A324189 -id:A324189 - cp's OEIS frontend

A324184 a(n) = sigma(A163511(n)).

1, 3, 7, 4, 15, 13, 12, 6, 31, 40, 39, 31, 28, 24, 18, 8, 63, 121, 120, 156, 91, 124, 93, 57, 60, 78, 72, 48, 42, 32, 24, 12, 127, 364, 363, 781, 280, 624, 468, 400, 195, 403, 372, 342, 217, 228, 171, 133, 124, 240, 234, 248, 168, 192, 144, 96, 90, 104, 96, 72, 56, 48, 36, 14, 255, 1093, 1092, 3906, 847, 3124, 2343, 2801, 600

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. A000203, A054429, A163511, A324054, A324183, A324185, A324186, A324187, A324188, A324189.

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)));

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(p+1)); n >>= 1); (t*p));
A324184(n) = sigma(A163511(n));

from sympy import nextprime
def A324184(n):
    if n:
        c, p = 1, 1
        while n:
            c *= ((p:=nextprime(p))**(s:=(~n&n-1).bit_length()+1)-1)//(p-1)
            n >>= s
        return c*(p**(s+1)-1)//(p**s-1)
    return 1 # Chai Wah Wu, Jul 25 2023

a(n) = A000203(A163511(n)).

For n >= 1, a(n) = A324054(A054429(n)).

A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 0, 4, 1, 14, -3, 19, -4, 6, 2, 6, 1, 41, -12, 94, -19, 26, 7, 41, -12, 12, -12, 22, -2, 10, 4, 10, 1, 122, -39, 469, -64, 126, 32, 286, -51, 47, -72, 148, -17, 66, 25, 109, -28, 30, -54, 102, -48, 18, -4, 58, -10, 22, -12, 38, 0, 18, 8, 12, 1, 365, -120, 2344, -199, 626, 157, 2001, -168, 222, -372, 1030, -92, 458, 172, 1198

Offset: 0

Antti Karttunen, Feb 17 2019

If there are no odd perfect numbers, then all n for which a(n) is 0 are given by sequence A324200.

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

Cf. A000203, A033879, A054429, A163511, A324055, A324182 (compare the scatter plot), A324184, A324187, A324188, A324189, A324199, A324200.

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));
A033879(n) = (2*n-sigma(n));
A324185(n) = A033879(A163511(n));

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)));
A324185(n) = (2*A163511(n)) - A324184(n);

a(n) = A033879(A163511(n)) = 2*A163511(n) - A324184(n) = 2*A163511(n) - A000203(A163511(n)).

For n > 0, a(n) = A324055(A054429(n)).

A324187 a(n) = A106315(A163511(n)).

Original entry on oeis.org

0, 1, 5, 2, 2, 1, 0, 4, 18, 28, 30, 13, 16, 12, 4, 6, 3, 42, 72, 32, 51, 78, 21, 33, 12, 36, 24, 44, 36, 20, 8, 10, 67, 2, 168, 1, 176, 504, 128, 172, 84, 10, 312, 102, 32, 198, 75, 97, 108, 120, 144, 58, 48, 72, 128, 20, 50, 66, 48, 4, 0, 36, 16, 12, 4, 731, 372, 3126, 625, 6, 785, 801, 456, 1332, 768, 1720, 540, 232, 688, 932, 145, 660

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

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

Cf. A054429, A106315, A163511, A324057, A324183, A324184, A324188, A324189.

Cf. A324199 (positions of zeros).

A106315(n) = (n*numdiv(n) % sigma(n));
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));
A324187(n) = A106315(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)));
A324187(n) = ((A163511(n)*A324183(n))%A324184(n));

a(n) = A106315(A163511(n)) = (A163511(n)*A324183(n)) mod A324184(n).

For n > 0, a(n) = A324057(A054429(n)).

A324188 a(n) = A324121(A163511(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 12, 2, 1, 4, 3, 1, 4, 12, 2, 2, 3, 1, 24, 4, 1, 2, 3, 3, 12, 6, 24, 4, 6, 4, 8, 2, 1, 2, 3, 1, 8, 24, 4, 4, 3, 1, 12, 6, 1, 6, 3, 1, 4, 120, 18, 2, 24, 24, 16, 4, 10, 2, 48, 4, 56, 12, 4, 2, 1, 1, 12, 6, 1, 2, 1, 1, 24, 12, 48, 40, 12, 8, 16, 4, 1, 20, 3, 3, 4, 36, 6, 14, 15, 3, 36, 6, 21, 2, 3, 3, 36, 6, 720, 8, 6, 4, 24, 6, 120, 12

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

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

Cf. A054429, A324058, A163511, A324121, A324183, A324184, A324187, A324189.

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));
A324121(n) = gcd(sigma(n),n*numdiv(n));
A324188(n) = A324121(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)));
A324188(n) = gcd(A324184(n), A163511(n)*A324183(n));

a(n) = A324121(A163511(n)) = gcd(A324184(n), A163511(n)*A324183(n)).

For n > 0, a(n) = A324058(A054429(n)).

A324199 Numbers n such that A324187(n) = 0.

Original entry on oeis.org

0, 6, 60, 98, 108, 928, 930, 946, 1874, 3506, 6688, 7496, 7980, 13640, 14476, 15148, 15956, 27168, 30290, 30292, 32752, 59528, 59986, 60556, 60580, 64338, 65432, 108680, 109732, 119972, 121108, 126280, 126872, 130856, 218276, 229160, 242210, 242306, 438914, 454552, 484420, 484640, 485512, 496904, 507032, 518688, 522848, 523400, 811556, 877636

Offset: 1

Antti Karttunen, Feb 17 2019

nonn

Sequence A243071(A001599(k)), k >= 1, sorted into ascending order. See also comments in A324185.

Also positions of zeros in A324189.

Antti Karttunen, Table of n, a(n) for n = 1..164 (all terms up to size 2^26)

Cf. A001599, A243071, A324185, A324187, A324189.

Cf. A324200 (subsequence).

for(n=0,2^21,if(0==A324187(n),print1(n,", "))); \\ Uses code from A324187.

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)).

cp's OEIS Frontend

A324184 a(n) = sigma(A163511(n)).

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A324185 Deficiency of n permuted by A163511: a(n) = A033879(A163511(n)) = 2*A163511(n) - sigma(A163511(n)).

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A324187 a(n) = A106315(A163511(n)).

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A324188 a(n) = A324121(A163511(n)).

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A324199 Numbers n such that A324187(n) = 0.

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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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