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 1-9 of 9 results.

A323243 a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

Original entry on oeis.org

0, 1, 3, 4, 7, 6, 15, 8, 12, 13, 31, 12, 63, 18, 18, 24, 127, 14, 255, 20, 39, 48, 511, 24, 28, 84, 24, 48, 1023, 32, 2047, 32, 54, 176, 42, 40, 4095, 258, 144, 56, 8191, 38, 16383, 68, 36, 800, 32767, 48, 60, 31, 252, 132, 65535, 30, 91, 72, 528, 1302, 131071, 44, 262143, 2736, 60, 104, 126, 96, 524287, 304, 774, 42, 1048575, 72, 2097151, 4356, 42
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2019

Keywords

Links

Crossrefs

Cf. A000203, A156552, A323244, A323247, A323248, A324118, A324543 (Möbius transform), A324396, A324823.
Cf. A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Programs

  • Mathematica
    Array[If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 75] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    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))));
A323243(n) = if(1==n, 0, sigma(A156552(n)));
  • PARI
    \\ For computing terms a(n), with n > ~4000 use Hans Havermann's factorization file https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt"); \\ First read it in as a PARI-vector.
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1))); \\ Then play sigma
\\ Antti Karttunen, Mar 15 2019
  • Python
    from sympy import divisor_sigma, primepi, factorint
def A323243(n): return divisor_sigma(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

Formula

a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).
a(n) = 2*A156552(n) - A323244(n).
a(n) = A323247(n) - A323248(n).
From Antti Karttunen, Mar 12 2019: (Start)
a(A000040(n)) = A000225(n).
a(n) = Sum_{d|n} A324543(d).
For n > 1, a(2*A246277(n)) = A324118(n).
gcd(a(n), A156552(n)) = A324396(n).
A000035(a(n)) = A324823(n).
(End)

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)).

A324186 Sum of odd divisors permuted by A163511: a(n) = A000593(A163511(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 13, 4, 6, 1, 40, 13, 31, 4, 24, 6, 8, 1, 121, 40, 156, 13, 124, 31, 57, 4, 78, 24, 48, 6, 32, 8, 12, 1, 364, 121, 781, 40, 624, 156, 400, 13, 403, 124, 342, 31, 228, 57, 133, 4, 240, 78, 248, 24, 192, 48, 96, 6, 104, 32, 72, 8, 48, 12, 14, 1, 1093, 364, 3906, 121, 3124, 781, 2801, 40, 2028, 624, 2400, 156, 1600, 400, 1464, 13, 1240, 403
Offset: 0

Views

Author

Antti Karttunen, Feb 17 2019

Keywords

Links

Crossrefs

Cf. A000593, A038712, A054429, A163511, A324056, A324184.

Programs

Formula

a(n) = A000593(A163511(n)).
For n > 0, a(n) = A324056(A054429(n)).

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

Original entry on oeis.org

1, 3, 7, 4, 31, 15, 511, 12, 13, 63, 131071, 28, 8589934591, 1023, 127, 6, 36893488147419103231, 39, 680564733841876926926749214863536422911, 124, 2047, 262143, 231584178474632390847141970017375815706539969331281128078915168015826259279871, 60, 121, 17179869183, 91, 2044
Offset: 1

Views

Author

Antti Karttunen, Feb 02 2020

Keywords

Links

Crossrefs

Cf. A000203, A048675, A225546, A331309, A331734, A331735, A331741.
Cf. A323243, A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Programs

  • Mathematica
    Array[If[# == 1, 1, DivisorSigma[1, #] &@ Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 28] (* Michael De Vlieger, Feb 08 2020 *)
  • PARI
    A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331733(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

Formula

a(n) = A000203(A225546(n)).
For all n >= 1, A000035(a(A016754(n))) = 1. [Result is odd for all odd squares]

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)).

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

Views

Author

Antti Karttunen, Feb 17 2019

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • 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)));
  • 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));
A324183(n) = numdiv(A163511(n));
  • PARI
    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));
  • Python
    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

Formula

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

A324182 a(n) = A083254(A163511(n)), where A083254(n) = 2*phi(n) - n, the Möbius transform of the deficiency of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 3, -2, 3, 0, 9, -6, 15, -4, 1, -2, 5, 0, 27, -18, 75, -12, 5, -10, 35, -8, 3, -14, 13, -4, 3, -2, 9, 0, 81, -54, 375, -36, 25, -50, 245, -24, 15, -70, 91, -20, 21, -14, 99, -16, 9, -42, 65, -28, -9, -22, 43, -8, 9, -18, 25, -4, 7, -2, 11, 0, 243, -162, 1875, -108, 125, -250, 1715, -72, 75, -350, 637, -100, 147, -98, 1089
Offset: 0

Views

Author

Antti Karttunen, Feb 18 2019

Keywords

Links

Crossrefs

Cf. A054429, A083254, A163511, A324052, A324183, A324184, A324185 (compare the scatter plot), A366804 (rgs-transform).
Cf. also A324103.

Programs

Formula

a(n) = A083254(A163511(n)).
For n > 0, a(n) = A324052(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

Views

Author

Antti Karttunen, Feb 17 2019

Keywords

Links

Crossrefs

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

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

Formula

a(n) = A324121(A163511(n)) = gcd(A324184(n), A163511(n)*A324183(n)).
For n > 0, a(n) = A324058(A054429(n)).
Showing 1-9 of 9 results.

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