A033879 -id:A033879 - cp's OEIS frontend

A074918 Highly imperfect numbers: n sets a record for the value of abs(sigma(n)-2*n) (absolute value of A033879).

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 180, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 240, 269, 271, 277, 281, 283, 293, 307

Offset: 1

Joseph L. Pe, Oct 01 2002

A perfect number n is defined by sigma(n) = 2n, so the value of i(n) = |sigma(n)-2n| measures the degree of perfection of n. The larger i(n) is, the more "imperfect" n is. I call the numbers n such that i(k) < i(n) for all k < n "highly-imperfect numbers".
RECORDS transform of |A033879|.
Initial terms are odd primes but then even numbers appear.
The last odd term is a(79) = 719. (Proof: sigma(27720n) >= 11080n, and so sigma(27720n) >= 4 * 27720(n + 1) for n >= 8, so there is no odd member of this sequence between 27720 * 8 and 27720 * 9, between 27720 * 9 and 2770 * 10, etc.; the remaining terms are checked by computer.) [Charles R Greathouse IV, Apr 12 2010]

R. J. Mathar and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 238 terms from R. J. Mathar)
Joseph L. Pe, Puzzle 241. Highly imperfect primes, and answers, on C. Rivera's primepuzzles.net. [From _M. F. Hasler_, May 31 2009]
N. J. A. Sloane, Transforms

Cf. A033879, A075728.

r = 0; l = {}; Do[ n = Abs[2 i - DivisorSigma[1, i]]; If[n > r, r = n; l = Append[l, i]], {i, 1, 10^4}]; l
DeleteDuplicates[Table[{n,Abs[DivisorSigma[1,n]-2n]},{n,350}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, Jan 16 2023 *)

A075728 Records in abs(sigma(n)-2*n) (absolute value of A033879).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 120, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 186, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 264, 268, 270, 276, 280, 282, 292, 306

Offset: 1

N. J. A. Sloane, Oct 03 2002

nonn

RECORDS transform of |A033879|.

R. J. Mathar and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 220 terms from R. J. Mathar)
N. J. A. Sloane, Transforms

Cf. A033879, A074918. Different from A006093.

lista(nn) = {rec = -1; for (n=1, nn, d = abs(sigma(n) - 2*n); if (d > rec, print1(d, ", "); rec = d;););} \\ Michel Marcus, Nov 02 2013

Corrected name, Michel Marcus, Nov 02 2013

A295881 Reversing binary representation of the deficiency of n, A033879(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 14, 1, 13, 2, 26, 12, 28, 4, 14, 1, 16, 5, 50, 6, 26, 8, 62, 20, 55, 26, 22, 0, 44, 20, 38, 1, 50, 22, 62, 53, 100, 16, 62, 30, 104, 20, 122, 4, 28, 52, 118, 36, 121, 11, 38, 14, 84, 20, 110, 24, 98, 42, 74, 80, 76, 44, 62, 1, 118, 20, 194, 26, 122, 12, 206, 85, 200, 98, 42, 28, 74, 20, 214, 46, 121

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

For all n, A010060(a(A005100(n))) = 1 and A010060(a(A023196(n))) = 0. That is, for the deficient numbers a(n) is an odious number (A000069) and for the nondeficient numbers a(n) is an evil number (A001969).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005100, A023196, A033879, A033880, A048724, A065620, A065621, A286449, A295882.

Cf. A000396 (gives the positions of zeros).

(define (A295881 n) (let ((x (A033879 n))) (if (<= x 0) (A048724 (- x)) (A065621 x))))

If A033879(n) <= 0, a(n) = A048724(-A033879(n)), otherwise a(n) = A065621(A033879(n)).

For all n >= 1, A065620(a(n)) = A033879(n).

A318441 a(n) = Sum_{d|n} [moebius(n/d) > 0]*A033879(d).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 5, 3, 10, -3, 12, 5, 7, 1, 16, -1, 18, -1, 11, 9, 22, -11, 19, 11, 14, 1, 28, -5, 30, 1, 19, 15, 23, -19, 36, 17, 23, -9, 40, -3, 42, 5, 14, 21, 46, -27, 41, 11, 31, 7, 52, -7, 39, -7, 35, 27, 58, -45, 60, 29, 24, 1, 47, 1, 66, 11, 43, 7, 70, -55, 72, 35, 30, 13, 59, 3, 78, -25, 41, 39, 82, -51, 63, 41, 55, -3, 88

Offset: 1

Antti Karttunen, Aug 26 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033879, A083254, A291784, A318325, A318442.

A318441(n) = sumdiv(n,d,(1==moebius(n/d))*(d+d-sigma(d)));

a(n) = Sum_{d|n} [A008683(n/d) == 1]*A033879(d).
a(n) = A291784(n) - A318325(n).
A083254(n) = a(n) - A318442(n).

A324530 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A033879(n), A318458(n)] for all other numbers, except f(1) = -1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 9, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 2, 16, 28, 19, 29, 30, 31, 19, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 2, 39, 56, 57, 58, 35, 59, 60, 61, 62, 63, 64, 65, 51, 66, 67, 68, 41, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 51, 80

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

Cf. A033879, A318458, A324389, A324400, A324531.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A033879(n) = (n+n-sigma(n));
A318458(n) = bitand(n,sigma(n)-n);
Aux324530(n) = if(1==n,-1,[A033879(n), A318458(n)]);
v324530 = rgs_transform(vector(up_to,n,Aux324530(n)));
A324530(n) = v324530[n];

a(2^n) = 2 for all n >= 1.

A325976 a(n) is the largest k <= n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 9, 11, 11, 13, 13, 13, 16, 17, 17, 19, 19, 21, 21, 23, 23, 25, 23, 27, 1, 29, 29, 31, 32, 31, 33, 35, 36, 37, 37, 39, 39, 41, 41, 43, 43, 43, 43, 47, 47, 49, 50, 49, 49, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 64, 65, 65, 67, 67, 67, 69, 71, 71, 73, 73, 75, 73, 77, 77, 79, 79, 81, 81, 83, 83, 85, 83

Offset: 1

Antti Karttunen, Jun 01 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A033879, A325817, A325818, A325826, A325959.

A325976(n) = { my(s=sigma(n)); forstep(k=n, 0, -1, if(1==gcd((n+n-s), k), return(k))); };

A325817(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(i))); };
A325976(n) = (n - A325817(n));

a(n) = n - A325817(n) = A033879(n) + A325826(n).
a(n) >= A325959(n).
gcd(a(n), A325826(n)) = 1.

A331734 a(n) = A033879(A225546(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 0, 5, 1, 1, -4, 1, 1, 1, 4, 1, -3, 1, -28, 1, 1, 1, -12, 41, 1, -19, -508, 1, 1, 1, 2, 1, 1, 1, 14, 1, 1, 1, -60, 1, 1, 1, -131068, -115, 1, 1, -2, 3281, -39, 1, -8589934588, 1, -51, 1, -1020, 1, 1, 1, -124, 1, 1, -2035, 6, 1, 1, 1, -36893488147419103228, 1, 1, 1, -12, 1, 1, -199, -680564733841876926926749214863536422908

Offset: 1

Antti Karttunen, Feb 02 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..136

Cf. A005117, A033879, A225546, A331733, A331735, A331741.

Cf. A323244, A323174, A324055, A324185, A324546 for other permutations of the deficiency, and also A324574, A324654.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331734(n) = if(issquarefree(n),1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); (2*prod(i=1,u,prime(i)^A048675(prods[i]))) - prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

a(n) = A033879(A225546(n)) = 2*A225546(n) - A331733(n).
For all n, a(A005117(n)) = 1. [It is not known if there are 1's in any other positions. See Jianing Song's Oct 13 2019 comment in A033879.]
For a necessary condition that a(s) would be zero for any square, see A331741.

A330901 Numbers k such that k and k+2 have the same deficiency (A033879).

Original entry on oeis.org

2, 6497, 12317, 91610, 133787, 181427, 404471, 439097, 485237, 1410119, 2696807, 6220607, 6827369, 6954767, 9770027, 10302419, 10449347, 10887977, 11014007, 16745387, 18959111, 25883519, 27334469, 39508037, 40311149, 40551617, 42561437, 44592209, 47717471, 48912107

Offset: 1

Amiram Eldar, May 01 2020

nonn

Are 2 and 91610 the only even terms?
Are there any abundant numbers (A005101) in this sequence?
Numbers k such that k and k+1 have the same deficiency are 1, 145215, and no more below 10^13 (they are a subset of A112645).
Up to a(2214) = 2001876242879 there are no further even terms nor abundant terms. - Giovanni Resta, May 01 2020

			2 is a term since 2 and 4 have the same deficiency: A033879(2) = 2*2 - sigma(2) = 4 - 3 = 1, and A033879(4) = 2*4 - sigma(4) = 8 - 7 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000203, A033879, A033880, A112645.

def[n_] := 2*n - DivisorSigma[1, n]; Select[Range[10^5], def[#] == def[# + 2] &]
SequencePosition[Table[2n-DivisorSigma[1,n],{n,48920000}],{x_,,x}][[;;,1]] (* Harvey P. Dale, Apr 26 2025 *)

j1=1;j2=1;for(k=3,50000000,j=k+k-sigma(k);if(j==j1,print1(k-2,", "));j1=j2;j2=j) \\ Hugo Pfoertner, May 01 2020

A331181 Number of values of k, 1 <= k <= n, with A033879(k) = A033879(n), where A033879(n) is the deficiency of n, 2n-sigma(n).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 2, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 6, 2, 2, 2, 1, 1, 2, 3, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 2, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 2, 5, 1, 4, 2, 2, 1, 1, 1, 2, 2, 3, 2, 6, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 3, 3, 1, 3, 1, 1, 1, 3, 1, 1, 7, 1, 2, 3

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A033879, or equally, of A033880, or of A286449.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A033879, A033880, A286449.

Cf. also A331180, A331182.

b[_] = 0;
a[n_] := With[{t = 2 n - DivisorSigma[1, n]}, b[t] = b[t] + 1];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A033879(n) = (2*n-sigma(n));
v331181 = ordinal_transform(vector(up_to,n,A033879(n)));
A331181(n) = v331181[n];

A335254 Numbers k such that the abundance (A033880) of k is equal to the deficiency (A033879) of k+1.

Original entry on oeis.org

672, 523776, 19327369215

Offset: 1

Amiram Eldar, May 28 2020

Equivalently, k and k+1 have the same absolute value of abundance (or deficiency) with opposite signs.
Equivalently, s(k) + s(k+1) = k + (k+1), where s(k) is the sum of proper divisors of k (A001065).
If k is a 3-perfect number (A005820) and k+1 is a prime, then k is in the sequence. Of the 6 known 3-perfect numbers only 672 and 523776 have this property.
a(4) > 10^11, if it exists.
a(4) > 10^13, if it exists. - Giovanni Resta, May 30 2020

			672 is a term since A033880(672) = sigma(672) - 2*672 = 2016 - 1344 = 672, and A033879(673) = 2*673 - sigma(673) = 1346 - 674 = 672.

Cf. A000203, A001065, A005820, A033879, A033880, A330901, A335253.

ab[n_] := DivisorSigma[1, n] - 2*n; Select[Range[6 * 10^5], ab[#] == -ab[# + 1] &]

cp's OEIS Frontend

A074918 Highly imperfect numbers: n sets a record for the value of abs(sigma(n)-2*n) (absolute value of A033879).

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A075728 Records in abs(sigma(n)-2*n) (absolute value of A033879).

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A295881 Reversing binary representation of the deficiency of n, A033879(n).

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A318441 a(n) = Sum_{d|n} [moebius(n/d) > 0]*A033879(d).

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A324530 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A033879(n), A318458(n)] for all other numbers, except f(1) = -1.

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A325976 a(n) is the largest k <= n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

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A331734 a(n) = A033879(A225546(n)).

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A330901 Numbers k such that k and k+2 have the same deficiency (A033879).

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A331181 Number of values of k, 1 <= k <= n, with A033879(k) = A033879(n), where A033879(n) is the deficiency of n, 2n-sigma(n).

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