A324187 -id:A324187 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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)).

A324051 a(n) = A106315(A156552(n)).

Original entry on oeis.org

0, 1, 2, 5, 4, 2, 6, 0, 1, 18, 10, 3, 16, 4, 12, 67, 12, 4, 18, 30, 36, 260, 22, 16, 8, 8, 44, 5, 20, 1029, 30, 28, 164, 36, 28, 6, 256, 96, 44, 4102, 36, 7, 66, 16, 104, 16391, 46, 12, 13, 32, 130, 8, 28, 51, 70, 480, 942, 65544, 42, 9, 2724, 32, 66, 30, 84, 262153, 124, 508, 40, 10, 4, 1048586, 3320, 20, 188, 50, 52, 11, 78, 24

Offset: 2

Antti Karttunen, Feb 19 2019

nonn

Positions of zeros, which is sequence A005940(1+A001599(n)) sorted into ascending order: 2, 9, 125, 325, 351, 4199, ..., has A324201 as its subsequence.

Antti Karttunen, Table of n, a(n) for n = 2..4473
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A005940, A106315, A156552, A158879, A323243, A323244, A324049, A324105, A324201.

Cf. also A324057, A324187.

A106315(n) = (n*numdiv(n) % sigma(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A324051(n) = A106315(A156552(n));

a(n) = A106315(A156552(n)).

a(n) = (A156552(n)*A324105(n)) mod A323243(n).

A324183 a(n) = d(A163511(n)), where d(n) is A000005, the number of divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 4, 2, 5, 4, 6, 3, 6, 4, 4, 2, 6, 5, 8, 4, 9, 6, 6, 3, 8, 6, 8, 4, 6, 4, 4, 2, 7, 6, 10, 5, 12, 8, 8, 4, 12, 9, 12, 6, 9, 6, 6, 3, 10, 8, 12, 6, 12, 8, 8, 4, 8, 6, 8, 4, 6, 4, 4, 2, 8, 7, 12, 6, 15, 10, 10, 5, 16, 12, 16, 8, 12, 8, 8, 4, 15, 12, 18, 9, 18, 12, 12, 6, 12, 9, 12, 6, 9, 6, 6, 3, 12, 10, 16, 8, 18, 12, 12, 6, 16, 12

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

For all i, j: A286531(i) = A286531(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000005, A054429, A106737, A163511, A286531, A324184, A324187.

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)));

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));
A324183(n) = numdiv(A163511(n));

A054429(n) = if(!n,n,((3<<#binary(n\2))-n-1)); \\ After code in A054429
A106737(n) = sum(k=0, n, (binomial(n+k, n-k)*binomial(n, k)) % 2);
A324183(n) = A106737(A054429(n));

def A324183(n):
    if n:
        c = 1
        while n:
            c *= (s:=(~n&n-1).bit_length()+1)
            n >>= s
        return c*(s+1)//s
    return 1 # Chai Wah Wu, Jul 25 2023

a(n) = A000005(A163511(n)).
a(n) = A106737(A054429(n)).
For all n >= 0, a(2^n) = n+2.

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)).

cp's OEIS Frontend

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

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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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A324051 a(n) = A106315(A156552(n)).

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A324183 a(n) = d(A163511(n)), where d(n) is A000005, the number of divisors of n.

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

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