A033879 -id:A033879 - cp's OEIS frontend

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

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).

A325826 a(n) is the largest k <= sigma(n)-n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 7, 4, 7, 1, 15, 1, 9, 7, 15, 1, 20, 1, 21, 11, 13, 1, 35, 6, 13, 13, 1, 1, 41, 1, 31, 13, 19, 13, 55, 1, 21, 17, 49, 1, 53, 1, 39, 31, 23, 1, 75, 8, 43, 19, 43, 1, 65, 17, 63, 23, 31, 1, 107, 1, 33, 41, 63, 19, 77, 1, 57, 25, 73, 1, 122, 1, 39, 49, 61, 19, 89, 1, 105, 40, 43, 1, 139, 23, 43, 31, 91, 1, 143, 19, 75

Offset: 1

Antti Karttunen, May 29 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000040, A000203, A000396, A001065, A033879, A324213, A325817, A325818, A325969, A325970, A325976.

A325826(n) = { my(s=sigma(n)); forstep(k=s-n, 0, -1, if(1==gcd((n+n-sigma(n)), k), return(k))); };

A325818(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(s-i))); };
A325826(n) = (A325818(n) - n);

a(n) = A325818(n) - n = A001065(n) - A325817(n) = A325976(n) - A033879(n).
a(A000040(n)) = a(A000396(n)) = 1.
a(n) >= A325969(n).
gcd(a(n), A325976(n)) = 1.

A323911 Sum of deficiency of n (A033879) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 8, 0, -2, 0, 12, 16, 1, 0, -2, 0, 0, 24, 20, 0, -8, 16, 24, 12, 2, 0, -28, 0, 1, 40, 32, 48, -15, 0, 36, 48, -6, 0, -36, 0, 6, 16, 44, 0, -20, 36, 6, 64, 8, 0, -12, 80, -4, 72, 56, 0, -46, 0, 60, 28, 1, 96, -52, 0, 12, 88, -44, 0, -39, 0, 72, 28, 14, 120, -60, 0, -18, 37, 80, 0, -58, 128, 84, 112, 0, 0, -52, 144, 18

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A033879, A297159, A323403, A323910, A323913.

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];
A323911(n) = (A033879(n)+A323910(n));

a(n) = A033879(n) + A323910(n).

A324574 a(1) = 0; for n > 1, a(n) = A033879(A087207(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 5, 0, 1, 1, 2, 1, 4, 2, 16, 1, 2, 1, 18, 1, 5, 1, 6, 1, 1, -3, 46, -4, 2, 1, 82, 14, 4, 1, 10, 1, 16, 0, 256, 1, 2, 1, 4, -12, 18, 1, 2, -2, 5, 8, 226, 1, 6, 1, 748, 2, 1, -19, 18, 1, 46, -12, 12, 1, 2, 1, 1362, 0, 82, -12, 22, 1, 4, 1, 3838, 1, 10, 10, 5458, 254, 16, 1, 6, -10, 256, -348, 12250

Offset: 1

Antti Karttunen, Mar 08 2019

sign

As A087207 is a surjective function that toggles the parity, it follows that if it can be proved/disproved that a(n) = 0 for some/any even n, then it also proves/disproves the existence of odd perfect numbers.

The positions (n > 1) of zeros in squarefree n, 15, 385, ..., can be obtained as A019565(A000396(n)).

Cf. A000396, A007947, A033879, A019565, A087207, A324573, A324575.

Cf. also A323244, A323174, A324546.

A033879(n) = (2*n-sigma(n));
A087207(n) = vecsum(apply(p->1<A087207
A324574(n) = if(1==n,0,A033879(A087207(n)));

a(1) = 0; for n > 1, a(n) = A033879(A087207(n)).

a(n) = a(A007947(n)) = A324575(A007947(n)).

A324575 a(1) = 0; for n > 1, a(n) = A033879(A048675(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 5, 0, 1, 1, 4, 1, 0, 2, 16, 1, 4, 1, 18, 0, 2, 1, 6, 1, 4, -3, 46, -4, 0, 1, 82, 14, 6, 1, 10, 1, -3, 1, 256, 1, 0, 1, 5, -12, 14, 1, 6, -2, 10, 8, 226, 1, 1, 1, 748, -4, 0, -19, 18, 1, -12, -12, 12, 1, 6, 1, 1362, 2, 8, -12, 22, 1, 1, 1, 3838, 1, -4, 10, 5458, 254, 18, 1, 5, -10, -12, -348, 12250

Offset: 1

Antti Karttunen, Mar 07 2019

sign

Cf. A007947, A033879, A048675, A324573, A324574.

Cf. also A323244, A323174, A324546.

A033879(n) = (2*n-sigma(n));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A324575(n) = if(1==n,0,A033879(A048675(n)));

a(1) = 0; for n > 1, a(n) = A033879(A048675(n)).
a(n) = 2*A048675(n) - A324573(n).
a(A007947(n)) = A324574(n).
a(p) = 1 for all primes p.

A325970 a(n) is the largest k <= A066503(n) such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

0, 0, -1, 2, -1, -1, -1, 6, 6, -1, -1, 5, -1, -1, -1, 14, -1, 11, -1, 9, -1, -1, -1, 17, 20, -1, 23, 1, -1, -1, -1, 30, -1, -1, -1, 30, -1, -1, -1, 29, -1, -1, -1, 21, 29, -1, -1, 41, 42, 40, -1, 25, -1, 47, -1, 41, -1, -1, -1, 29, -1, -1, 41, 62, -1, -1, -1, 33, -1, -1, -1, 65, -1, -1, 59, 37, -1, -1, -1, 69, 78

Offset: 1

Antti Karttunen, May 31 2019

sign

a(n) = n-k for the least k >= A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A005117, A007947, A033879, A066503, A324213, A325826, A325960, A325967, A325971, A325972.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325970(n) = { my(s=sigma(n)); forstep(k=n-A007947(n), -oo, -1, if(1==gcd(k, n+n-sigma(n)), return(k))); };
\\ Or alternatively:
A325970(n) = { my(s=sigma(n)); for(i=A007947(n), s, if(1==gcd(n-i, n-(s-i)), return(n-i))); };

a(n) = n - A325971(n).

For n >= 3, a(A005117(n)) = -1.

A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 0, 12, 4, 6, 1, 16, -1, 18, 0, 10, 8, 22, 0, 19, 10, 14, 0, 28, 0, 30, 1, 18, 14, 22, -1, 36, 16, 22, 0, 40, 0, 42, 4, 12, 20, 46, 0, 41, 7, 30, 6, 52, 0, 38, 0, 34, 26, 58, 0, 60, 28, 22, 1, 46, 0, 66, 10, 42, 0, 70, -1, 72, 34, 26, 12, 58, 0, 78, 0, 41, 38, 82, 0, 62, 40, 54, 0, 88, 0, 70

Offset: 1

Antti Karttunen, Dec 04 2024

sign

If n is abundant, then negate the value of A103977(n), otherwise use as it is.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A033879, A103977.

Cf. A005100 (positions of terms > 0), A083207 (positions of 0's), A083211 (positions of negative terms), A156903 (positions of odd negative terms), A171641 (of even negative terms).

A033879(n) = (n+n-sigma(n));
nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1,#v,if(v[i], listput(lista,i))); Vec(lista); };
A103977(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); my(plist=nonzerocoefpositions(p), m = #plist, d); if(!(m%2), plist[1+(m/2)]-plist[m/2], d = plist[(m+1)/2]-plist[(m-1)/2]; if(1==d,0,d)); };
A378600(n) = { my(d=A033879(n)); if(d>=0, d, -A103977(n)); };

If A033879(n) >= 0, a(n) = A033879(n), otherwise a(n) = -A103977(n).

A318442 a(n) = Sum_{d|n} [moebius(n/d) < 0]*A033879(d).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 2, 5, 1, 1, 1, 7, 6, 1, 1, 5, 1, 3, 8, 11, 1, -3, 4, 13, 5, 5, 1, 9, 1, 1, 12, 17, 10, -7, 1, 19, 14, -1, 1, 15, 1, 9, 11, 23, 1, -11, 6, 21, 18, 11, 1, 11, 14, 1, 20, 29, 1, -17, 1, 31, 15, 1, 16, 27, 1, 15, 24, 29, 1, -31, 1, 37, 25, 17, 16, 33, 1, -9, 14, 41, 1, -15, 20, 43, 30, 5, 1, -1, 18, 21, 32, 47, 22, -27, 1

Offset: 1

Antti Karttunen, Aug 26 2018

sign

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

Cf. A033879, A083254, A318320, A318325, A318441.

A318442(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) = A318320(n) - A318326(n).
A083254(n) = A318441(n) - a(n).

A323892 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A033879(i) = A033879(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 8, 9, 10, 3, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 8, 21, 22, 23, 24, 25, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 35, 47, 48, 49, 50, 51, 52, 53, 54, 55, 1, 56, 57, 58, 8, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 47, 72, 73, 74, 75, 76, 77, 78, 79, 80, 31, 81

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A033879(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A033879.

Cf. also A318310, A323889.

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A033879(n) = (2*n-sigma(n));
A323892aux(n) = [A002487(n), A033879(n)];
v323892 = rgs_transform(vector(up_to,n,A323892aux(n)));
A323892(n) = v323892[n];

a(2^n) = 1 for all n >= 0.

A324654 a(n) = A033879(A276086(n)).

Original entry on oeis.org

1, 1, 2, 0, 5, -3, 4, 2, 6, -12, 12, -54, 19, 7, 26, -72, 47, -309, 94, 32, 126, -372, 222, -1584, 469, 157, 626, -1872, 1097, -7959, 6, 4, 10, -12, 22, -60, 22, -4, 18, -156, 6, -612, 102, -44, 58, -876, -74, -3372, 502, -244, 258, -4476, -474, -17172, 2502, -1244, 1258, -22476, -2474, -86172, 41, 25, 66, -96, 141, -459, 148, -46, 102

Offset: 0

Antti Karttunen, Mar 10 2019

sign

Interestingly, this kind of sampling of deficiency (A033879; recall that the range of A276086 does not cover the whole N) biases it strongly towards negative values: of the first 2310 terms, 1565 are negative (~ 68%) and of the first 30030 terms, 22507 are negative (~ 75%). Compare also to A323174, which covers whole N.

Cf. A033879, A276086, A324653.

Cf. also A323174, A324055.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A033879(n) = (2*n-sigma(n));
A324654(n) = A033879(A276086(n));

a(n) = A033879(A276086(n)).

a(n) = 2*A276086(n) - A324653(n).

cp's OEIS Frontend

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

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A325826 a(n) is the largest k <= sigma(n)-n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

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A323911 Sum of deficiency of n (A033879) and its Dirichlet inverse.

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A324574 a(1) = 0; for n > 1, a(n) = A033879(A087207(n)).

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A324575 a(1) = 0; for n > 1, a(n) = A033879(A048675(n)).

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A325970 a(n) is the largest k <= A066503(n) such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

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A378600 Signed variant of Zumkeller deficiency: a(n) = signum(A033879(n)) * A103977(n).

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A318442 a(n) = Sum_{d|n} [moebius(n/d) < 0]*A033879(d).

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A323892 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A033879(i) = A033879(j), for all i, j >= 1.

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