cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Feb 17 2019

Keywords

Links

Crossrefs

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

Programs

  • PARI
    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)));
  • PARI
    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));
  • Python
    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

Formula

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

Views

Author

Antti Karttunen, Feb 17 2019

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • PARI
    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));
  • PARI
    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);

Formula

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

Views

Author

Antti Karttunen, Feb 17 2019

Keywords

Links

Crossrefs

Cf. A054429, A106315, A163511, A324057, A324183, A324184, A324188, A324189.
Cf. A324199 (positions of zeros).

Programs

  • PARI
    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));
  • PARI
    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));

Formula

a(n) = A106315(A163511(n)) = (A163511(n)*A324183(n)) mod A324184(n).
For n > 0, a(n) = A324057(A054429(n)).

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 17 2019

Keywords

Links

Crossrefs

Cf. A163511, A324183, A324184, A324188.
Cf. A324199 (positions of zeros).

Programs

  • PARI
    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));
  • PARI
    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)));

Formula

a(n) = A324184(n) - A324188(n) = A324184(n) - gcd(A324184(n),A163511(n)*A324183(n)).
Showing 1-4 of 4 results.

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