A324122 -id:A324122 - cp's OEIS frontend

A324189 a(n) = A324122(A163511(n)).

0, 2, 6, 2, 14, 12, 0, 4, 30, 36, 36, 30, 24, 12, 16, 6, 60, 120, 96, 152, 90, 122, 90, 54, 48, 72, 48, 44, 36, 28, 16, 10, 126, 362, 360, 780, 272, 600, 464, 396, 192, 402, 360, 336, 216, 222, 168, 132, 120, 120, 216, 246, 144, 168, 128, 92, 80, 102, 48, 68, 0, 36, 32, 12, 254, 1092, 1080, 3900, 846, 3122, 2342, 2800, 576, 2016, 1824, 2360, 1080

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A163511, A324183, A324184, A324188.

Cf. A324199 (positions of zeros).

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));
A324122(n) = (sigma(n) - gcd(sigma(n),n*numdiv(n)));
A324189(n) = A324122(A163511(n));

A324183(n) = if(!n,1,n = ((3<<#binary(n\2))-n-1); my(e=0,m=1); while(n>0, if(!(n%2), m *= (1+e); e=0, e++); n >>= 1); (m*(1+e)));
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)));
A324189(n) = (A324184(n) - gcd(A324184(n), A163511(n)*A324183(n)));

a(n) = A324184(n) - A324188(n) = A324184(n) - gcd(A324184(n),A163511(n)*A324183(n)).

A324349 a(n) = A324122(A005940(1+n)), where A005940 is the Doudna sequence and A324122(n) = sigma(n) - gcd(n*d(n), sigma(n)).

Original entry on oeis.org

0, 2, 2, 6, 4, 0, 12, 14, 6, 16, 12, 24, 30, 36, 36, 30, 10, 16, 28, 36, 44, 48, 72, 48, 54, 90, 122, 90, 152, 96, 120, 60, 12, 32, 36, 0, 68, 48, 102, 80, 92, 128, 168, 144, 246, 216, 120, 120, 132, 168, 222, 216, 336, 360, 402, 192, 396, 464, 600, 272, 780, 360, 362, 126, 16, 40, 52, 72, 80, 96, 150, 112, 84, 208, 264, 112, 366, 288, 312, 184, 164, 272, 360, 0, 568

Offset: 0

Antti Karttunen, Feb 24 2019

nonn

Zeros occur in the same positions as in A324057, and can be obtained by sorting into ascending order the terms obtained with A156552(A001599(n)), n >= 1.

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

Cf. A005940, A054429, A324054, A324057, A324058, A324189

Cf. also A001599.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324122(n) = (sigma(n) - gcd(sigma(n),n*numdiv(n)));
A324349(n)  = A324122(A005940(1+n));

a(n) = A324122(A005940(1+n)).
a(n) = A324054(n) - A324058(n).
For n > 0, a(n) = A324189(A054429(n)).

A324121 a(n) = gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 2, 1, 1, 2, 2, 4, 2, 8, 12, 1, 2, 3, 2, 6, 4, 4, 2, 12, 1, 2, 4, 56, 2, 24, 2, 3, 12, 2, 4, 1, 2, 4, 4, 10, 2, 48, 2, 12, 6, 8, 2, 4, 3, 3, 12, 2, 2, 24, 4, 8, 4, 2, 2, 24, 2, 8, 2, 1, 4, 48, 2, 6, 12, 16, 2, 3, 2, 2, 2, 4, 4, 24, 2, 2, 1, 2, 2, 112, 4, 4, 12, 4, 2, 18, 28, 24, 4, 8, 20, 36, 2, 3, 6, 1, 2, 24, 2, 2, 24

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Records 1, 2, 12, 56, 112, 120, 336, 720, 992, 2016, 4368, 8640, 14880, 16256, 26208, 59520, 78624, 120960, 131040, 191520, 227584, 297600, ... occur at positions: 1, 3, 6, 28, 84, 120, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 27846, 30240, 32760, 55860, 105664, 117800, ... . Note that A001599 is not a subsequence of the latter, as at least 18620 (present in A001599) is missing.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..117800

Cf. A000005, A000203, A001599, A038040, A106315, A106316, A156552, A324045, A324046, A324047, A324058, A324122.

Table[GCD[n DivisorSigma[0,n],DivisorSigma[1,n]],{n,120}] (* Harvey P. Dale, Feb 17 2023 *)

PARI
```
A324121(n) = gcd(sigma(n),n*numdiv(n));
```

a(n) = gcd(A000203(n), A038040(n)).

a(n) = A324058(A156552(n)).

cp's OEIS Frontend

A324189 a(n) = A324122(A163511(n)).

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A324349 a(n) = A324122(A005940(1+n)), where A005940 is the Doudna sequence and A324122(n) = sigma(n) - gcd(n*d(n), sigma(n)).

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PARI

Formula

A324121 a(n) = gcd(n*d(n), sigma(n)), where d(n) = number of divisors of n (A000005) and sigma(n) = sum of divisors of n (A000203).

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