A229110 -id:A229110 - cp's OEIS frontend

A239876 Partial sums of A229110, where A229110(n) = antisigma(n) mod n = A024816(n) mod n.

0, 0, 2, 5, 9, 12, 18, 23, 28, 35, 45, 47, 59, 70, 76, 85, 101, 107, 125, 133, 143, 162, 184, 184, 203, 226, 240, 254, 282, 285, 315, 332, 350, 381, 403, 438, 474, 509, 531, 541, 581, 590, 632, 658, 670, 713, 759, 803, 844, 876, 906, 938, 990, 1005, 1043, 1063

Offset: 1

Jaroslav Krizek, Mar 29 2014

nonn

Antisigma(n) = A024816(n) = sum of non-divisors of n.

See A239877 - values of n for which a(n)/n is an integer.

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

Cf. A024816, A229110, A239877.

[&+[(k*(k+1)div 2 - SumOfDivisors (k)) mod k: k in [1..n]]: n in [1..1000]]

Accumulate[Table[Mod[Total[Complement[Range[n],Divisors[n]]],n],{n,60}]] (* Harvey P. Dale, Jul 05 2024 *)

a(n) = Sum_{k = 1...n} antisigma(k) mod k = Sum_{k = 1...n} A229110(k).

a(n) = a(n-1) for numbers n from A159907 if there are no odd multiply perfect numbers (A007691).

A007691 Multiply-perfect numbers: n divides sigma(n).

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

sigma(n)/n is in A054030.
Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001
Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007
Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009
A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013
Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013
Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013
Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013
Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014
The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021

			120 is OK because divisors of 120 are {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, the sum of which is 360=120*3.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 135-136.

T. D. Noe, Table of n, a(n) for n=1..1600 (using Flammenkamp's data)
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Anonymous, Multiply Perfect Numbers [broken link]
Eric Bach, Gary Miller, and Jeffrey Shallit, Sums of divisors perfect numbers and factoring, SIAM J. Comput. 15:4 (1986), pp. 1143-1154.
Robert D. Carmichael, A table of multiply perfect numbers, Bull. Amer. Math. Soc. 13 (1907), 383-386.
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.
Walter Nissen, Abundancy : Some Resources
Kaitlin Rafferty and Judy Holdener, On the form of perfect and multiperfect numbers, Pi Mu Epsilon Journal, Vol. 13, No. 5 (Fall 2011), pp. 291-298.
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See p. 11.
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Hyperperfect Number.
Index entries for sequences where any odd perfect numbers must occur

Complement is A054027. Cf. A000203, A054030.
Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.
Other subsequences: A046985, A046986, A046987, A047728, A065997, A066289, (A076231, A076233, A076234), A088844, A088845, A088846, A091443, A114887, A166069, A245782, A260508, A306667, (A325021 U A325022), (A325023 U A325024), (A325025 U A325026), A325637, A323653, A330532, (A330533 U A331724), A336702, A341045.
Subsequence of the following sequences: A011775, A071707, A083865, A089748 (after the initial 1), A102783, A166070, A175200, A225110, A226476, A237719, A245774, A246454, A259307, A263928, A282775, A323652, A336745, A340864. Also conjectured to be a subsequence of A005153, of A307740, and after 1 also of A295078.
Various number-theoretical functions applied to these numbers: A088843 [tau], A098203 [phi], A098204 [gcd(a(n),phi(a(n)))], A134665 [2-adic valuation], A307741 [sigma], A308423 [product of divisors], A320024 [the odd part], A134740 [omega], A342658 [bigomega], A342659 [smallest prime not dividing], A342660 [largest prime divisor].
Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.
Cf. A069926, A330746 (left inverses, when applied to a(n) give n).
Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).
Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.

a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012

Do[If[Mod[DivisorSigma[1, n], n] == 0, Print[n]], {n, 2, 2*10^11}] (* or *)
Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]
(* Third program: *)
Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* Michael De Vlieger, Mar 19 2021 *)

for(n=1,1e6,if(sigma(n)%n==0, print1(n", ")))

from sympy import divisor_sigma as sigma
def ok(n): return sigma(n, 1)%n == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

More terms from Jud McCranie and then from David W. Wilson.

Incorrect comment removed and the crossrefs-section reorganized by Antti Karttunen, Mar 20 2021

A242480 a(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n.

Original entry on oeis.org

0, 2, 3, 8, 5, 6, 7, 16, 9, 20, 11, 12, 13, 28, 15, 32, 17, 18, 19, 20, 21, 44, 23, 24, 25, 52, 27, 28, 29, 30, 31, 64, 33, 68, 35, 72, 37, 76, 39, 40, 41, 42, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 54, 55, 56, 57, 116, 59, 120, 61, 124, 63, 128, 65, 66

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

a(n) / n = 1 for numbers n from A242482, a(n) / n = 2 for numbers n from A242483.
If there are any odd multiply-perfect numbers n > 1 then a(n) = 0.
Possible values of a(n) in increasing order = A242485. Numbers m such that a(n) = m has no solution = A242486.

			a(8) = (8*(8+1)/2) mod 8 + sigma(8) mod 8 + antisigma(8) mod 8 = 36 mod 8 + 15 mod 8 + 21 mod 8 = 4 + 7 + 5 = 16.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A242481, A242482, A242483, A242484, A242485, A242486.

[((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n)): n in [1..1000]]

a(n) = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

a(n) = A242481(n) * n.

A242481 a(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

a(1) = 0. If there is no odd multiply-perfect number then a(n) = 1 or 2 for n >= 2. See A242482 = numbers m such that a(n) = 1, A242483 = numbers m such that a(n) = 2. If there are any odd multiply-perfect numbers m > 1 then a(m) = 0.

			a(8) = [(8*(8+1)/2) mod 8 + sigma(8) mod 8 + antisigma(8) mod 8] / 8 = (36 mod 8 + 15 mod 8 + 21 mod 8) / 8 = (4 + 7 + 5 ) / 8 = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A242480, A242482, A242483, A242484, A242485, A242486.

[((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n))div n: n in [1..1000]]

a(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n.

a(n) = A242480(n) / n.

A242482 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 51, 53, 54, 55, 56, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 97, 99

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

Numbers n such that A242480(n) = (1/2*n*(n+1)) mod n + sigma(n) mod n + antisigma(n) mod n = (A142150(n) + A054024(n) + A229110(n)) = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) = n. Numbers n such that A242481(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n = 1.
Conjecture: with number 1 complement of A242483.
Supersequence of primes (A000040).
If there is no odd multiply-perfect number, then:
(1) a(n) = union of odd numbers >= 3 and even numbers from A239719.
(2) a(n) = supersequence of odd numbers (A005408).

			6 is in sequence because [(6*(6+1)/2) mod 6 + sigma(6) mod 6 + antisigma(6) mod 6] / 6 = (21 mod 6 + 12 mod 6 + 9 mod 6) / 6 = (3 + 0 + 3 ) / 6 = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A242480, A242481, A242483, A242484, A242485, A242486.

[n: n in [1..1000] | n eq ((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n))]

A242483 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 2.

Original entry on oeis.org

4, 8, 10, 14, 16, 22, 26, 32, 34, 36, 38, 44, 46, 48, 50, 52, 58, 60, 62, 64, 68, 72, 74, 76, 82, 84, 86, 90, 92, 94, 96, 98, 106, 108, 110, 116, 118, 122, 124, 128, 130, 132, 134, 136, 142, 144, 146, 148, 152, 154, 156, 158, 164, 166, 168, 170, 172, 178, 182

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

Numbers n such that A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = (A142150(n) + A054024(n) + A229110(n)) = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) = 2n. Numbers n such that A242481(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n = 2.

Conjecture: with number 1 complement of A242482.

			8 is in sequence because [(8*(8+1)/2) mod 8 + sigma(8) mod 8 + antisigma(8) mod 8] / 8 = (36 mod 8 + 15 mod 8 + 21 mod 8) / 8 = (4 + 7 + 5 ) / 8 = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A242480, A242481, A242482, A242484, A242485, A242486.

[n: n in [1..1000] | 2 eq (((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n)))div n]

A242484 Numbers n such that antisigma(n) mod n = 0, where antisigma(n) = A024816(n) = sum of numbers less than n which do not divide n.

Original entry on oeis.org

1, 2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, 8583644160, 10200236032, 21857648640, 57575890944, 57629644800, 206166804480, 17116004505600, 1416963251404800, 15338300494970880

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

Numbers n such that antisigma(n) mod n = A229110(n) = 0.
If there are any odd multiply-perfect numbers, they are members of this sequence.
If there is no odd multiply-perfect number, then a(n) = A159907(n-1) for n >= 2.

			24 is in sequence because antisigma(24) mod 24 = 240 mod 24 = 0.

Cf. A242480, A242481, A242482, A242483, A242485, A242486.

[n: n in [1..1000000] | 0 eq ((n*(n+1))div 2 - SumOfDivisors(n)) mod n];

A242485 Possible values of A242480(n) in increasing order.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83

Offset: 1

Jaroslav Krizek, May 27 2014

nonn

A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

Supersequence of odd numbers > 1. Complement of A242486.

			16 is in the sequence because there is a number m such that A242480(m) = 16; m = 8.

Cf. A242480, A242481, A242482, A242483, A242484, A242486.

A242486 Numbers n such that A242480(x) = n has no solution.

Original entry on oeis.org

1, 4, 10, 14, 22, 26, 34, 36, 38, 46, 48, 50, 58, 60, 62, 74, 82, 84, 86, 90, 94, 98, 106, 108, 110, 118, 122, 130, 132, 134, 142, 146, 154, 156, 158, 166, 170, 178, 182, 190, 194, 202, 206, 210, 214, 218, 226, 230, 238, 242, 250, 252, 254, 262, 266, 270, 274

Offset: 1

Jaroslav Krizek, May 27 2014

nonn

A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

All values of a(n) are even for n > 1. Complement of A242485.

			14 is in the sequence because there is no x whose A242480(x) = 14.

Cf. A242480, A242481, A242482, A242483, A242484, A242485.

A229087 a(n) = sigma(n) mod n - antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisor of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

Original entry on oeis.org

0, 1, -1, 0, -3, -3, -5, 2, -1, 1, -9, 2, -11, -1, 3, 6, -15, -3, -17, -6, 1, -5, -21, 12, -13, -7, -1, -14, -27, 9, -29, 14, -3, -11, -9, -16, -35, -13, -5, 0, -39, 3, -41, 14, 21, -17, -45, -16, -33, 11, -9, 14, -51, -3, -21, -12, -11, -23, -57, 6, -59, -25

Offset: 1

Jaroslav Krizek, Oct 24 2013

sign

Sequence contains anomalous increased frequency of values 14 (see A229115), a(n) = 14 for n = 32, 44, 52, 68, 76, 92, ... ).

			For n = 32; a(32 ) = sigma(32) mod 32 - antisigma(32) mod 32 = 63 mod 32 - 465 mod 32 = 31 - 17 = 14.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)), A024816 (antisigma(n)).
Cf. A054024 (sigma(n) mod n), A229110(antisigma(n) mod n).
Cf. A229088 (numbers n such that sigma(n) mod n = antisigma(n) mod n).
Cf. A229089 (numbers n such that sigma(n) mod n < antisigma(n) mod n).
Cf. A229090 (numbers n such that sigma(n) mod n > antisigma(n) mod n).

a(n) = A000203(n) mod n - A024816(n) mod n = A054024(n) - A229110(n).

cp's OEIS Frontend

A239876 Partial sums of A229110, where A229110(n) = antisigma(n) mod n = A024816(n) mod n.

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A007691 Multiply-perfect numbers: n divides sigma(n).

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A242480 a(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n.

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A242481 a(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n.

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A242482 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 1.

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A242483 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 2.

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A242484 Numbers n such that antisigma(n) mod n = 0, where antisigma(n) = A024816(n) = sum of numbers less than n which do not divide n.

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A242485 Possible values of A242480(n) in increasing order.

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