A326305 -id:A326305 - cp's OEIS frontend

A377584 E.g.f.: exp(Sum_{k>=1} A326305(k) * x^k/k).

1, 1, 1, 5, 23, 167, 907, 8647, 84625, 840401, 8917289, 122748749, 1753750759, 26047588855, 401961006787, 6422475692063, 124830139084193, 2445151343123873, 48495757104590545, 1038849234759346069, 23966120552360409271, 545230613480963786951, 13288745250263697838331

Offset: 0

Vaclav Kotesovec, Nov 02 2024

nonn

Cf. A326305.

nmax = 25; CoefficientList[Series[Exp[Sum[If[OddQ[k], EulerPhi[k], EulerPhi[k] - EulerPhi[k/2]]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ sqrt(3) * exp(3*sqrt(2*n)/Pi - n) * n^(n - 1/4) / (2^(3/4) * sqrt(Pi)).

A348045 Möbius transform of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 2, 2, 2, 6, 2, 10, 2, 2, 4, 12, 4, 16, 4, 4, 4, 18, 4, 6, 2, 4, 6, 22, 6, 28, 8, 6, 4, 8, 6, 30, 2, 10, 8, 36, 8, 40, 10, 4, 4, 42, 8, 20, 14, 12, 12, 46, 14, 12, 12, 16, 6, 52, 8, 58, 2, 8, 16, 20, 14, 60, 16, 18, 16, 66, 12, 70, 6, 6, 18, 24, 14, 72, 16, 8, 4, 78, 12, 24, 2, 22, 20, 82, 20

Offset: 1

Antti Karttunen, Oct 12 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A008683, A064989, A252463, A285702 (odd bisection), A348046 (positions of 2's).

Cf. also A023022, A326305, A347115.

A064989(n) = {my(f); 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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A348045(n) = sumdiv(n,d,moebius(n/d)*A252463(d));

a(n) = Sum_{d|n} A008683(n/d) * A252463(d).

A334087 Draw the lines with equations y=kx (k=1..n) on the R^2/Z^2 square flat torus. a(n) is the number of intersection points.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 17, 25, 39, 55, 77, 99, 131, 165, 211, 257, 311, 369, 443, 517, 609, 705, 813, 921, 1051, 1185, 1339, 1493, 1665, 1843, 2049, 2255, 2491, 2735, 2999, 3263, 3551, 3845, 4175, 4505, 4859, 5221, 5623, 6025, 6469, 6923, 7401, 7879, 8403, 8935, 9509

Offset: 0

Luc Rousseau, Apr 15 2020

nonn

It appears that the second differences of this sequence yield A326305.

Luc Rousseau, Illustration of the fact that a(7) = 25

Cf. A326305.

f(i,j,c,d)=my(L=List(),x,y);x=(d-c)/(j-i);if(max(c/i,d/j)<=x&&x1)+#h(n)
for(n=0,60,print1(a(n), ", "))

cp's OEIS Frontend

A377584 E.g.f.: exp(Sum_{k>=1} A326305(k) * x^k/k).

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A348045 Möbius transform of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

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Author

Keywords

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Crossrefs

Programs

PARI

Formula

A334087 Draw the lines with equations y=kx (k=1..n) on the R^2/Z^2 square flat torus. a(n) is the number of intersection points.

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Author

Keywords

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Programs

PARI