A033879 -id:A033879 - cp's OEIS frontend

A323244 a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

0, 1, 1, 2, 1, 4, 1, 6, 0, 5, 1, 10, 1, 16, 2, 6, 1, 12, 1, 18, -3, 18, 1, 22, -4, 46, 4, 22, 1, 10, 1, 30, 14, 82, -2, 14, 1, 256, -12, 22, 1, 36, 1, 66, 8, 226, 1, 46, -12, 19, 8, 130, 1, 28, -19, 70, -12, 748, 1, 42, 1, 1362, 16, 22, 10, 42, 1, 214, 254, 40, 1, 38, 1, 3838, 10, 406, -10, 106, 1, 78, -12, 5458, 1, 26, -72, 12250, -348, 30, 1, 12

Offset: 1

Antti Karttunen, Jan 10 2019

sign

After a(1) = 0, the other zeros occur for k >= 1, at A005940(1+A000396(k)), which, provided no odd perfect numbers exist, is equal to A324201(k) = A062457(A000043(k)): 9, 125, 161051, 410338673, ..., etc.

There are 2321 negative terms among the first 10000 terms.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000043, A000396, A033879, A064989, A156552, A297112, A323240, A323243, A323245, A323248, A324115, A324051, A324103, A324396, A324398, A324543, A324713.
Cf. A324201 (positions of zeros, conjectured), A324551 (of negative terms), A324720 (of nonnegative terms), A324721 (of positive terms), A324731, A324732.
Cf. A329644 (Möbius transform).
Cf. A323174, A324055, A324185, A324546 for other permutations of deficiency, and also A324574, A324575, A324654.

Array[2 # - If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 90] (* Michael De Vlieger, Apr 21 2019 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323244(n) = if(1==n, 0, my(k=A156552(n)); (2*k)-sigma(k));

from sympy import divisor_sigma, primepi, factorint
def A323244(n): return (lambda n: (n<<1)-divisor_sigma(n))(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

a(n) = 2*A156552(n) - A323243(n).
a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).
a(n) = A323248(n) + A001222(n) = (A323247(n) - A323243(n)) + A001222(n).
From Antti Karttunen, Mar 12 2019 & Nov 23 2019: (Start)
a(n) = Sum_{d|n} (2*A297112(d) - A324543(d)) = Sum_{d|n} A329644(d).
A002487(a(n)) = A324115(n).
a(n) = A329638(n) - A329639(n).
a(n) = A329645(n) - A329646(n).
(End)

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 7, -6, 9, 1, 2, 0, 14, -5, 15, -4, 7, 5, 18, -14, 16, 7, 10, -3, 24, -16, 25, 0, 16, 12, 19, -21, 33, 13, 18, -12, 37, -15, 38, 1, 8, 16, 41, -30, 38, 4, 26, 3, 48, -16, 33, -11, 30, 22, 53, -52, 55, 23, 16, 0, 44, -14, 63, 8, 39, -7, 66, -53, 69, 31, 22, 9, 54, -16, 73, -28, 38, 35, 78, -59, 58

Offset: 1

Antti Karttunen, Nov 25 2017

sign

"Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - Antti Karttunen, Dec 16 2024

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.

Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

Array[IntegerExponent[(2 #)!, 2] - DivisorSigma[1, #] &, 85] (* Michael De Vlieger, Nov 26 2017 *)

A294898(n) = ((2*n-sigma(n))-hammingweight(n)); \\ Antti Karttunen, Dec 16 2024

(define (A294898 n) (- (A005187 n) (A000203 n)))

a(n) = A005187(n) - A000203(n).
a(n) = A011371(n) - A001065(n).
a(n) = A033879(n) - A000120(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - Amiram Eldar, Feb 22 2024

Name edited by Antti Karttunen, Dec 16 2024

A323910 Dirichlet inverse of the deficiency of n, A033879.

Original entry on oeis.org

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323910, Sequence Machine.

Cf. A033879, A323911, A323912, A359549 (parity of terms).

Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A324055 Deficiency of Doudna-sequence: a(n) = A033879(A005940(1+n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 5, 1, 6, 2, 6, -4, 19, -3, 14, 1, 10, 4, 10, -2, 22, -12, 12, -12, 41, 7, 26, -19, 94, -12, 41, 1, 12, 8, 18, 0, 38, -12, 22, -10, 58, -4, 18, -48, 102, -54, 30, -28, 109, 25, 66, -17, 148, -72, 47, -51, 286, 32, 126, -64, 469, -39, 122, 1, 16, 10, 22, 4, 46, -12, 42, -8, 70, 4, 42, -56, 178, -60, 58, -26, 118, 20

Offset: 0

Antti Karttunen, Feb 14 2019

Both here and in the mirror image sequence A324185, the lowermost (asinh) scatter plot shows on the y = 0 line the numbers that correspond to the perfect numbers. Compare also to the scatter plot of A243492.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A000120, A033879, A054429.
See A106737, A290077, A323915, A324052, A324054, A324056, A324057, A324058, A324114, A324335, A324340, A324348, A324349, A324394, A324395 for other sequences as permuted by A005940, and compare their scatter plots.
Cf. also A243492, A323174, A323244, A324185, A324346, A324347, A324654.

Array[Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2]}, 2 # - DivisorSigma[1, #] &[Times @@ Flatten@ Table[Prime[Count[Flatten@ #, 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]] &, 82, 0] (* Michael De Vlieger, Mar 11 2019, after Robert G. Wilson v at A005940 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A033879(n) = (2*n-sigma(n));
A324055(n) = A033879(A005940(1+n));

A324055(n) = { my(m1=2,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); (m1-m2); };

a(n) = A033879(A005940(1+n)).
a(n) = 2*A005940(1+n) - A324054(n).
For n > 0, a(n) = A324185(A054429(n)).
a(n) = A324348(n) + A000120(A005940(1+n)).

A286449 Restricted growth sequence computed for A033879 (deficiency), or equally, for A033880 (abundance of n).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 2, 7, 8, 9, 3, 5, 1, 10, 11, 12, 13, 7, 14, 15, 16, 17, 7, 18, 4, 19, 16, 20, 1, 12, 18, 15, 21, 22, 10, 15, 23, 24, 16, 25, 3, 9, 26, 27, 28, 29, 30, 20, 5, 31, 16, 32, 33, 34, 35, 36, 37, 38, 19, 15, 1, 27, 16, 39, 7, 25, 8, 40, 41, 42, 34, 35, 9, 36, 16, 43, 44, 29, 32, 45, 46, 47, 24, 48, 8, 49, 50, 40, 10, 36, 51, 40, 52, 53

Offset: 1

Antti Karttunen, May 13 2017

nonn

			We start by setting a(1) = 1 for A033879(1) = 1. Then, whenever A033879(k) is equal to some A033879(m) with m < k, we set a(k) = a(m). Otherwise (when the value is a new one, not encountered before), we allot for a(k) the least natural number not present among a(1) .. a(k-1).
For n=2, as A033879(2) = 1, which was already present at A033879(1), we set a(2) = a(1) = 1.
For n=3, as A033879(3) = 2, which is a new value not encountered before, we set a(3) = 1 + max(a(1),a(2)) = 2.
For n=4, as A033879(4) = 1, which was already present at n = 2 and n = 1, we set a(4) = a(1) = 1.
For n=5, as A033879(5) = 4, which is a new value not encountered before, we set a(5) = 1 + max(a(1),a(2),a(3),a(4)) = 3.
For n=12, as A033879(12) = -4, which is a new value not encountered before, we set a(12) = 1 + max(a(1),...,a(11)) = 8. Note that the sign matters here; -4 is not equal to +4, which was encountered already at n=5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A033879, A033880, A286448, A286450, A286603.

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])); }
A033879(n) = ((2*n)-sigma(n));
write_to_bfile(1,rgs_transform(vector(10000,n,A033879(n))),"b286449.txt");

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

A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 4, 5, -4, 1, 2, 1, -12, -3, 6, 1, 6, 1, -2, -19, -28, 1, 4, 14, -60, 19, -10, 1, -12, 1, 10, -51, -124, -12, 10, 1, -252, -115, 0, 1, -48, 1, -26, 7, -508, 1, 8, 41, 12, -243, -58, 1, 22, -64, -8, -499, -1020, 1, -12, 1, -2044, -17, 12, -168, -120, 1, -122, -1011, -54, 1, 18, 1, -4092, 26, -250, -39, -264, 1, 4

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Zeros occur at A122111(A000396(k)), k >= 1: 6, 40, 11264, 18253611008, ...

Cf. A033879, A122111, A323167, A323168.

Cf. also A323244.

A122111[n_] := Product[Prime[Sum[If[jA122111[n]}, 2k - DivisorSigma[1, k]];
Array[a, 80] (* Jean-François Alcover, Sep 23 2020 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A323174(n) = { my(k=A122111(n)); ((2*k)-sigma(k)); }

a(n) = A033879(A122111(n)).

a(n) = 2*A122111(n) - A323173(n).

A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 0, 4, 1, 14, -3, 19, -4, 6, 2, 6, 1, 41, -12, 94, -19, 26, 7, 41, -12, 12, -12, 22, -2, 10, 4, 10, 1, 122, -39, 469, -64, 126, 32, 286, -51, 47, -72, 148, -17, 66, 25, 109, -28, 30, -54, 102, -48, 18, -4, 58, -10, 22, -12, 38, 0, 18, 8, 12, 1, 365, -120, 2344, -199, 626, 157, 2001, -168, 222, -372, 1030, -92, 458, 172, 1198

Offset: 0

Antti Karttunen, Feb 17 2019

If there are no odd perfect numbers, then all n for which a(n) is 0 are given by sequence A324200.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000203, A033879, A054429, A163511, A324055, A324182 (compare the scatter plot), A324184, A324187, A324188, A324189, A324199, A324200.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A033879(n) = (2*n-sigma(n));
A324185(n) = A033879(A163511(n));

A324184(n) = if(!n,1,my(p=2,mp=p*p,m=1); while(n>1, if(n%2, p=nextprime(1+p); mp = p*p, if((2==n)||!(n%4),mp *= p,m *= (mp-1)/(p-1))); n >>= 1); (m*(mp-1)/(p-1)));
A324185(n) = (2*A163511(n)) - A324184(n);

a(n) = A033879(A163511(n)) = 2*A163511(n) - A324184(n) = 2*A163511(n) - A000203(A163511(n)).

For n > 0, a(n) = A324055(A054429(n)).

A325379 a(n) = A033879(A228058(n)).

Original entry on oeis.org

12, 52, 72, 148, 132, 216, 172, 192, 84, 292, 252, 292, 412, 476, 352, 520, 432, 640, 592, 472, 492, 672, 532, 552, 748, 412, 672, 976, 732, 576, 772, 1132, 1048, 1128, 852, 1284, 892, 952, 972, 1324, 1460, 1356, 1624, 1720, 1132, 1152, 1192, -36, 1660, 1272, 1068, 1332, 1812, 1372, 1888, 1392, 2116, 1452, 1972, 2040, 1552, 2116

Offset: 1

Antti Karttunen, Apr 22 2019

sign

The negative terms -36, -1692, -2388, -34944, -16596, -38628, -512, ..., occur at n = 48, 378, 1744, 2255, 2745, 2870, 3555, ..., where A228058(n) is 2205, 19845, 108045, 143325, 178605, 187425, 236925, ..., one of the odd abundant numbers, A005231.

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A005231, A033879, A228058, A228059, A325310, A325319, A325320, A325378.

A033879(n) = (n+n-sigma(n));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(A033879(n), ", ")));

a(n) = A033879(A228058(n)).
a(n) = A325319(n) - A325320(n).
A001511(abs(a(n))) = A325310(A228058(n)), assuming there are no odd perfect numbers, in which case A001511(abs(a(n))) >= 3 for all n. That is, all terms are multiples of 4.

A324546 An analog of deficiency (A033879) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

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, 14, 8, 22, -12, 19, 10, 10, 0, 28, -12, 30, 1, 12, 14, 22, -19, 36, 16, 18, -10, 40, -12, 42, 4, 41, 20, 46, -28, 41, 7, 26, 6, 52, -12, 94, -8, 22, 26, 58, -48, 60, 28, 22, 1, 38, -54, 66, 10, 30, -4, 70, -51, 72, 34, 30, 12, 58, -12, 78, -26, 42, 38, 82, -64, 102, 40, 18, -4, 88

Offset: 1

Antti Karttunen, Mar 06 2019

sign

Even positions for zeros is given by the even terms of A000396, because they are among the fixed points of permutation A250246. Whether there are any zeros in odd positions depends on whether there are any odd perfect numbers. If such zeros exist, they would not necessarily be in the same positions as in A033879.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539
Index entries for sequences generated by sieves
Index entries for sequences related to sigma(n)

Cf. A000396, A033879, A083221, A250246, A324535, A324545.

Cf. also A323244, A323174, A324574, A324575.

up_to = 65539;
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A324546(n) = { my(k=A250246(n)); (k+k - sigma(k)); };

a(n) = A033879(A250246(n)) = 2*A250246(n) - A324545(n).

a(n) = A250246(n) - A324535(n).

cp's OEIS Frontend

A323244 a(1) = 0; and for n > 1, a(n) = A033879(A156552(n)).

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A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

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A323910 Dirichlet inverse of the deficiency of n, A033879.

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A324055 Deficiency of Doudna-sequence: a(n) = A033879(A005940(1+n)).

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A286449 Restricted growth sequence computed for A033879 (deficiency), or equally, for A033880 (abundance 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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A323174 Deficiency computed for conjugated prime factorization: a(n) = A033879(A122111(n)).

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A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

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