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.

A353749 a(n) = phi(n) * A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 1, 4, 2, 12, 4, 30, 4, 24, 12, 70, 8, 132, 30, 48, 8, 208, 24, 306, 24, 120, 70, 418, 16, 180, 132, 144, 60, 644, 48, 870, 16, 280, 208, 360, 48, 1116, 306, 528, 48, 1480, 120, 1722, 140, 288, 418, 1978, 32, 1050, 180, 832, 264, 2444, 144, 840, 120, 1224, 644, 3074, 96, 3540, 870, 720, 32, 1584, 280, 4026, 416
Offset: 1

Views

Author

Antti Karttunen, May 07 2022

Keywords

Links

Crossrefs

Cf. A000010, A064989, A151800, A353747, A353748, A353750.

Programs

  • Mathematica
    f[p_, e_] := (p - 1)*p^(e - 1)*If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, May 07 2022 *)
  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));

Formula

Multiplicative with a(p^e) = (p - 1) * p^(e-1) * q^e, where q is the largest prime less than p, and 1 if p = 2.
a(n) = A000010(n) * A064989(n).
For n >= 0, a(4n+2) = a(2n+1).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) * Product_{p prime} ((p^3-q)/((p+1)*(p^2-q))) = 0.1118576617..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022

A353750 a(n) = phi(sigma(n)) * A064989(sigma(n)), where A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

1, 4, 2, 30, 4, 8, 4, 48, 132, 24, 8, 60, 30, 16, 16, 870, 24, 528, 24, 120, 16, 48, 16, 96, 870, 120, 48, 120, 48, 96, 16, 720, 32, 144, 32, 3960, 306, 96, 120, 288, 120, 64, 140, 240, 528, 96, 32, 1740, 1224, 3480, 96, 1050, 144, 192, 96, 192, 96, 288, 96, 480, 870, 64, 528, 14238, 240, 192, 416, 720, 64, 192, 96
Offset: 1

Views

Author

Antti Karttunen, May 07 2022

Keywords

Comments

In contrast to A353749, this is not multiplicative, except on positions given by A336547.
It seems that a(n) = A353749(n) only on n=1. This would then imply that the intersection of A006872 and A336702 = {1}.

Links

Crossrefs

Cf. A000010, A000203, A006872, A062401, A064989, A336547, A336548, A336549, A336702, A350073, A353747, A353748, A353749.
Cf. A353757, A353758 (where a(n) < A353749(n)), A353759 (where a(n) >= A353749(n)), A353760, A353790 [= a(A003961(n))].
Cf. also A353792.

Programs

  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353750(n) = { my(s=sigma(n)); (eulerphi(s)*A064989(s)); };

Formula

a(n) = A353749(A000203(n)) = A062401(n) * A350073(n).
a(n) = A353749(n) + A353757(n).

Extensions

Dubious comment deleted by Antti Karttunen, Jan 26 2023

A353748 a(n) = phi(n) - A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 3, 2, 1, 3, 2, 1, 1, 2, 7, 3, 2, 1, 5, 2, 3, 3, 6, 11, 1, 10, 7, 5, 2, 1, 15, 6, 3, 9, 8, 5, 1, 2, 13, 3, 2, 1, 13, 12, 3, 3, 14, 17, 11, 6, 13, 5, 10, 19, 19, 2, 5, 5, 10, 1, 1, 16, 31, 15, 6, 5, 19, 6, 9, 3, 20, 1, 5, 22, 19, 25, 2, 5, 29, 38, 3, 3, 14, 25, 1, 10, 33, 5, 12, 17, 25, 2, 3, 21
Offset: 1

Views

Author

Antti Karttunen, May 06 2022

Keywords

Links

Crossrefs

Cf. A000010, A064989, A140434, A151799, A353747.

Programs

  • Mathematica
    f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, May 07 2022 *)
  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353748(n) = (eulerphi(n)-A064989(n));

Formula

a(n) = A000010(n) - A064989(n).
For n >= 0, a(4n+2) = a(2n+1).
For n > 1, a(n) = A140434(n) - A353747(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/Pi^2 - (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.0832596219... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A353746 Numbers k for which phi(sigma(k)) + A064989(sigma(k)) is equal to phi(k) + A064989(k).

Original entry on oeis.org

1, 114, 618, 2428, 3868, 11706, 16012, 23946, 2031108, 2938902, 3531102, 10475862, 98250486, 205600756
Offset: 1

Views

Author

Antti Karttunen, May 06 2022

Keywords

Comments

Numbers k such that A353747(k) = A353747(sigma(k)), or equally that A062401(k) + A350073(k) = A000010(k) + A064989(k).
When A003961(x) is substituted for k, the condition becomes: numbers x such that phi(A003973(x))+A326042(x) = A003972(x)+x, i.e. A353747(A003973(x)) - x = A003972(x).
a(15) > 1442840576, if it exists.

Links

Crossrefs

Cf. A000010, A000203, A062401, A064989, A350073, A353747.
Cf. A003961, A003972, A003973, A326042.
Cf. also A348748, A348749, A351440.

Programs

  • PARI
    A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
isA353746(n) = { my(s=sigma(n)); ((eulerphi(s)+A064989(s))==(eulerphi(n)+A064989(n))); };
Showing 1-4 of 4 results.

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