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.

A380340 a(n) = phi(1 + phi(2 + phi(3 + ... phi(n)))).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
Offset: 1

Views

Author

Michel Marcus, Jan 22 2025

Keywords

Links

Crossrefs

Cf. A000010, A132857, A380341, A380342, A380354, A380414, A380415.

Programs

  • Mathematica
    PadRight[{1, 1, 2, 2}, 100, 4] (* Paolo Xausa, Jan 22 2025 *)
  • PARI
    a(n) = my(x=0); forstep(k=n, 1, -1, x = eulerphi(x+k)); x;
  • Python
    from functools import reduce
from sympy import totient
def A380340(n): return reduce(lambda x,y:totient(x)+y,range(n,-1,-1)) # Chai Wah Wu, Jan 22 2025

Formula

a(n) = 4 for n >= 5 (see Vela and Wolird). - Paolo Xausa, Jan 22 2025
G.f.: x*(2*x^4+x^2+1)/(1-x). - Alois P. Heinz, Jan 22 2025

A380341 a(n) = phi(1 + phi(4 + phi(9 + ... phi(n^2)))).

Original entry on oeis.org

1, 2, 4, 6, 6, 6, 16, 16, 22, 16, 16, 16, 16, 22, 22, 16, 22, 16, 16, 16, 16, 16, 16, 22, 16, 22, 16, 16, 16, 22, 22, 22, 22, 22, 22, 16, 16, 22, 16, 22, 16, 16, 16, 16, 16, 16, 22, 16, 16, 16, 16, 16, 22, 16, 16, 16, 16, 16, 22, 16, 16, 16, 16, 16, 16, 16, 16, 16, 22
Offset: 1

Views

Author

Michel Marcus, Jan 22 2025

Keywords

Links

Crossrefs

Cf. A000010, A380340, A380342, A380354, A380414, A380415.

Programs

  • Mathematica
    A380341[n_] := Fold[EulerPhi[#2 + #] &, 0, Range[n, 1, -1]^2]; Array[A380341, 100] (* or *)
A380341[n_] := Which[n <= 2, n, n == 3, 4, 4 <= n <= 6, 6, MemberQ[{9, 14, 15, 17, 24, 26, 30, 31, 32, 33, 34, 35, 38, 40, 47, 53, 59, 69}, n], 22, True, 16]; Array[A380341, 100] (* Paolo Xausa, Jan 22 2025 *)
  • PARI
    a(n) = my(x=0); forstep(k=n, 1, -1, x = eulerphi(x+k^2)); x;
  • Python
    from functools import reduce
from sympy import totient
def A380341(n): return reduce(lambda x,y:totient(x)+y,(m**2 for m in range(n,-1,-1))) # Chai Wah Wu, Jan 22 2025

Formula

From Paolo Xausa, Jan 22 2025: (Start)
a(n) = 1 for n = 1;
a(n) = 2 for n = 2;
a(n) = 4 for n = 3;
a(n) = 6 for n = 4, 5, 6;
a(n) = 22 for n = 9, 14, 15, 17, 24, 26, 30, 31, 32, 33, 34, 35, 38, 40, 47, 53, 59, 69;
a(n) = 16 otherwise (see Vela and Wolird). (End)
G.f.: x*(6*x^69 - 6*x^68 + 6*x^59 - 6*x^58 + 6*x^53 - 6*x^52 + 6*x^47 - 6*x^46 + 6*x^40 - 6*x^39 + 6*x^38 - 6*x^37 + 6*x^35 - 6*x^29 + 6*x^26 - 6*x^25 + 6*x^24 - 6*x^23 + 6*x^17 - 6*x^16 + 6*x^15 - 6*x^13 + 6*x^9 - 6*x^8 - 10*x^6 - 2*x^3 - 2*x^2 - x - 1)/(x - 1). - Chai Wah Wu, Jan 22 2025

A380342 a(n) = phi(1 + phi(8 + phi(27 + ... phi(n^3)))).

Original entry on oeis.org

1, 4, 12, 12, 36, 40, 36, 72, 156, 112, 48, 112, 110, 116, 72, 72, 36, 88, 72, 88, 116, 88, 88, 36, 48, 72, 88, 96, 88, 88, 88, 116, 116, 72, 36, 88, 116, 116, 36, 88, 116, 112, 48, 72, 112, 116, 116, 116, 116, 116, 88, 88, 72, 88, 116, 116, 88, 72, 88, 88, 88, 36, 116
Offset: 1

Views

Author

Michel Marcus, Jan 22 2025

Keywords

Comments

a(n) is either 1, 4, 12, 36, 40, 48, 72, 88, 96, 110, 112, 116, or 156 (see Vela and Wolird). - Chai Wah Wu, Jan 22 2025

Links

Crossrefs

Cf. A000010, A380340, A380341, A380354, A380414, A380415.

Programs

  • Mathematica
    A380342[n_] := Fold[EulerPhi[#2 + #] &, 0, Range[n, 1, -1]^3];
Array[A380342, 100] (* Paolo Xausa, Jan 22 2025 *)
  • PARI
    a(n) = my(x=0); forstep(k=n, 1, -1, x = eulerphi(x+k^3)); x;
  • Python
    from functools import reduce
from sympy import totient
def A380342(n): return reduce(lambda x,y:totient(x)+y,(m**3 for m in range(n,-1,-1))) # Chai Wah Wu, Jan 22 2025

A380415 a(n) = phi(1 + phi(3 + phi(5 + ... + phi(2*n-1)))), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 6, 12, 18, 22, 42, 42, 72, 20, 48, 18, 12, 108, 20, 42, 20, 42, 20, 36, 42, 42, 36, 36, 36, 42, 20, 42, 42, 36, 36, 42, 20, 48, 48, 18, 36, 36, 36, 36, 48, 48, 20, 48, 48, 36, 20, 48, 96, 20, 96, 36, 20, 20, 42, 36, 36, 20, 36, 36, 36, 20, 20, 36, 36, 20
Offset: 1

Views

Author

Paolo Xausa, Jan 24 2025

Keywords

Comments

Inspired by A380340, A380341 and A380342.
Conjecture 1: a(n) can be only 1, 2, 6, 12, 18, 20, 22, 36, 42, 48, 72, 96 or 108.
Conjecture 2: for n >= 320, a(n) = 20.

Links

Crossrefs

Cf. A000010, A005408, A380340, A380341, A380342, A380414.

Programs

  • Mathematica
    A380415[n_] := Fold[EulerPhi[#2 + #] &, 0, Range[2*n - 1, 1, -2]];
Array[A380415, 100]
  • PARI
    a(n) = my(x=0); forstep(k=n, 1, -1, x = eulerphi(2*k-1+x)); x; \\ Michel Marcus, Jan 24 2025
  • Python
    from functools import reduce
from sympy import totient
def A380415(n): return totient(reduce(lambda x,y:totient(x)+y,range((n<<1)-1,0,-2))) # Chai Wah Wu, Jan 25 2025

A380354 a(n) = phi(2 + phi(3 + phi(5 + ... + phi(prime(n))))), where phi is Euler totient function (A000010).

Original entry on oeis.org

1, 2, 4, 6, 4, 8, 8, 12, 12, 16, 20, 20, 18, 40, 40, 16, 18, 18, 16, 72, 40, 16, 40, 18, 96, 96, 18, 64, 20, 40, 20, 48, 42, 40, 42, 20, 20, 40, 40, 20, 18, 20, 64, 64, 20, 40, 40, 40, 40, 20, 40, 20, 18, 64, 64, 40, 40, 20, 40, 20, 40, 64, 20, 40, 40, 20, 20, 64, 64, 64
Offset: 1

Views

Author

Paolo Xausa, Jan 22 2025

Keywords

Comments

Inspired by A380340, A380341 and A380342.
Conjecture 1: a(n) can be only 1, 2, 4, 6, 8, 12, 16, 20, 18, 40, 72, 96, 64, 48 or 42.
Conjecture 2: for n >= 187, a(n) can be only 20 or 64.

Links

Crossrefs

Cf. A000010, A380340, A380341, A380342, A380414, A380415.

Programs

  • Mathematica
    A380354[n_] := Fold[EulerPhi[#2 + #] &, 0, Prime[Range[n, 1, -1]]];
Array[A380354, 100]
  • PARI
    a(n) = my(x=0); forstep(k=n, 1, -1, x = eulerphi(prime(k)+x)); x; \\ Michel Marcus, Jan 22 2025
  • Python
    from functools import reduce
from sympy import totient, primerange
def A380354(n): return totient(reduce(lambda x,y:totient(x)+y,tuple(reversed(tuple(primerange(prime(n)+1)))))) # Chai Wah Wu, Jan 23 2025
Showing 1-5 of 5 results.

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