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 1-4 of 4 results.

A349631 Dirichlet convolution of A003961 with A346479, which is Dirichlet inverse of A250469.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 18, 0, 24, 0, 24, 0, -24, 0, 0, 0, -24, 60, 36, 0, 48, 0, 42, -20, -42, 0, -12, 0, -42, -10, 12, 0, 72, 0, 60, 60, -48, 0, -24, 0, 42, -30, 72, 0, -84, 0, 12, -30, -78, 0, -120, 0, -72, 120, 126, 0, 180, 0, 96, -30, 132, 0, -48, 0, -96, 60, 108, 0, 174, 0, -84, 120
Offset: 1

Views

Author

Antti Karttunen, Nov 27 2021

Keywords

Comments

Note that for n = 2..36, a(n) = -A349632(n).
Dirichlet convolution of this sequence with A347376 is A003972.

Links

Crossrefs

Cf. A003961, A250469, A346479, A349632 (Dirichlet inverse).
Cf. also A003972, A347376, A349381.
Cf. also arrays A083221, A246278, A249821, A249822 and permutations A250245, A250246.

Programs

  • PARI
    up_to = 20000;
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA250469(n)));
A346479(n) = v346479[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349631(n) = sumdiv(n,d,A003961(d)*A346479(n/d));

Formula

a(n) = Sum_{d|n} A003961(d) * A346479(n/d).

A346477 Dirichlet inverse of A346476.

Original entry on oeis.org

1, -1, -1, 2, -3, 5, -3, 2, 8, 13, -9, -2, -9, 17, 11, 8, -15, -8, -15, -12, 19, 37, -17, 18, 8, 41, -4, -12, -27, -33, -25, 20, 37, 61, 25, 56, -33, 65, 35, 38, -39, -45, -39, -42, -36, 77, -41, 32, 32, -20, 53, -42, -47, 96, 35, 58, 61, 109, -57, 132, -55, 109, -48, 56, 43, -121, -63, -72, 71, -109, -69, 56
Offset: 1

Views

Author

Antti Karttunen, Jul 29 2021

Keywords

Links

Crossrefs

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346478.
Cf. also A323910, A346248, A346479.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];

Formula

a(1) = 1; and for n > 2, a(n) = -Sum_{d|n, dA346476(n/d).
a(n) = A346478(n) - A346476(n).
a(p) = A252748(p) = A346248(p) = -A346476(p) = -A062234(A000720(p)), for any prime p.

A346480 Sum of A250469 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 30, 0, 27, 25, 42, 0, 45, 0, 66, 70, 45, 0, 75, 0, 99, 110, 78, 0, 3, 49, 102, 125, 135, 0, 60, 0, 81, 130, 114, 154, -39, 0, 138, 170, 15, 0, 60, 0, 261, 175, 174, 0, 117, 121, 231, 190, 297, 0, -225, 182, 3, 230, 186, 0, -381, 0, 222, 275, 189, 238, 360, 0, 423, 290, 216, 0, 381, 0, 246, 245, 459
Offset: 1

Views

Author

Antti Karttunen, Jul 30 2021

Keywords

Links

Crossrefs

Cf. A250469, A346479.
Cf. also A346478.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA250469(n)));
A346479(n) = v346479[n];
A346480(n) = (A250469(n)+A346479(n));
  • PARI
    A346480(n) = if(1==n, 2, -sumdiv(n,d,if((1==d)||n==d,0,A250469(d)*A346479(n/d)))); \\ (Demonstrates the convolution formula).

Formula

a(n) = A250469(n) + A346479(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A250469(d) * A346479(n/d).

A347376 Möbius transform of A250469.

Original entry on oeis.org

1, 2, 4, 6, 6, 8, 10, 12, 20, 18, 12, 12, 16, 26, 24, 24, 18, 16, 22, 24, 40, 48, 28, 24, 42, 56, 40, 36, 30, 24, 36, 48, 68, 78, 60, 36, 40, 86, 74, 48, 42, 32, 46, 60, 60, 104, 52, 48, 110, 78, 102, 72, 58, 68, 72, 72, 118, 138, 60, 48, 66, 144, 80, 96, 96, 52, 70, 96, 142, 84, 72, 72, 78, 176, 108, 108, 120, 70
Offset: 1

Views

Author

Antti Karttunen, Sep 01 2021

Keywords

Comments

Question: Are all terms positive?

Links

Crossrefs

Cf. A003972, A008683, A020639, A078898, A250469, A347377.
Cf. also A346479.

Programs

  • PARI
    up_to = 10000;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
A347376(n) = sumdiv(n,d,moebius(n/d)*A250469(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A250469(d).
a(n) = A003972(n) - A347377(n).
Showing 1-4 of 4 results.

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