqrt[n]//N]&& OddQ[n], Print[n]], {n, 1, 100000}]

isok(k) = k % 2 && (sigma(k)-2*k)^2 <= k; \\ Amiram Eldar, Mar 02 2025

A088818 Numbers k whose abundance-radius does not exceed log(log(k)), i.e., abs(sigma(k)-2*k) <= log(log(k)).

Original entry on oeis.org

6, 16, 28, 32, 64, 128, 256, 496, 512, 1024, 1952, 2048, 4096, 8128, 8192, 16384, 32768, 32896, 65536, 130304, 131072, 262144, 522752, 524288, 1048576, 2097152, 4194304, 8382464, 8388608, 16777216, 33550336, 33554432, 67108864, 134193152, 134217728, 268435456

Offset: 1

Labos Elemer, Oct 20 2003

nonn

			The perfect numbers are all terms.
It is an infinite sequence since it includes all the powers of 2 (A000079) that are larger than 8 (as almost-perfect numbers).

Cf. A033880, A088009, A088010, A088011, A088012, A077374, A000396, A000079.

abu[x_] := Abs[DivisorSigma[1, x]-2*x]; Do[If[ !Greater[abu[n], Log[Log[n]]//N], Print[n]], {n, 1, 1000000}]

is(n) = n > 1 && abs(sigma(n)-2*n) < log(log(n)); \\ Amiram Eldar, Jul 24 2024

a(28)-a(34) from Donovan Johnson, Dec 21 2008

a(35)-a(36) from Amiram Eldar, Jul 24 2024

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

A117349 Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

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

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A077371 Fibonacci numbers whose internal digits form a Fibonacci number. Equivalently, Fibonacci numbers from which deleting the MSD and LSD leaves a Fibonacci number.

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

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A088006 Abundance values of numbers whose abundance is (+-1) times a prime.

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A301859 Abundant numbers whose abundance is a perfect number.

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A087415 Odd numbers k such that abs(sigma(k)-2k) <= sqrt(k). Abundance-radius = abs(sigma(k)-2k) does not exceed square root of k and k is odd.

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