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.

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A324544 a(n) = A009194(A250246(n)) = gcd(A250246(n), A324545(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 3, 10, 1, 6, 1, 4, 1, 2, 1, 4, 1, 1, 1, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 18, 1, 2, 15, 2, 1, 3, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 1, 2, 1, 12, 1, 1, 1, 1, 1, 6, 1, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 06 2019

Keywords

Comments

Fixed points are: 1, 6, 28, 120, 496, 8128, etc,
Positions where a(n) == A250246(n) are: 1, 6, 28, 120, 496, 864, 8128, 11424, 15240, ..., which is sequence A250245(A007691(n)) sorted into ascending order.

Links

Crossrefs

Cf. A007691, A009194, A250246, A252754, A324394, A324545, A324546.
Differs from A009194 for the first time at n=39. Here a(39) = 3.

Programs

  • PARI
    up_to = 65537;
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
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A009194(n) = gcd(n, sigma(n));
A324544(n) = A009194(A250246(n));

Formula

a(n) = A009194(A250246(n)) = gcd(A250246(n), A324545(n)).
a(n) = A324394(A252754(n)).

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).

A349632 Dirichlet convolution of A250469 with A346234, which is Dirichlet inverse of A003961.

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, -90, 0, -180, 0, -96, 30, -132, 0, 48, 0, 96, -60, -108, 0, -174, 0, 12, -120
Offset: 1

Views

Author

Antti Karttunen, Nov 27 2021

Keywords

Comments

Note that for n = 2..36, a(n) = -A349631(n).
Dirichlet convolution of this sequence with A003972 is A347376.

Links

Crossrefs

Cf. A003961, A250469, A346234, A349631 (Dirichlet inverse).
Cf. also A003972, A347376, A349382.
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));
A346234(n) = (moebius(n)*A003961(n));
A349632(n) = sumdiv(n,d,A250469(n/d)*A346234(d));

Formula

a(n) = Sum_{d|n} A250469(d) * A346234(n/d).
Previous Showing 21-23 of 23 results.

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