A340073 -id:A340073 - cp's OEIS frontend

A340071 a(n) = gcd(A003961(n)-1, phi(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes.

1, 2, 4, 2, 6, 2, 10, 2, 4, 4, 12, 4, 16, 4, 2, 2, 18, 2, 22, 2, 2, 2, 28, 2, 6, 2, 4, 2, 30, 8, 36, 2, 16, 4, 4, 8, 40, 4, 4, 4, 42, 4, 46, 4, 6, 2, 52, 4, 10, 2, 2, 8, 58, 2, 18, 4, 2, 4, 60, 2, 66, 2, 2, 2, 2, 2, 70, 2, 16, 10, 72, 2, 78, 2, 4, 2, 2, 2, 82, 2, 4, 4, 88, 2, 12, 4, 2, 2, 96, 4, 2, 4, 8, 2, 4, 2, 100

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A049559.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A049559, A253885, A340072, A340073.

Cf. also A340081.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340071(n) = { my(u=A003961(n)); gcd(u-1, eulerphi(u)); };

a(n) = A049559(A003961(n)).
a(n) = gcd(A253885(n-1), A003972(n)) = gcd(A003961(n)-1, A000010(A003961(n))).
a(n) = A003972(n) / A340072(n).
For n > 1, a(n) = (A003961(n)-1) / A340073(n).

A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 9, 5, 3, 1, 6, 1, 5, 12, 27, 1, 20, 1, 18, 20, 12, 1, 36, 7, 16, 25, 30, 1, 6, 1, 81, 3, 9, 15, 15, 1, 11, 16, 27, 1, 20, 1, 18, 20, 28, 1, 54, 11, 42, 36, 12, 1, 100, 4, 45, 44, 15, 1, 72, 1, 36, 100, 243, 48, 48, 1, 54, 7, 12, 1, 180, 1, 40, 42, 66, 60, 64, 1, 162, 125, 21, 1, 120, 9, 23, 60, 108

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Prime shifted analog of A160595.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000010, A003961, A003972, A160595, A253885, A340071, A340073, A340075 (gives the odd part).

Cf. also A340082.

f:= proc(n) local F,x,p,t;
  F:= ifactors(n)[2];
  x:= mul(nextprime(t[1])^t[2],t=F);
  p:= numtheory:-phi(x);
  p/igcd(x-1,p)
end proc:
map(f,[$1..100]); # Robert Israel, Dec 28 2020

a[n_] := Module[{x, p, e, phi}, x = Product[{p, e} = pe; NextPrime[p]^e, {pe, FactorInteger[n]}]; phi = EulerPhi[x]; phi/GCD[x-1, phi]];
Array[a, 100] (* Jean-François Alcover, Jan 04 2022 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A340072(n) = { my(x=A003961(n), u=eulerphi(x)); u/gcd(x-1, u); };

a(n) = A160595(A003961(n)).

a(n) = A003972(n) / A340071(n).

A340083 a(n) = (n-1) / gcd(n-1, A003958(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 2, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 1, 23, 3, 25, 13, 9, 1, 29, 1, 31, 8, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 4, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 63, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 5, 81, 1, 83, 21, 85, 43, 87, 1, 89, 5, 91, 23

Offset: 1

Antti Karttunen, Dec 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003958, A340081, A340082.

Cf. also A160596, A340073.

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340083(n) = ((n-1)/gcd(n-1, A003958(n)));

a(n) = (n-1) / A340081(n) = (n-1) / gcd(n-1, A003958(n)).

cp's OEIS Frontend

A340071 a(n) = gcd(A003961(n)-1, phi(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A340072 a(n) = phi(x) / gcd(x-1, phi(x)), where x = A003961(n), i.e., n with its prime factorization shifted one step towards larger primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A340083 a(n) = (n-1) / gcd(n-1, A003958(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula