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.

A094162 Where n first appears in A093320.

Original entry on oeis.org

1, 6, 26, 78, 390, 2262, 7878, 30966, 154830, 890214, 3127566, 15637830
Offset: 1

Views

Author

Robert G. Wilson v, May 04 2004

Keywords

Crossrefs

Cf. A093320.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; a[1] = 1; a[n_] := a[n] = (Plus @@ (a[ # ] & /@ PrimePi[ PrimeFactors[n]])); b = Table[0, {12}]; Do[c = a[n]; If[ b[[c]] == 0, b[[c]] = n], {n, 11500000}]; b

Extensions

a(12) from Michael S. Branicky, Jun 24 2025

A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 2, 1, 2, 2, 2, 2, 6, 1, 2, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 11 2020

Keywords

Comments

Number of ordered prime factorizations of radical of n.
Number of permutations of the prime indices of n (counting multiplicity) avoiding the patterns (1,2,1) and (2,1,2). These are permutations with all equal parts contiguous. Depends only on sorted prime signature (A118914). - Gus Wiseman, Jun 27 2020

Examples

			From _Gus Wiseman_, Jun 27 2020 (Start)
The a(n) permutations of prime indices for n = 2, 12, 60:
  (1)  (112)  (1123)
       (211)  (1132)
              (2113)
              (2311)
              (3112)
              (3211)
(End)

Links

Crossrefs

Cf. A000142, A000961 (positions of 1's), A001221, A050363, A066504, A069513, A064372, A093320, A292586.
Dominates A335451.
Permutations of prime indices are A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
(1,2,1)-avoiding permutations of prime indices are A335449.
(2,1,2)-avoiding permutations of prime indices are A335450.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Cf. A056239, A112798, A181796, A333221, A335452, A335463, A335521.

Programs

  • Maple
    f:= n -> nops(numtheory:-factorset(n))!:
map(f, [$1..100]); # Robert Israel, Mar 12 2020
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Plus @@ (a[n/#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n/d, d] == 1 && d < n, Boole[PrimePowerQ[n/d]] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
Table[PrimeNu[n]!, {n, 1, 100}]

Formula

a(1) = 1; a(n) = Sum_{d|n, d < n, gcd(d, n/d) = 1} A069513(n/d) * a(d).
a(n) = A000142(A001221(n)).

A093321 a(1) = 1; for m >= 2, a(m) =sum_{primes p

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 0, 2, 2, 3, 3, 3, 3, 4, 5, 6, 5, 7, 5, 7, 8, 8, 7, 9, 8, 9, 7, 9, 10, 10, 8, 10, 10, 11, 11, 12, 11, 13, 15, 16, 17, 18, 16, 19, 21, 22, 20, 20, 20, 22, 24, 23, 24, 24, 24, 25, 27, 29, 31, 34, 33, 32, 32, 34, 37, 38, 38, 40, 44, 45, 49, 51, 50, 51, 50, 52, 55
Offset: 1

Views

Author

Leroy Quet, Apr 26 2004

Keywords

Crossrefs

Cf. A093320, A094163.

Programs

  • Mathematica
    PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; a[1] = 1; a[n_] := a[n] = (Plus @@ (a[ # ] & /@ Complement[ Range[ PrimePi[n]], PrimePi[ PrimeFactors[n]]])); Table[ a[n], {n, 80}] (* Robert G. Wilson v, May 04 2004 *)

Extensions

More terms from Robert G. Wilson v, May 04 2004

A341041 If n = Product (p_j^k_j) then a(n) = 1 + Product (a(pi(p_j))), where pi = A000720, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 2, 3, 4, 5, 3, 4, 3, 7, 2, 4, 3, 3, 4, 5, 5, 4, 3, 4, 4, 3, 3, 5, 7, 6, 2, 9, 4, 7, 3, 4, 3, 7, 4, 5, 5, 4, 5, 7, 4, 8, 3, 3, 4, 7, 4, 3, 3, 13, 3, 5, 5, 5, 7, 4, 6, 5, 2, 10, 9, 4, 4, 7, 7, 5, 3, 6, 4, 7, 3, 9, 7, 6, 4, 3, 5, 5, 5, 10, 4, 9, 5, 4, 7, 7, 4, 11, 8, 7, 3, 5, 3, 9, 4
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 03 2021

Keywords

Examples

			a(45) = a(3^2 * 5) = a(prime(2)^2 * prime(3)) = 1 + a(2) * a(3) = 1 + 2 * 3 = 7.

Links

Crossrefs

Cf. A000720, A093320, A156061, A214567, A328880.

Programs

  • Maple
    a:= n-> `if`(n=1, 1, 1+mul(a(numtheory[pi](i[1])), i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 03 2021
  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + Times @@ (a[PrimePi[#[[1]]]] & /@ FactorInteger[n]); Table[a[n], {n, 100}]

Formula

a(n) = a(prime(n)^k) - 1 for k > 0.
a(2*n) = a(n) for n > 1.
Showing 1-4 of 4 results.

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