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.

A347097 a(1) = 2; and for n > 1, a(n) = A341512(n) + A347096(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 21, 4, 4, 0, 110, 0, 8, 8, 259, 0, 224, 0, 154, 16, 4, 0, 1548, 4, 8, 176, 316, 0, 592, 0, 2445, 8, 4, 16, 4312, 0, 8, 16, 2450, 0, 1216, 0, 382, 640, 12, 0, 15532, 16, 408, 8, 616, 0, 6708, 8, 5064, 16, 4, 0, 12272, 0, 12, 1312, 19543, 16, 1504, 0, 754, 24, 1568, 0, 50561, 0, 8, 832, 1060, 16
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Sum of {the pointwise sum of A341512 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.
The first negative term is a(5760) = -1223227750.

Links

Crossrefs

Cf. A000203, A001223, A001248, A003961, A063524, A341512, A347096.
Cf. also A346236, A346240, A347099.

Programs

  • PARI
    up_to = 16384;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341512(n) = { my(u=A003961(n)); ((sigma(n)*u) - (n*sigma(u))); };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA341512(n));
v347096 = DirInverseCorrect(vector(up_to,n,Aux347096(n)));
A347096(n) = v347096[n];
A347097(n) = if(1==n,2,A341512(n) + A347096(n));

Formula

a(1) = 2, and for n>1, a(n) = -Sum_{d|n, 1A341512(d) * A347096(n/d).
For all n >= 1, a(A001248(n)) = A001223(n)^2.

A346239 Möbius transform of A341512, sigma(n)*A003961(n) - n*sigma(A003961(n)).

Original entry on oeis.org

0, 1, 2, 10, 2, 33, 4, 74, 44, 55, 2, 278, 4, 115, 116, 490, 2, 613, 4, 498, 242, 169, 6, 1942, 92, 265, 742, 1046, 2, 1591, 6, 3086, 344, 355, 330, 4986, 4, 487, 542, 3570, 2, 3347, 4, 1638, 2326, 737, 6, 12542, 376, 2121, 716, 2546, 6, 9869, 388, 7510, 986, 943, 2, 12894, 6, 1225, 4872, 18970, 630, 5353, 4, 3498, 1492
Offset: 1

Views

Author

Antti Karttunen, Jul 13 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346240, A347096, A347124, A347125.
Cf. also the sequences A001359, A029710, A031924 that give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.

Programs

Formula

a(n) = Sum_{d|n} A008683(n/d) * A341512(d).
a(n) = A341512(n) - A346240(n).
a(n) = A347125(n) - A347124(n). - Antti Karttunen, Aug 25 2021

A347098 a(1) = 1; a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d), where A336853(n) = A003961(n) - n.

Original entry on oeis.org

1, -1, -2, -4, -2, -5, -4, -10, -12, -7, -2, -1, -4, -11, -12, -16, -2, -1, -4, -7, -18, -13, -6, 42, -20, -17, -42, -5, -2, 21, -6, -4, -24, -19, -26, 106, -4, -23, -30, 38, -2, 45, -4, -25, -10, -29, -6, 196, -56, -17, -36, -23, -6, 123, -28, 82, -42, -31, -2, 225, -6, -37, 4, 80, -38, 15, -4, -43, -52, 39, -2, 413
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Dirichlet inverse of the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...).

Links

Crossrefs

Cf. A000040, A001223, A003961, A336853, A347099, A347100.
Cf. also A346248, A347096.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];

Formula

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A336853(n/d) * a(d).
For all n >= 1, a(A000040(n)) = -A001223(n).
Showing 1-3 of 3 results.

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