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 1-3 of 3 results.

A345053 Positions of zeros in A345055, which is the Dirichlet inverse of A011772.

Original entry on oeis.org

8, 16, 32, 64, 98, 128, 147, 256, 512, 1024, 1552, 2048, 2597, 2752, 3088, 4064, 4096, 4112, 5648, 6112, 6176, 7184, 7399, 8128, 8192, 8224, 9232, 9344, 10256, 10768, 12256, 12304, 14368, 14864, 16384, 16448, 17003, 18448, 18464, 18688, 19472, 19984, 20512, 20992, 22544, 24512, 24608, 25616, 27152, 30224, 31409, 32272, 32768
Offset: 1

Views

Author

Antti Karttunen, Jul 01 2021

Keywords

Links

Crossrefs

Cf. A011772, A345055.

Programs

Formula

From Chai Wah Wu, Jul 05 2021: (Start)
Theorem: 2^i for i >= 3 are terms.
Proof: This can be shown by induction on i. For the inductive step, A345055(1)=1, A345055(2)=-3, A345055(3)=2, and A011772(2^i)=2^(i+1)-1.
So for the divisors 1,2,4 for 2^i, A011772(2^i)*A345055(1)+A011772(2^(i-1))*A345055(2)+A011772(2^(i-2))*A345055(4)=0.
A345055(d)=0 for the other proper divisors d of 2^i by the inductive hypothesis as d is a power of 2 and this implies A345033(2^i)=0 for i>=3.
(End)
Conjecture: all terms are of the form 2^i, 2^i*p, 2^i*p*q or 7^2*p for some primes p and q. - Chai Wah Wu, Jul 05 2021

A354875 Dirichlet inverse of A344005, the smallest positive m such that n divides the oblong number m*(m+1).

Original entry on oeis.org

1, -1, -2, -2, -4, 2, -6, -2, -4, 4, -10, 7, -12, 6, 11, 0, -16, 4, -18, 16, 18, 10, -22, 4, -8, 12, -2, 23, -28, -11, -30, 4, 29, 16, 34, 12, -36, 18, 36, 5, -40, -18, -42, 39, 27, 22, -46, -6, -12, 8, 47, 48, -52, 2, 70, 25, 54, 28, -58, -78, -60, 30, 21, 8, 71, -29, -66, 64, 65, -34, -70, 24, -72, 36, 16, 71, 99
Offset: 1

Views

Author

Antti Karttunen, Jun 12 2022

Keywords

Links

Crossrefs

Cf. A002378, A344005, A354876, A354877 (positions of 0's).
Cf. also A345055.

Programs

  • Mathematica
    f[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*f[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)
  • PARI
    A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
memoA354875 = Map();
A354875(n) = if(1==n,1,my(v); if(mapisdefined(memoA354875,n,&v), v, v = -sumdiv(n,d,if(dA344005(n/d)*A354875(d),0)); mapput(memoA354875,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA344005(n/d) * a(d).
a(n) = A354876(n) - A344005(n).

A345065 Sum of A011772 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 12, 0, 15, 4, 24, 0, -8, 0, 36, 16, 31, 0, 24, 0, -28, 24, 60, 0, 33, 16, 72, 24, -36, 0, -74, 0, 63, 40, 96, 48, 9, 0, 108, 48, 86, 0, -116, 0, -64, 36, 132, 0, 14, 36, 32, 64, -84, 0, 8, 80, 23, 72, 168, 0, 341, 0, 180, 48, 127, 96, -190, 0, -112, 88, -244, 0, -77, 0, 216, 40, -120, 120, -240, 0, -73, 88
Offset: 1

Views

Author

Antti Karttunen, Jun 20 2021

Keywords

Links

Crossrefs

Cf. A011772, A345053, A345055.

Programs

  • PARI
    up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
v345055 = DirInverseCorrect(vector(up_to,n,A011772(n)));
A345055(n) = v345055[n];
A345065(n) = (A011772(n)+A345055(n));

Formula

a(n) = A011772(n) + A345055(n).
a(2^i) = 2^(i+1)-1 for i >= 3. See A345053. - Chai Wah Wu, Jul 05 2021
Showing 1-3 of 3 results.

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