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 1-5 of 5 results.

A354365 Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Original entry on oeis.org

1, -2, -3, 0, -5, 3, -7, 0, 0, 10, -11, 0, -13, 14, 5, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -5, -31, 0, 33, 34, 7, 0, -37, 38, 39, 0, -41, -21, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -33, -67, 0, 69, -14, -71, 0, -73, 74, 0, 0, 11, -39, -79, 0, 0, 82
Offset: 1

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Comments

Because the ratio n / A064989(n) = A319626(n) / A319627(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354366 gives the denominators. See the examples.

Examples

			The ratio a(n)/A354366(n) for n = 1..22: 1, -2, -3/2, 0, -5/3, 3, -7/5, 0, 0, 10/3, -11/7, 0, -13/11, 14/5, 5/2, 0, -17/13, 0, -19/17, 0, 21/10, 22/7.

Links

Crossrefs

Cf. A013929 (positions of 0's), A055615, A319626, A319627, A354350.
Cf. A354366 (denominators).
Cf. also A349629, A354351, A354827.

Programs

  • PARI
    A064989(n) = { my(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); };
A354365(n) = numerator((moebius(n)*n)/A064989(n));

Formula

a(n) = A055615(n) / gcd(A055615(n), A064989(n)).

A354359 Dirichlet inverse of A124859.

Original entry on oeis.org

1, -2, -2, -2, -2, 4, -2, -14, -2, 4, -2, 4, -2, 4, 4, -110, -2, 4, -2, 4, 4, 4, -2, 28, -2, 4, -14, 4, -2, -8, -2, -1526, 4, 4, 4, 4, -2, 4, 4, 28, -2, -8, -2, 4, 4, 4, -2, 220, -2, 4, 4, 4, -2, 28, 4, 28, 4, 4, -2, -8, -2, 4, 4, -20858, 4, -8, -2, 4, 4, -8, -2, 28, -2, 4, 4, 4, 4, -8, -2, 220, -110, 4, -2, -8
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2022

Keywords

Comments

Multiplicative because A124859 is.

Links

Crossrefs

Cf. A002110, A124859.
Cf. also A354186, A354349, A354351, A354358.

Programs

  • PARI
    A124859(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1] = prod(j=1, f[k, 2], prime(j)); f[k, 2] = 1); factorback(f); }; \\ From A124859
memoA354359 = Map();
A354359(n) = if(1==n,1,my(v); if(mapisdefined(memoA354359,n,&v), v, v = -sumdiv(n,d,if(dA124859(n/d)*A354359(d),0)); mapput(memoA354359,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A124859(n/d) * a(d).

A354349 Dirichlet inverse of A181819, prime shadow of n.

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -2, -1, 1, 4, -2, -2, -2, 4, 4, 2, -2, -2, -2, -2, 4, 4, -2, 2, 1, 4, -1, -2, -2, -8, -2, -3, 4, 4, 4, 1, -2, 4, 4, 2, -2, -8, -2, -2, -2, 4, -2, -4, 1, -2, 4, -2, -2, 2, 4, 2, 4, 4, -2, 4, -2, 4, -2, 7, 4, -8, -2, -2, 4, -8, -2, -1, -2, 4, -2, -2, 4, -8, -2, -4, 2, 4, -2, 4, 4, 4, 4, 2, -2, 4, 4
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2022

Keywords

Comments

Multiplicative because A181819 is.

Links

Crossrefs

Cf. A181819.
Cf. also A354186, A354351, A354359.

Programs

  • PARI
    A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
memoA354349 = Map();
A354349(n) = if(1==n,1,my(v); if(mapisdefined(memoA354349,n,&v), v, v = -sumdiv(n,d,if(dA181819(n/d)*A354349(d),0)); mapput(memoA354349,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A181819(n/d) * a(d).

A354866 Dirichlet inverse of A122111.

Original entry on oeis.org

1, -2, -4, 1, -8, 10, -16, -1, 7, 20, -32, -10, -64, 40, 46, 2, -128, -27, -256, -20, 92, 80, -512, 14, 37, 160, -17, -40, -1024, -150, -2048, -3, 184, 320, 202, 53, -4096, 640, 368, 28, -8192, -300, -16384, -80, -146, 1280, -32768, -26, 175, -129, 736, -160, -65536, 85, 404, 56, 1472, 2560, -131072, 242, -262144
Offset: 1

Views

Author

Antti Karttunen, Jun 09 2022

Keywords

Links

Crossrefs

Cf. A122111, A354867, A354868 (parity), A354869 (positions of odd terms).
Cf. also A354186, A354349, A354351, A354359, A354826.

Programs

  • PARI
    A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
memoA354866 = Map();
A354866(n) = if(1==n,1,my(v); if(mapisdefined(memoA354866,n,&v), v, v = -sumdiv(n,d,if(dA122111(n/d)*A354866(d),0)); mapput(memoA354866,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA122111(n/d) * a(d).
a(n) = A354867(n) - A122111(n).

A354352 Sum of primorial inflation (A108951) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 24, 0, 8, 36, 120, 0, 24, 0, 840, 360, 16, 0, 72, 0, 120, 2520, 9240, 0, 48, 900, 120120, 216, 840, 0, 0, 0, 32, 27720, 2042040, 12600, 144, 0, 38798760, 360360, 240, 0, 0, 0, 9240, 1080, 892371480, 0, 96, 44100, 1800, 6126120, 120120, 0, 432, 138600, 1680, 116396280, 25878772920, 0, 720, 0, 802241960520
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2022

Keywords

Links

Crossrefs

Cf. A001248, A002110, A061742, A108951, A354351.

Programs

Formula

a(n) = A108951(n) + A354351(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A108951(d) * A354351(n/d).
For all n >= 1, a(A001248(n)) = A061742(n).
Showing 1-5 of 5 results.

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