A286449 -id:A286449 - cp's OEIS frontend

A286592 Compound filter (prime signature & deficiency/abundance): a(n) = P(A046523(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 3, 5, 10, 8, 42, 17, 36, 40, 27, 30, 183, 47, 34, 51, 136, 57, 243, 80, 288, 72, 177, 122, 765, 194, 72, 308, 117, 192, 1020, 212, 528, 142, 259, 196, 1576, 255, 111, 196, 1059, 302, 1020, 327, 103, 202, 471, 380, 2823, 500, 832, 306, 132, 498, 765, 672, 1564, 747, 786, 668, 4620, 743, 282, 337, 2080, 502, 1020, 782, 165, 441, 696, 822, 6288, 905, 747, 1047, 202

Offset: 1

Antti Karttunen, May 21 2017

nonn

The lowermost conspicuous horizontal line in the scatter plot (at about log 3) is caused by value 1020, which corresponds to the prime signature 30 (p*q*r) and deficiency -12 packed together with the pairing function (as A002260(1020) = 30 and A004736(1020) = 16, A286449(24) = 16 and A033879(24) = -12). This value occurs in this sequence (at least) in the positions given by A138636, from its third term 30 onward.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A033879, A033880, A046523, A138636, A286449, A286593, A286595.

(define (A286592 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A286449 n)) 2) (- (A046523 n)) (- (* 3 (A286449 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A286449(n))^2) - A046523(n) - 3*A286449(n)).

A286593 Compound filter (the length of rightmost run of 1's in base-2 & deficiency/abundance): a(n) = P(A089309(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 5, 1, 4, 12, 24, 1, 16, 2, 30, 38, 37, 13, 32, 1, 46, 56, 80, 79, 22, 107, 139, 138, 137, 22, 173, 18, 172, 175, 281, 1, 67, 154, 122, 211, 232, 57, 139, 254, 277, 121, 327, 8, 37, 381, 439, 408, 407, 436, 212, 11, 466, 138, 564, 598, 562, 596, 668, 784, 704, 258, 196, 1, 352, 121, 782, 22, 301, 38, 864, 821, 862, 562, 632, 47, 631, 156, 1039, 947, 407

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A033879, A033880, A089309, A286449, A286592, A286595.

(define (A286593 n) (* (/ 1 2) (+ (expt (+ (A089309 n) (A286449 n)) 2) (- (A089309 n)) (- (* 3 (A286449 n))) 2)))

a(n) = (1/2)*(2 + ((A089309(n)+A286449(n))^2) - A089309(n) - 3*A286449(n)).

A286595 Compound filter (2-adic valuation & deficiency/abundance): a(n) = P(A001511(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 2, 6, 4, 12, 11, 10, 16, 5, 22, 48, 37, 8, 11, 15, 46, 68, 67, 108, 22, 107, 106, 175, 137, 30, 154, 18, 172, 138, 191, 21, 67, 173, 106, 256, 232, 57, 106, 329, 277, 138, 301, 13, 37, 353, 352, 501, 407, 467, 191, 24, 466, 138, 497, 634, 562, 632, 631, 744, 704, 192, 106, 28, 352, 138, 742, 39, 301, 38, 781, 950, 862, 597, 596, 58, 631, 138, 904, 1133, 407

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A001511, A033879, A033880, A286449, A286260, A286451, A286460, A286592, A286593.

(define (A286595 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A286449 n)) 2) (- (A001511 n)) (- (* 3 (A286449 n))) 2)))

a(n) = (1/2)*(2 + ((A001511(n)+A286449(n))^2) - A001511(n) - 3*A286449(n)).

A286458 Compound filter: a(n) = P(A286448(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 2, 3, 13, 25, 24, 15, 61, 5, 85, 113, 112, 64, 87, 55, 201, 243, 242, 313, 204, 393, 451, 513, 137, 22, 613, 250, 723, 651, 842, 276, 67, 844, 196, 1015, 1107, 657, 196, 1253, 1355, 1020, 1407, 559, 812, 795, 1744, 1864, 833, 2051, 1062, 101, 2181, 1363, 2384, 2524, 597, 2741, 2891, 3045, 3203, 1935, 1756, 1081, 1249, 1938, 3703, 1534, 441

Offset: 1

Antti Karttunen, May 14 2017

nonn

For all i, j: a(i) = a(j) => A286385(i) = A286385(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A033879, A048673, A252748, A286385, A286448, A286449, A286459.

(define (A286458 n) (* (/ 1 2) (+ (expt (+ (A286448 n) (A286449 n)) 2) (- (A286448 n)) (- (* 3 (A286449 n))) 2)))

a(n) = (1/2)*(2 + ((A286448(n)+A286449(n))^2) - A286448(n) - 3*A286449(n)).

a(n) = A286459(A048673(n)).

A033879 Deficiency of n, or 2n - (sum of divisors of n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, -4, 12, 4, 6, 1, 16, -3, 18, -2, 10, 8, 22, -12, 19, 10, 14, 0, 28, -12, 30, 1, 18, 14, 22, -19, 36, 16, 22, -10, 40, -12, 42, 4, 12, 20, 46, -28, 41, 7, 30, 6, 52, -12, 38, -8, 34, 26, 58, -48, 60, 28, 22, 1, 46, -12, 66, 10, 42, -4, 70, -51

Offset: 1

N. J. A. Sloane

Records for the sequence of the absolute values are in A075728 and the indices of these records in A074918. - R. J. Mathar, Mar 02 2007
a(n) = 1 iff n is a power of 2. a(n) = n - 1 iff n is prime. - Omar E. Pol, Jan 30 2014
If a(n) = 1 then n is called a least deficient number or an almost perfect number. All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2. See A000079. - Jianing Song, Oct 13 2019
It is not known whether there are any -1's in this sequence. See comment in A033880. - Antti Karttunen, Feb 02 2020

			For n = 10 the divisors of 10 are 1, 2, 5, 10, so the deficiency of 10 is 10 minus the sum of its proper divisors or simply 10 - 5 - 2 - 1 = 2. - _Omar E. Pol_, Dec 27 2013

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

N. J. A. Sloane, Table of n, a(n) for n = 1..25000 [First 2000 terms from T. D. Noe, terms up to 16384 from Antti Karttunen]
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.
Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1610.01868 [math.NT], 2016.
Jose Arnaldo B. Dris, Analysis of the Ratio D(n)/n, arXiv:1703.09077 [math.NT], 2017.
Jose Arnaldo Bebita Dris, On a curious biconditional involving the divisors of odd perfect numbers, Notes on Number Theory and Discrete Mathematics, 23(4) (2017), 1-13.
Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego, Some Modular Considerations Regarding Odd Perfect Numbers, arXiv:2002.12139 [math.NT], 2020.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, Conditions equivalent to the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers - Part II, Notes on Number Theory and Discrete Mathematics (2018) Vol. 24, No. 3, 62-67.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
Index entries for sequences related to sigma(n).

Cf. A000396 (positions of zeros), A005100 (of positive terms), A005101 (of negative terms).
Cf. A000203, A033880, A074918, A075728, A192895, A286385, A286449, A295881, A295882.
Cf. A083254 (Möbius transform), A228058, A296074, A296075, A323910, A325636, A325826, A325970, A325976.
Cf. A141545 (positions of a(n) = -12).
For this sequence applied to various permutations of natural numbers and some other sequences, see A323174, A323244, A324055, A324185, A324546, A324574, A324575, A324654, A325379.

with(numtheory): A033879:=n->2*n-sigma(n): seq(A033879(n), n=1..100);

Table[2n-DivisorSigma[1,n],{n,80}] (* Harvey P. Dale, Oct 24 2011 *)

a(n)=2*n-sigma(n) \\ Charles R Greathouse IV, Oct 13 2016

from sympy import divisor_sigma
def A033879(n): return (n<<1)-divisor_sigma(n) # Chai Wah Wu, Apr 13 2024

[2*n-sigma(n, 1) for n in range(1, 73)] # Stefano Spezia, Jul 18 2025

a(n) = -A033880(n).
a(n) = A005843(n) - A000203(n). - Omar E. Pol, Dec 14 2008
a(n) = n - A001065(n). - Omar E. Pol, Dec 27 2013
G.f.: 2*x/(1 - x)^2 - Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 24 2017
a(n) = A286385(n) - A252748(n). - Antti Karttunen, May 13 2017
From Antti Karttunen, Dec 29 2017: (Start)
a(n) = Sum_{d|n} A083254(d).
a(n) = Sum_{d|n} A008683(n/d)*A296075(d).
a(n) = A065620(A295881(n)) = A117966(A295882(n)).
a(n) = A294898(n) + A000120(n).
(End)
From Antti Karttunen, Jun 03 2019: (Start)
Sequence can be represented in arbitrarily many ways as a difference of the form (n - f(n)) - (g(n) - n), where f and g are any two sequences whose sum f(n)+g(n) = sigma(n). Here are few examples:
a(n) = A325314(n) - A325313(n) = A325814(n) - A034460(n) = A325978(n) - A325977(n).
a(n) = A325976(n) - A325826(n) = A325959(n) - A325969(n) = A003958(n) - A324044(n).
a(n) = A326049(n) - A326050(n) = A326055(n) - A326054(n) = A326044(n) - A326045(n).
a(n) = A326058(n) - A326059(n) = A326068(n) - A326067(n).
a(n) = A326128(n) - A326127(n) = A066503(n) - A326143(n).
a(n) = A318878(n) - A318879(n).
a(A228058(n)) = A325379(n). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Pi^2/12 = 0.177532... . - Amiram Eldar, Dec 07 2023

Definition corrected by N. J. A. Sloane, Jul 04 2005

A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 25 2018

Restricted growth sequence transform of ordered pair [A000120(n), A033879(n)], or equally, of ordered pair [A000120(n), A294898(n)].
For all i, j:
  A318311(i) = A318311(j) => a(i) = a(j),
  a(i) = a(j) => A286449(i) = A286449(j),
  a(i) = a(j) => A294898(i) = A294898(j).
In the scatter plot one can see the effects of both base-2 related A000120 (binary weight of n) and prime factorization related A033879 (deficiency of n) graphically mixed: from the former, a square grid pattern, and from the latter the black rays that emanate from the origin. The same is true for A323898, while in the ordinal transform of this sequence, A331184, such effects are harder to visually discern. - Antti Karttunen, Jan 13 2020

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

Cf. A000120, A033879, A286449, A295881, A295882, A294898.
Cf. A318311, A323889, A323892, A323898, A324344, A324380, A324390 for similar constructions.
Cf. A331184 (ordinal transform).

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; };
A318310aux(n) = [hammingweight(n), (2*n) - sigma(n)];
v318310 = rgs_transform(vector(up_to,n,A318310aux(n)));
A318310(n) = v318310[n];

Name changed by Antti Karttunen, Jan 13 2020

A295882 Balanced ternary representation of the deficiency of n, A033879(n).

Original entry on oeis.org

1, 1, 5, 1, 4, 0, 15, 1, 17, 5, 10, 8, 12, 4, 15, 1, 52, 6, 45, 7, 10, 11, 49, 24, 46, 10, 53, 0, 28, 24, 30, 1, 45, 53, 49, 65, 36, 52, 49, 20, 40, 24, 159, 4, 12, 50, 154, 56, 161, 16, 30, 15, 142, 24, 41, 19, 43, 29, 139, 204, 150, 28, 49, 1, 154, 24, 147, 10, 159, 8, 106, 192, 99, 43, 29, 12, 139, 24, 87, 55

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

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

Cf. A033879, A033880, A117966, A117967, A117968, A286449, A295881, A296071, A296072, A296073.

Cf. A000396 (gives the positions of zeros).

A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };

(define (A295882 n) (let ((x (A033879 n))) (if (>= x 0) (A117967 x) (A117968 (- x)))))

If A033879(n) >= 0, then a(n) = A117967(A033879(n)), otherwise a(n) = A117968(-A033879(n)).

For all n >= 1, A117966(a(n)) = A033879(n).

A286448 Restricted growth sequence computed for A252748 (= A003961(n) - 2*n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003961, A252748, A286449, A286450.

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A252748(n) = (A003961(n) - (2*n));
write_to_bfile(1,rgs_transform(vector(10000,n,A252748(n))),"b286448.txt");

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).

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];

cp's OEIS Frontend

A286592 Compound filter (prime signature & deficiency/abundance): a(n) = P(A046523(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286593 Compound filter (the length of rightmost run of 1's in base-2 & deficiency/abundance): a(n) = P(A089309(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286595 Compound filter (2-adic valuation & deficiency/abundance): a(n) = P(A001511(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286458 Compound filter: a(n) = P(A286448(n), A286449(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A033879 Deficiency of n, or 2n - (sum of divisors of n).

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A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.

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A295882 Balanced ternary representation of the deficiency of n, A033879(n).

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A286448 Restricted growth sequence computed for A252748 (= A003961(n) - 2*n).

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