A332459 -id:A332459 - cp's OEIS frontend

A336694 a(n) = A329697(1+sigma(n)), where A329697 is totally additive with a(2) = 0 and a(p) = 1 + a(p-1) for odd primes.

0, 0, 1, 0, 2, 2, 2, 0, 2, 3, 2, 3, 2, 2, 2, 0, 3, 1, 3, 4, 3, 3, 2, 3, 0, 4, 2, 4, 3, 3, 3, 0, 4, 3, 4, 3, 3, 3, 4, 4, 4, 2, 3, 2, 4, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 3, 4, 4, 2, 4, 0, 2, 4, 4, 5, 2, 4, 3, 4, 3, 4, 3, 5, 2, 4, 4, 3, 3, 5, 2, 4, 4, 5, 4, 4, 4, 5, 3, 4, 5, 4, 4, 5, 4, 4, 4, 4, 3, 5, 4, 5, 2

Offset: 1

Antti Karttunen, Jul 31 2020

nonn

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

Cf. A000203, A088580, A324294, A329697, A332459, A336691, A336692, A336693, A336695, A336696.

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)))); };
A336694(n) = A329697(1+sigma(n));

a(n) = A329697(1+A000203(n)) = A329697(A088580(n)) = A329697(A332459(n)).

A336691 Number of distinct prime factors of 1+sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 3, 1, 1

Offset: 1

Antti Karttunen, Jul 31 2020

nonn

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

Cf. A000203, A001221, A088580, A058062, A332459, A336692, A336693, A336694, A336695.

PARI
```
A336691(n) = omega(1+sigma(n));
```

a(n) = A001221(1+A000203(n)) = A001221(A088580(n)).

A336692 Binary weight of 1+sigma(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 31 2020

nonn

Cf. A000120, A000203, A088580, A175548, A324294, A332459, A336691, A336693, A336694, A336695, A336696.

PARI
```
A336692(n) = hammingweight(1+sigma(n));
```

a(n) = A000120(1+A000203(n)) = A000120(A088580(n)) = A000120(A332459(n)).

A336695 a(n) = A331410(1+sigma(n)), where A331410 is totally additive with a(2) = 0 and a(p) = 1 + a(p+1) for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 31 2020

nonn

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

Cf. A000203, A088580, A324294, A331410, A332459, A336691, A336692, A336693, A336694, A336696.

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)))); };
A336695(n) = A331410(1+sigma(n));

a(n) = A331410(1+A000203(n)) = A331410(A088580(n)) = A331410(A332459(n)).

A336693 Period of binary representation of 1/(1+sigma(n)).

Original entry on oeis.org

1, 1, 4, 1, 3, 12, 6, 1, 3, 18, 12, 28, 4, 20, 20, 1, 18, 4, 6, 14, 10, 36, 20, 60, 1, 14, 20, 18, 5, 9, 10, 1, 21, 20, 21, 11, 12, 60, 18, 12, 14, 48, 12, 8, 39, 9, 21, 100, 28, 23, 9, 30, 20, 110, 9, 110, 54, 12, 60, 156, 6, 48, 12, 1, 8, 28, 22, 7, 48, 28, 9, 21, 20, 44, 100, 46, 48, 156, 54, 40, 60, 7, 8, 60

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, A007733, A088580, A324294, A332459, A336691, A336692, A336694, A336695.

A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
A336693(n) = A007733(1+sigma(n));

a(n) = A007733(1+A000203(n)) = A007733(A088580(n)) = A007733(A332459(n)).

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

Original entry on oeis.org

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)).

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]

A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

Original entry on oeis.org

1, 2, 4, 8, 16, 25, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472

Offset: 1

Jaroslav Krizek, Dec 25 2011

nonn

Original definition, now conjectural, is "Positive integers m in increasing order determined by these rules: a(1) = 1, for n>=1, if m is in the sequence then also are numbers h such that sigma(h) = 4m-1". This is certainly equal to the new definition if 25 is the only term that is not a power of 2, as there is no x such that sigma(x) = 99 = 4*25-1. - Antti Karttunen, Dec 13 2024
If 31 is only number h of form 2^k-1 for any k>=1 such that sigma(x) = h has solution for more than one value of x then a(n) is union number 25 with A000079 (powers of 2).
Numbers k such that A000203(k) is in A000225. If Goormaghtigh conjecture is valid, then it is certain that 25 is the only odd prime power (after 1) in this sequence. - Antti Karttunen, Dec 13 2024

			These examples relate to the original definition:
m=1, 4m-1=3, sigma(h)=3 for h=2; number 2 is in sequence.
m=2, 4m-1=7, sigma(h)=7 for h=4; number 4 is in sequence.
m=4, 4m-1=15, sigma(h)=15 for h=8; number 8 is in sequence.
m=8, 4m-1=31, sigma(h)=31 for h=16 and 25; numbers 16 and 25 are in sequence.

Wikipedia, Goormaghtigh conjecture

Cf. A000079 (subsequence), A000203, A000225, A002191, A292369 (conjectured subsequence).
Subsequence of A028982. Conjectured intersection of A028982 and A378983.
Positions of 0's in A336694, A336695.
Positions of 1's in A324294, A332459, A336692, A336693, A336696.
Cf. also A202273.

is_A202274(n) = ((x->!bitand(x,x+1))(sigma(n)));
for(n=1,2^20,if(is_A202274(n^2), print1(n^2,", ")); if(n>1 && is_A202274(2*((n-1)^2)), print1(2*((n-1)^2),", "))); \\ Remember to sort! - Antti Karttunen, Dec 13 2024

Data section corrected (terms 1024, 2048 were duplicated), more terms added, and the name replaced with a new definition, with the original definition moved to the comments - Antti Karttunen, Dec 13 2024

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];

A332230 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(2^k) = 0 for k >= 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 8, 9, 10, 11, 12, 13, 13, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 4, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 41, 46, 47, 48, 41, 49, 50, 51, 52, 53, 54, 55, 56, 4, 57, 58, 59, 60, 55, 58, 61, 62, 63, 64, 65, 66, 55, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 76

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  a(i) = a(j) => A048250(i) = A048250(j),
  a(i) = a(j) => A332455(i) = A332455(j),
  a(i) = a(j) => A332459(i) = A332459(j).

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

Cf. A003557, A046523, A048250, A295300, A332455, A332459.

Cf. also A326199.

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; };
A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A209229(n) = (n && !bitand(n,n-1));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux332230(n) = if((n>2)&&A209229(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v332230 = rgs_transform(vector(up_to,n,Aux332230(n)));
A332230(n) = v332230[n];

cp's OEIS Frontend

A336694 a(n) = A329697(1+sigma(n)), where A329697 is totally additive with a(2) = 0 and a(p) = 1 + a(p-1) for odd primes.

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A336691 Number of distinct prime factors of 1+sigma(n).

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A336692 Binary weight of 1+sigma(n).

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A336695 a(n) = A331410(1+sigma(n)), where A331410 is totally additive with a(2) = 0 and a(p) = 1 + a(p+1) for odd primes.

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A336693 Period of binary representation of 1/(1+sigma(n)).

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

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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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A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

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

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