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.

A340085 a(n) = A336466(n) / gcd(n-1, A336466(n)); Odd part of A340082(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 9, 3, 1, 1, 3, 1, 5, 1, 11, 1, 1, 3, 1, 1, 1, 1, 1, 5, 3, 9, 7, 1, 1, 1, 15, 3, 1, 3, 1, 1, 1, 11, 1, 1, 1, 1, 9, 1, 3, 15, 3, 1, 1, 1, 5, 1, 3, 1, 21, 7, 5, 1, 1, 1, 11, 15, 23, 9, 1, 1, 9, 5, 1, 1, 1, 1, 3, 3
Offset: 1

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Links

Crossrefs

Cf. A000265, A003958, A003961, A019565, A336466, A339901, A340082, A340084, A340086.
Cf. also A160595.

Programs

  • Mathematica
    Array[#2/GCD[#1 - 1, #2] & @@ {#, Times @@ Map[If[# <= 2, 1, (# - 1)/2^IntegerExponent[# - 1, 2]] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]]} &, 105] (* Michael De Vlieger, Dec 29 2020 *)
  • PARI
    A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
A340085(n) = { my(u=A336466(n)); u/gcd(n-1, u); };

Formula

a(n) = A000265(A340082(n)).
a(n) = A336466(n) / A340084(n) = A336466(n) / gcd(n-1, A336466(n)).
For all n >= 0, a(A003961(A019565(n))) = a(A019565(2*n)) = A339901(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

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Comments

Prime shifted analog of A160595.

Links

Crossrefs

Cf. A000010, A003961, A003972, A160595, A253885, A340071, A340073, A340075 (gives the odd part).
Cf. also A340082.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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); };

Formula

a(n) = A160595(A003961(n)).
a(n) = A003972(n) / A340071(n).

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

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 8, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 12, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 16, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4, 1, 2, 1, 96, 1, 2, 1, 100
Offset: 1

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Links

Crossrefs

Cf. A003958, A340082, A340083.
Cf. also A049559, A340071.

Programs

  • PARI
    A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A340081(n) = gcd(n-1, A003958(n));

Formula

a(n) = gcd(n-1, A003958(n)).
a(n) = A003958(n) / A340082(n).
For n > 1, a(n) = (n-1) / A340083(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

Views

Author

Antti Karttunen, Dec 28 2020

Keywords

Links

Crossrefs

Cf. A003958, A340081, A340082.
Cf. also A160596, A340073.

Programs

  • PARI
    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)));

Formula

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

A340093 Composite numbers k such that A003958(k) divides k-1.

Original entry on oeis.org

4, 8, 9, 16, 32, 64, 81, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 180225, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648
Offset: 1

Views

Author

Antti Karttunen, Dec 31 2020

Keywords

Comments

Composite numbers k for which A340082(k) = 1.
Are there any other non-powers of 2 apart from 9, 81, 180225 (= 3^4 * 5^2 * 89) present?
If there are no squarefree numbers in this sequence, then Lehmer's Totient problem has no composite solutions.

Links

Crossrefs

Cf. A003958, A005117, A340082.
Cf. A000079 (subsequence from its term a(2)=4 onward).
Cf. also A160595.

Programs

  • Mathematica
    f[n_] := Times @@ (((fct = FactorInteger[n])[[;; , 1]] - 1)^fct[[;; , 2]]); Select[Range[10^7], CompositeQ[#] && Divisible[# - 1, f[#]] &] (* Amiram Eldar, Dec 31 2020 *)
  • PARI
    A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
isA340093(n) = ((n>1)&&!isprime(n)&&!((n-1)%A003958(n)));

Extensions

More terms from Amiram Eldar, Dec 31 2020
Showing 1-5 of 5 results.

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