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 1-6 of 6 results.

A349136 Möbius transform of Kimberling's paraphrases, A003602.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 5, 0, 6, 0, 4, 0, 8, 0, 9, 0, 6, 0, 11, 0, 10, 0, 9, 0, 14, 0, 15, 0, 10, 0, 12, 0, 18, 0, 12, 0, 20, 0, 21, 0, 12, 0, 23, 0, 21, 0, 16, 0, 26, 0, 20, 0, 18, 0, 29, 0, 30, 0, 18, 0, 24, 0, 33, 0, 22, 0, 35, 0, 36, 0, 20, 0, 30, 0, 39, 0, 27, 0, 41, 0, 32, 0, 28, 0, 44, 0, 36, 0, 30, 0, 36
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2021

Keywords

Links

Crossrefs

Agrees with A055034 on odd arguments.
Cf. A000010, A003602, A008683, A349134.
Cf. A000004, A072451 (even and odd bisection).
Cf. also A347233, A349127, A349137.

Programs

  • Maple
    with(numtheory): a:=proc(n) if n=1 then 1; elif n mod 2 = 0 then 0; else phi(n)/2; fi: end proc: seq(a(n), n=1..60); # Ridouane Oudra, Jul 13 2023
  • Mathematica
    k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, MoebiusMu[#] * k[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)
  • PARI
    A349136(n) = if(1==n,1, if(n%2, eulerphi(n)/2, 0));
  • PARI
    A003602(n) = (1+(n>>valuation(n,2)))/2;
A349136(n) = sumdiv(n,d,moebius(d)*A003602(n/d));
  • Python
    from sympy import totient
def A349136(n): return totient(n)+1>>1 if n&1 else 0 # Chai Wah Wu, Nov 24 2023

Formula

a(n) = Sum_{d|n} A008683(d) * A003602(n/d).
a(1) = 1, a(n) = A000010(n)/2 for odd n > 1, a(n) = 0 for even n.
For all n >= 1, a(2*n-1) = A055034(2*n-1) = A072451(n).
a(n) = phi(n) - (1/2)*phi(2n), for n>1. - Ridouane Oudra, Jul 13 2023
Sum_{k=1..n} a(k) ~ (1/Pi^2)*n^2. - Amiram Eldar, Jul 15 2023

A285702 a(n) = A000010(A064216(n)).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 10, 2, 12, 16, 4, 18, 6, 4, 22, 28, 6, 8, 30, 10, 36, 40, 4, 42, 20, 12, 46, 12, 16, 52, 58, 8, 20, 60, 18, 66, 70, 6, 24, 72, 8, 78, 24, 22, 82, 40, 28, 32, 88, 12, 96, 100, 8, 102, 106, 30, 108, 36, 20, 48, 42, 36, 18, 112, 40, 126, 64, 8, 130, 136, 42, 60, 44, 20, 138, 148, 24, 56, 150, 46, 72, 156, 12, 162, 110, 32, 166, 24, 52, 172, 178
Offset: 1

Views

Author

Antti Karttunen, Apr 26 2017

Keywords

Links

Crossrefs

Cf. A000010, A064216, A064989, A099774, A151799, A285703.
Odd bisection of the following sequences: A347115, A348045, A349127, A349128.

Programs

  • Mathematica
    Table[EulerPhi@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 91}] (* Michael De Vlieger, Apr 26 2017 *)
  • Scheme
    (define (A285702 n) (A000010 (A064216 n)))

Formula

a(n) = A000010(A064216(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.5366875995..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A349125 Dirichlet inverse of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, -1, -2, 0, -3, 2, -5, 0, 0, 3, -7, 0, -11, 5, 6, 0, -13, 0, -17, 0, 10, 7, -19, 0, 0, 11, 0, 0, -23, -6, -29, 0, 14, 13, 15, 0, -31, 17, 22, 0, -37, -10, -41, 0, 0, 19, -43, 0, 0, 0, 26, 0, -47, 0, 21, 0, 34, 23, -53, 0, -59, 29, 0, 0, 33, -14, -61, 0, 38, -15, -67, 0, -71, 31, 0, 0, 35, -22, -73, 0, 0, 37, -79
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A008683, A064989, A151800, A349126, A349127.
Cf. also A055615, A346234, A349134.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1, If[p == 2, -1, -NextPrime[p, -1]], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)
  • PARI
    A064989(n) = { my(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); };
A349125(n) = (moebius(n)*A064989(n));
  • PARI
    A349125(n) = { my(f = factor(n)); prod(i=1, #f~, if(1
  • Python
    from sympy import prevprime, factorint, prod
def f(p, e):
    return 0 if e > 1 else -1 if p == 2 else -prevprime(p)
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 13 2021

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A064989(n/d) * a(d).
a(n) = A349126(n) - A064989(n).
Multiplicative with a(p^e) = 0 if e > 1, -1 if p = 2 and -prevprime(p) otherwise. - Sebastian Karlsson, Nov 13 2021
a(n) = A008683(n) * A064989(n). [Because A064989 is fully multiplicative. See "Properties" section in the Wikipedia article]

A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

Original entry on oeis.org

1, 2, 3, 6, 4, 6, 6, 18, 15, 8, 6, 18, 6, 12, 12, 54, 6, 30, 6, 24, 18, 12, 10, 54, 28, 12, 75, 36, 8, 24, 8, 162, 18, 12, 24, 90, 10, 12, 18, 72, 6, 36, 6, 36, 60, 20, 10, 162, 66, 56, 18, 36, 12, 150, 24, 108, 18, 16, 8, 72, 8, 16, 90, 486, 24, 36, 10, 36, 30, 48, 6, 270, 8, 20, 84, 36, 36, 36, 10, 216, 375, 12
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Multiplicative because both A003961 and A349125 are.
Convolving this with A349127 gives A003972.

Links

Crossrefs

Cf. A003961, A064989, A349125, A349382 (Dirichlet inverse), A349383 (sum with it).
Cf. also A003972, A349127, and A349355, A349356 and A349384, A349385, and A349387.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(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); };
A349125(n) = (moebius(n)*A064989(n));
A349381(n) = sumdiv(n,d,A003961(n/d)*A349125(d));

Formula

a(n) = Sum_{d|n} A003961(n/d) * A349125(d).
a(n) = A349383(n) - A349382(n).

A349128 a(n) = phi(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 13 2021

Keywords

Comments

See comments in A349127.

Links

Crossrefs

Agrees with A347115, A348045 and A349127 on odd numbers.
Cf. A285702 (odd bisection).
Cf. A000010, A064989, A151799, A349122 (inverse Möbius transform).
Cf. also A003972.

Programs

  • Mathematica
    f[p_, e_] := If[p == 2, 1, Module[{q = NextPrime[p, -1]}, (q - 1)*q^(e - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)
  • PARI
    A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i,1], 1, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1)))); };

Formula

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (q - 1)*q^(e-1), where q = prevprime(p), where prevprime is A151799.
For odd n, a(n) = A349127(n), for even n, a(n) = a(n/2).
For all n >= 1, a(n) = a(2*n) = a(A000265(n)).
For all n >= 1, a(A000040(1+n)) = A006093(n) = A000040(n)-1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (64/(3*Pi^4)) / Product_{p prime > 2} (1+1/p-q(p)/p^2-q(p)/p^3) = 0.17889586..., where q(p) = prevprime(p) = A151799(p). - Amiram Eldar, Dec 24 2022

A349122 Inverse Möbius transform of A349128, where A349128(n) = phi(A064989(n)), A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 4, 4, 6, 7, 6, 11, 10, 6, 5, 13, 8, 17, 9, 10, 14, 19, 8, 9, 22, 8, 15, 23, 12, 29, 6, 14, 26, 15, 12, 31, 34, 22, 12, 37, 20, 41, 21, 12, 38, 43, 10, 25, 18, 26, 33, 47, 16, 21, 20, 34, 46, 53, 18, 59, 58, 20, 7, 33, 28, 61, 39, 38, 30, 67, 16, 71, 62, 18, 51, 35, 44, 73, 15, 16, 74, 79, 30, 39
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2021

Keywords

Comments

Multiplicative because A349128 is.

Links

Crossrefs

Cf. A000010, A003961, A064216, A064989, A151799, A285702, A349127, A349128, A349138.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p, -1]^e; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)
  • PARI
    A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i,1], 1, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1)))); };
A349122(n) = sumdiv(n,d,A349128(d));
  • Python
    from sympy import prevprime, factorint, prod
def f(p, e):
    return e+1 if p == 2 else prevprime(p)**e
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 15 2021

Formula

a(n) = Sum_{d|n} A349128(d).
For all n >= 1, a(A003961(n)) = n, a(2*n-1) = A064216(n).
From Sebastian Karlsson, Nov 15 2021: (Start)
a(2*n-1) = A064989(2*n-1).
Multiplicative with a(2^e) = e + 1 and a(p^e) = prevprime(p)^e for odd primes p. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/9) * Product_{p prime > 2} ((p^2-p)/(p^2-prevprime(p))) = 0.2942719052..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022
Showing 1-6 of 6 results.

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