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 1-6 of 6 results.

A286357 One more than the exponent of the highest power of 2 dividing sigma(n): a(n) = A001511(A000203(n)).

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 4, 1, 1, 2, 3, 3, 2, 4, 4, 1, 2, 1, 3, 2, 6, 3, 4, 3, 1, 2, 4, 4, 2, 4, 6, 1, 5, 2, 5, 1, 2, 3, 4, 2, 2, 6, 3, 3, 2, 4, 5, 3, 1, 1, 4, 2, 2, 4, 4, 4, 5, 2, 3, 4, 2, 6, 4, 1, 3, 5, 3, 2, 6, 5, 4, 1, 2, 2, 3, 3, 6, 4, 5, 2, 1, 2, 3, 6, 3, 3, 4, 3, 2, 2, 5, 4, 8, 5, 4, 3, 2, 1, 3, 1, 2, 4, 4, 2, 7, 2, 3, 4, 2, 4, 4, 4, 2, 5, 5, 2, 2, 3, 5, 4
Offset: 1

Views

Author

Antti Karttunen, May 10 2017

Keywords

Links

Crossrefs

Cf. A000203, A000523, A001511, A053866, A070939, A082903, A161942, A286358.

Programs

  • Mathematica
    Table[IntegerExponent[DivisorSigma[1,n],2]+1,{n,120}] (* Harvey P. Dale, Sep 04 2023 *)
  • PARI
    A001511(n) = (1+valuation(n,2));
A286357(n) = A001511(sigma(n));
for(n=1, 10000, write("b286357.txt", n, " ", A286357(n)));
  • Python
    from sympy import divisor_sigma as D
def a001511(n): return bin(n)[2:][::-1].index("1") + 1
def a(n): return a001511(D(n)) # Indranil Ghosh, May 12 2017
  • Python
    from sympy import divisor_sigma
def A286357(n): return ((m:=int(divisor_sigma(n)))&-m).bit_length() # Chai Wah Wu, Jul 10 2022
  • Scheme
    (define (A286357 n) (A001511 (A000203 n)))
(define (A286357 n) (A070939 (/ (A000203 n) (A161942 n))))

Formula

a(n) = A001511(A000203(n)).
a(n) = 1 + A000523(A000203(n)/A161942(n)). [See also A082903.]
a(n) = 1 iff A053866(n) = 1.

A336699 a(n) = A000265(1+A000265(sigma(A000265(n)))), where A000265(k) gives the odd part of k, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 7, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 7, 5, 3, 1, 1, 11, 1, 3, 1, 5, 1, 1, 1, 29, 1, 5, 1, 7, 3, 5, 1, 3, 1, 1, 1, 1, 1, 7, 1, 11, 1, 9, 5, 1, 1, 5, 7, 19, 5, 1, 3, 1, 1, 3, 1, 61, 11, 11, 1, 7, 3, 1, 1, 23, 5, 1, 1, 1, 1, 1, 1, 25, 29, 5, 1, 13, 5, 7, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 02 2020

Keywords

Comments

See the "lacunae" in the scatter plot. - Antti Karttunen, Mar 27 2022

Links

Crossrefs

Cf. A000203, A000265, A000593, A002131, A082903, A161942, A171435, A322250, A336698, A336700, A336701, A336844 [= a(A003961(n))], A351565.

Programs

Formula

a(n) = A000265(1+A000265(A000593(n))) = A000265(1+A161942(A000265(n))).
a(n) = A336698(A000265(n)).
From Antti Karttunen, Mar 27 2022: (Start)
a(n) = A351565(A000593(n)).
[The following formulas were discovered by Sequence Machine]:
a(n) = A351565(A002131(n)) = A000265(1+A000265(A002131(n))).
a(n) = A336698(1+A322250(n)).
a(n) = A171435(A000593(n)+A082903(n)).
(End)

A366889 Dirichlet inverse of the highest power of two that divides the sum of divisors of n.

Original entry on oeis.org

1, -1, -4, 0, -2, 4, -8, 0, 15, 2, -4, 0, -2, 8, 8, 0, -2, -15, -4, 0, 32, 4, -8, 0, 3, 2, -64, 0, -2, -8, -32, 0, 16, 2, 16, 0, -2, 4, 8, 0, -2, -32, -4, 0, -30, 8, -16, 0, 63, -3, 8, 0, -2, 64, 8, 0, 16, 2, -4, 0, -2, 32, -120, 0, 4, -16, -4, 0, 32, -16, -8, 0, -2, 2, -12, 0, 32, -8, -16, 0, 272, 2, -4, 0, 4, 4, 8, 0
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2024

Keywords

Comments

Multiplicative because A082903 is.

Links

Crossrefs

Cf. A000203, A082903, A336937, A359548, A359549 (parity of terms).

Programs

  • PARI
    A082903(n) = (2^valuation(sigma(n), 2));
memoA366889 = Map();
A366889(n) = if(1==n,1,my(v); if(mapisdefined(memoA366889,n,&v), v, v = -sumdiv(n,d,if(dA082903(n/d)*A366889(d),0)); mapput(memoA366889,n,v); (v)));
  • Python
    from functools import lru_cache
from sympy import divisor_sigma, divisors
@lru_cache(maxsize=None)
def A366889(n): return 1 if n==1 else -sum((1<<(~(m:=int(divisor_sigma(d))) & m-1).bit_length())*A366889(n//d) for d in divisors(n,generator=True) if d>1) # Chai Wah Wu, Jan 03 2024

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA082903(n/d) * a(d).

A379473 a(n) is the highest power of 3 dividing the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 3, 3, 1, 3, 1, 9, 3, 1, 1, 3, 3, 1, 9, 3, 1, 3, 1, 9, 3, 3, 1, 3, 1, 1, 3, 9, 1, 9, 3, 27, 3, 1, 1, 3, 1, 9, 3, 3, 1, 3, 3, 9, 3, 1, 3, 3, 9, 1, 27, 3, 9, 3, 1, 9, 3, 3, 1, 3, 1, 1, 3, 9, 1, 9, 3, 9, 9, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 9, 3, 1, 27, 3, 3, 9, 9, 9, 1, 3, 1, 9, 3, 9, 1, 9, 3, 1, 3, 27, 1, 3, 3
Offset: 1

Views

Author

Antti Karttunen, Dec 27 2024

Keywords

Links

Crossrefs

Cf. A000203, A000244, A038500, A354100, A379484 [= a(A379481(n))].
Cf. A329963 (positions of 1's), A087943 (of terms > 1).
Cf. also A082903.

Programs

  • Mathematica
    a[n_] := 3^IntegerExponent[DivisorSigma[1, n], 3]; Array[a, 100] (* Amiram Eldar, Dec 27 2024 *)
  • PARI
    A379473(n) = (3^valuation(sigma(n),3));

Formula

Multiplicative with a(p^e) = A038500((p^(e+1)-1)/(p-1)).
a(n) = A038500(A000203(n)).
a(n) = A000244(A354100(n)).

A368694 Dirichlet inverse of the highest power of two that divides sigma(n), applied to A163511(n).

Original entry on oeis.org

1, -1, 0, -4, 0, 15, 4, -2, 0, -64, -15, 3, 0, 8, 2, -8, 0, 272, 64, -8, 0, -12, -3, 63, 0, -30, -8, 16, 0, 32, 8, -4, 0, -1144, -272, 20, 0, 32, 8, -512, 0, 45, 12, -126, 0, -252, -63, 15, 0, 128, 30, -24, 0, -64, -16, 32, 0, -120, -32, 8, 0, 16, 4, -2, 0, 4816, 1144, -44, 0, -80, -20, 4160, 0, -120, -32, 1024, 0
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2024

Keywords

Comments

Scatter plot: "Sailboard congestion".

Links

Crossrefs

Cf. A163511, A082903, A366889, A368695 (rgs-transform).

Programs

  • PARI
    A082903(n) = (2^valuation(sigma(n), 2));
memoA366889 = Map();
A366889(n) = if(1==n,1,my(v); if(mapisdefined(memoA366889,n,&v), v, v = -sumdiv(n,d,if(dA082903(n/d)*A366889(d),0)); mapput(memoA366889,n,v); (v)));
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));
A368694(n) = A366889(A163511(n));

Formula

a(n) = A366889(A163511(n)).

A368695 Lexicographically earliest infinite sequence such that a(i) = a(j) => A368694(i) = A368694(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 3, 8, 9, 10, 3, 11, 12, 13, 3, 14, 15, 13, 3, 16, 17, 18, 3, 19, 13, 20, 3, 21, 11, 4, 3, 22, 23, 24, 3, 21, 11, 25, 3, 26, 27, 28, 3, 29, 30, 5, 3, 31, 32, 33, 3, 8, 34, 21, 3, 35, 36, 11, 3, 20, 6, 7, 3, 37, 38, 39, 3, 40, 41, 42, 3, 35, 36, 43, 3, 44, 45, 8, 3, 46, 47, 48, 3, 49, 50, 35, 3
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2024

Keywords

Comments

Restricted growth sequence transform of A368694, where A368694 is the Dirichlet inverse of the highest power of two that divides sigma(n), applied to A163511(n).

Links

Crossrefs

Cf. A163511, A082903, A366889, A368694.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A082903(n) = (2^valuation(sigma(n), 2));
memoA366889 = Map();
A366889(n) = if(1==n,1,my(v); if(mapisdefined(memoA366889,n,&v), v, v = -sumdiv(n,d,if(dA082903(n/d)*A366889(d),0)); mapput(memoA366889,n,v); (v)));
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));
A368694(n) = A366889(A163511(n));
v368695 = rgs_transform(vector(1+up_to,n,A368694(n-1)));
A368695(n) = v368695[1+n];
Showing 1-6 of 6 results.

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