A088580 -id:A088580 - cp's OEIS frontend

A336696 Sum of odd divisors of 1+sigma(n).

1, 1, 6, 1, 8, 14, 13, 1, 8, 20, 14, 30, 24, 31, 31, 1, 20, 6, 32, 44, 48, 38, 31, 62, 1, 44, 42, 80, 32, 74, 48, 1, 57, 72, 57, 24, 56, 62, 80, 112, 44, 98, 78, 108, 80, 74, 57, 156, 30, 48, 74, 156, 72, 133, 74, 133, 121, 112, 62, 183, 104, 98, 192, 1, 108, 180, 96, 128, 98, 180, 74, 57, 124, 144, 156, 192, 98

Offset: 1

Antti Karttunen, Jul 31 2020

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. A000203, A000593, A088580, A193337, A324294, A332459, A336691, A336692, A336693, A336694, A336696.

Table[Total[Select[Divisors[1+DivisorSigma[1,n]],OddQ]],{n,80}] (* Harvey P. Dale, Jan 01 2022 *)

A000593(n) = sigma(n>>valuation(n, 2));
A336696(n) = A000593(1+sigma(n));

a(n) = A000593(1+A000203(n)) = A000593(A088580(n)) = A000593(A332459(n)).

A296092 Least number with the same prime signature as sigma(n)+1.

Original entry on oeis.org

2, 4, 2, 8, 2, 2, 4, 16, 6, 2, 2, 2, 6, 4, 4, 32, 2, 24, 6, 2, 6, 2, 4, 2, 32, 2, 2, 6, 2, 2, 6, 64, 4, 6, 4, 12, 6, 2, 6, 6, 2, 2, 12, 6, 2, 2, 4, 8, 6, 6, 2, 12, 6, 4, 2, 4, 16, 6, 2, 4, 12, 2, 30, 128, 6, 6, 6, 2, 2, 6, 2, 36, 12, 6, 8, 6, 2, 4, 16, 6, 6, 2, 6, 36, 2, 6, 4, 2, 6, 6, 2, 4, 6, 6, 4, 6, 12, 12, 2, 6, 2, 6, 30, 2, 2

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

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

Cf. A000203, A046523, A088580, A296212.

Cf. also A276528, A296076, A296078, A296085, A296091.

(define (A296092 n) (A046523 (+ 1 (A000203 n))))

a(n) = A046523(A088580(n)) = A046523(1+A000203(n)).

A332455 Starting from sigma(n)+1, number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

Original entry on oeis.org

0, 0, 1, 0, 5, 2, 6, 0, 5, 6, 2, 5, 5, 7, 7, 0, 6, 1, 1, 9, 8, 6, 7, 5, 0, 9, 40, 10, 39, 42, 8, 0, 7, 41, 7, 4, 11, 5, 10, 33, 9, 43, 4, 1, 11, 42, 7, 39, 5, 38, 42, 7, 41, 34, 42, 34, 6, 33, 5, 16, 39, 43, 12, 0, 1, 42, 3, 15, 43, 42, 42, 7, 3, 10, 39, 3, 43, 16, 6, 14, 5, 15, 1, 17, 41, 8, 34, 4, 33, 46, 2, 16, 44, 42, 34, 39

Offset: 1

Antti Karttunen, Feb 16 2020

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 3x+1 (or Collatz) problem
Index entries for sequences related to sigma(n)

Cf. A000203, A006667, A088580, A202274, A332453, A332454.

Cf. also A324294, A332459.

A006667(n) = { my(t=0); while(n>1, if(n%2, t++; n=3*n+1, n>>=1)); (t); };
A332455(n) = A006667(1+sigma(n));

a(n) = A006667(A088580(n)) = A006667(1+sigma(n)).

a(2^n) = 0 for all n >= 0. [Zero occurs at least also at a(25). See A202274]

A152771 a(n) = sigma(n) - 2*d(n) + 1.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 5, 8, 8, 11, 9, 17, 11, 17, 17, 22, 15, 28, 17, 31, 25, 29, 21, 45, 26, 35, 33, 45, 27, 57, 29, 52, 41, 47, 41, 74, 35, 53, 49, 75, 39, 81, 41, 73, 67, 65, 45, 105, 52, 82, 65, 87, 51, 105, 65, 105, 73, 83, 57, 145

Offset: 1

Omar E. Pol, Dec 14 2008

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A152770, A152772.

Table[DivisorSigma[1, n] - 2 DivisorSigma[0, n] + 1, {n, 60}] (* Ivan Neretin, Sep 30 2017 *)

a(n) = sigma(n) - 2*numdiv(n) + 1; \\ Michel Marcus, Sep 30 2017

a(n) = A000203(n) - 2*A000005(n) + 1 = A000203(n) - A114003(n) = A088580(n) - A062011(n). - Omar E. Pol, Sep 30 2017

G.f.: Sum_{k>=1} x^(3*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Apr 24 2021

A332454 Starting from sigma(n)+1, number of halving steps to reach 1 in '3x+1' problem, or -1 if this never happens.

Original entry on oeis.org

1, 2, 4, 3, 11, 7, 13, 4, 12, 14, 7, 13, 12, 16, 16, 5, 14, 7, 6, 20, 18, 15, 16, 14, 5, 20, 69, 22, 67, 73, 18, 6, 17, 71, 17, 13, 23, 14, 22, 59, 20, 75, 12, 8, 24, 73, 17, 69, 14, 67, 73, 18, 71, 61, 73, 61, 16, 59, 14, 33, 68, 75, 26, 7, 8, 74, 11, 31, 75, 74, 73, 19, 11, 23, 69, 12, 75, 33, 16, 30, 15, 31, 8, 35, 72, 20

Offset: 1

Antti Karttunen, Feb 16 2020

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 3x+1 (or Collatz) problem
Index entries for sequences related to sigma(n)

Cf. A000203, A006666, A088580, A332452, A332455.

Cf. also A324294.

A006666(n) = { my(t=0); while(n>1, if(n%2, n=3*n+1, n>>=1; t++)); (t); }; \\ From A006666
A332454(n) = A006666(1+sigma(n));

a(n) = A006666(A088580(n)) = A006666(1+sigma(n)).

A378998 Number of trailing 1-bits in the binary representation of sigma(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 5, 0, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Dec 16 2024

nonn

Cf. A000203, A007814, A088580, A028982 (positions of terms > 0), A028983 (of 0's), A072461 (of 1's), A072462 (of terms > 1), A337195, A378999 [= a(n^2)].

IntegerExponent[DivisorSigma[1, Range[100]] + 1, 2] (* Paolo Xausa, Dec 19 2024 *)

PARI
```
A378998(n) = valuation(sigma(n)+1,2);
```

a(n) = A007814(A088580(n)). [the 2-adic valuation of 1+sigma(n)]

For all n in A028982, a(n) = A337195(n).

A237588 Sigma(n) - 2n + 1.

Original entry on oeis.org

0, 0, -1, 0, -3, 1, -5, 0, -4, -1, -9, 5, -11, -3, -5, 0, -15, 4, -17, 3, -9, -7, -21, 13, -18, -9, -13, 1, -27, 13, -29, 0, -17, -13, -21, 20, -35, -15, -21, 11, -39, 13, -41, -3, -11, -19, -45, 29, -40, -6, -29, -5, -51, 13, -37, 9, -33, -25, -57, 49, -59, -27, -21, 0

Offset: 1

Omar E. Pol, Feb 20 2014

sign

Also we can write Sigma(n) - (2n - 1).
a(n) = 2 - n iff n is prime.
a(n) = 1 iff n is a perfect number.
Conjecture: a(n) = 0 iff n is a power of 2.
The problem is not new. In fact, the following comments appeared on page 74 of Guy's book: "If Sigma(n) = 2*n - 1, n has been called almost perfect. Powers of 2 are almost perfect; it is not known if any other numbers are.". - Zhi-Wei Sun, Feb 23 2014

			-----------------------------------------------
.     The sum of       The positive
n    divisors of n     odd numbers        a(n)
-----------------------------------------------
1          1                1               0
2          3                3               0
3          4                5              -1
4          7                7               0
5          6                9              -3
6         12               11               1
7          8               13              -5
8         15               15               0
9         13               17              -4
10        18               19              -1
...

R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, New York, 2004.

Cf. A000079, A000203, A000396, A001065, A004125, A005408, A033879, A033880, A039653, A120444, A196020, A235796, A236104.

[1-2*n+SumOfDivisors(n): n in [1..100]]; // Vincenzo Librandi, Feb 25 2014

Table[DivisorSigma[1,n]-2n+1,{n,70}] (* Harvey P. Dale, Nov 15 2014 *)

vector(100, n, sigma(n)-2*n+1) \\ Colin Barker, Feb 21 2014

a(n) = A000203(n) - A005408(n-1) = 1 - n + A001065(n) = 1 - A033879(n) = 1 + A033880(n) = (-1)*A235796(n).

a(n) = A088580(n) - 2*n. - Omar E. Pol, Mar 23 2014

A296212 a(n) = 1 if sigma(n) + 1 is prime, 0 otherwise.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

Characteristic function of A065512, numbers n such that sigma(n) + 1 is prime.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Cf. A000203, A010051, A065512, A088580, A296092.

Cf. also A296077, A296079, A296211.

(define (A296212 n) (A010051 (+ 1 (A000203 n))))

a(n) = A010051(1+A000203(n)) = A010051(A088580(n)).

A334954 a(n) is 1 plus the number of divisors of n.

Original entry on oeis.org

2, 3, 3, 4, 3, 5, 3, 5, 4, 5, 3, 7, 3, 5, 5, 6, 3, 7, 3, 7, 5, 5, 3, 9, 4, 5, 5, 7, 3, 9, 3, 7, 5, 5, 5, 10, 3, 5, 5, 9, 3, 9, 3, 7, 7, 5, 3, 11, 4, 7, 5, 7, 3, 9, 5, 9, 5, 5, 3, 13, 3, 5, 7, 8, 5, 9, 3, 7, 5, 9, 3, 13, 3, 5, 7, 7, 5, 9, 3, 11, 6, 5, 3, 13, 5, 5, 5, 9, 3, 13, 5, 7, 5, 5, 5, 13, 3, 7, 7, 10, 3, 9, 3, 9

Offset: 1

David A. Corneth and Omar E. Pol, Aug 25 2020

a(n) is the number of times that every divisor of n occurs in the coordinates of divisors of n mentioned in A337360 (Corneth).
a(n) = 3 if and only if n is prime.
a(n) is even if and only if n is a square.
a(n) is the number of characteristic subgroups of the dihedral group D_2n. - Firdous Ahmad Mala, Dec 25 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Partial sums give A156745.

Cf. A000005, A088580, A000203, A212356, A337360.

1 + DivisorSigma[0, Range[105]] (* Michael De Vlieger, Sep 11 2020 *)

PARI
```
a(n) = numdiv(n) + 1
```

a(n) = 1 + A000005(n).
a(n) = A337360(n)/A000203(n).
a(n) = A212356(n) for n >= 3. - Ilya Gutkovskiy, Aug 27 2020

A336926 Lexicographically earliest infinite sequence such that a(i) = a(j) => A335880(1+sigma(i)) = A335880(1+sigma(j)), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 4, 1, 3, 5, 4, 6, 7, 8, 8, 1, 5, 2, 9, 10, 5, 6, 8, 9, 1, 10, 7, 11, 12, 13, 5, 1, 14, 6, 14, 9, 5, 9, 11, 10, 10, 7, 6, 15, 10, 13, 14, 16, 6, 14, 13, 11, 6, 11, 13, 11, 11, 10, 9, 11, 10, 7, 11, 1, 15, 17, 10, 18, 7, 17, 13, 14, 13, 11, 16, 19, 7, 11, 11, 13, 9, 18, 15, 17, 20, 21, 11, 11, 10, 21, 13, 11, 21, 17, 11, 21, 11, 10, 11, 20, 5

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the function f(n) = A335880(A088580(n)).
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A336694(i) = A336694(j),
  a(i) = a(j) => A336695(i) = A336695(j).

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

Cf. A000203, A088580, A329697, A331410, A332459, A335880, A336694, A336695.

Cf. also A336925.

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; };
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
Aux335880(n) = [A329697(n),A331410(n)];
v336926 = rgs_transform(vector(up_to, n, Aux335880(1+sigma(n))));
A336926(n) = v336926[n];

cp's OEIS Frontend

A336696 Sum of odd divisors of 1+sigma(n).

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A296092 Least number with the same prime signature as sigma(n)+1.

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A332455 Starting from sigma(n)+1, number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

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A152771 a(n) = sigma(n) - 2*d(n) + 1.

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A332454 Starting from sigma(n)+1, number of halving steps to reach 1 in '3x+1' problem, or -1 if this never happens.

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A378998 Number of trailing 1-bits in the binary representation of sigma(n).

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A237588 Sigma(n) - 2n + 1.

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A296212 a(n) = 1 if sigma(n) + 1 is prime, 0 otherwise.

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