A336691 -id:A336691 - 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)).

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)).

A337198 Number of distinct prime factors in A337194(n) = 1+A000265(sigma(n)), where A000265(k) gives the odd part of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Antti Karttunen, Aug 20 2020

nonn

The first 3 occurs at a(137).

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

Cf. A000203, A000265, A001221, A161942, A336698, A337194.

Cf. also A058062, A336691, A337199.

A000265(n) = (n>>valuation(n,2));
A337198(n) = (omega(1+A000265(sigma(n))));

a(n) = A001221(A337194(n)) = 1 + A001221(A336698(n)).

A336924 a(n) = spf(1+sigma(n)), where spf is the smallest prime factor and sigma is the sum of divisors function.

Original entry on oeis.org

2, 2, 5, 2, 7, 13, 3, 2, 2, 19, 13, 29, 3, 5, 5, 2, 19, 2, 3, 43, 3, 37, 5, 61, 2, 43, 41, 3, 31, 73, 3, 2, 7, 5, 7, 2, 3, 61, 3, 7, 43, 97, 3, 5, 79, 73, 7, 5, 2, 2, 73, 3, 5, 11, 73, 11, 3, 7, 61, 13, 3, 97, 3, 2, 5, 5, 3, 127, 97, 5, 73, 2, 3, 5, 5, 3, 97, 13, 3, 11, 2, 127, 5, 3, 109, 7, 11, 181, 7, 5, 113, 13

Offset: 1

Antti Karttunen, Aug 09 2020

nonn

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

Cf. A000203, A020639, A071189, A088580.

Cf. also A336691, A336692, A336693, A336694, A336695, A336696.

A336924(n) = (factor((1+sigma(n)))[1, 1]);

a(n) = A020639(1+A000203(n)) = A020639(A088580(n)).

A336925 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336147(1+sigma(i)) = A336147(1+sigma(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the function f(n) = A336147(A088580(n)).
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A336691(i) = A336691(j),
  a(i) = a(j) => A336924(i) = A336924(j).

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

Cf. also A336926.

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; };
A020639(n) = if(1==n, n, factor(n)[1, 1]);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
Aux336147(n) = [A020639(n),A278221(n)];
v336925 = rgs_transform(vector(up_to, n, Aux336147(1+sigma(n))));
A336925(n) = v336925[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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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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A337198 Number of distinct prime factors in A337194(n) = 1+A000265(sigma(n)), where A000265(k) gives the odd part of k.

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A336924 a(n) = spf(1+sigma(n)), where spf is the smallest prime factor and sigma is the sum of divisors function.

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A336925 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336147(1+sigma(i)) = A336147(1+sigma(j)), for all i, j >= 1.

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