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.

A346068 Numbers that are the product of distinct primes with prime subscripts raised to prime powers.

Original entry on oeis.org

1, 9, 25, 27, 121, 125, 225, 243, 289, 675, 961, 1089, 1125, 1331, 1681, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 6075, 6889, 7225, 7803, 8649, 11881, 11979, 15125, 15129, 16129, 24025, 24649, 25947, 27225, 28125, 29403, 29791, 30375, 31329, 32041, 33275, 34969
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

Examples

			675 = 3^3 * 5^2 = prime(prime(1))^prime(2) * prime(prime(2))^prime(1), therefore 675 is a term.

Links

Crossrefs

Intersection of A056166 and A076610.
Cf. A006450, A302590, A302596, A321874.

Programs

  • Mathematica
    Join[{1}, Select[Range[35000], AllTrue[Join[PrimePi[(t = Transpose @ FactorInteger[#])[[1]]], t[[2]]], PrimeQ] &]] (* Amiram Eldar, Jul 30 2021 *)
  • Python
    from sympy import factorint, isprime, primepi
def ok(n):
    f = factorint(n)
    if not all(isprime(e) for e in f.values()): return False
    return all(isprime(primepi(p)) for p in f)
print(list(filter(ok, range(35000)))) # Michael S. Branicky, Jul 30 2021

Formula

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + Sum_{q prime} 1/p^q) = 1.2271874... - Amiram Eldar, Jul 31 2021

A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).

Original entry on oeis.org

1, 4, 9, 8, 25, 36, 49, 32, 27, 100, 121, 72, 169, 196, 225, 128, 289, 108, 361, 200, 441, 484, 529, 288, 125, 676, 243, 392, 841, 900, 961, 2048, 1089, 1156, 1225, 216, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 675, 2116, 2209, 1152, 343, 500, 2601, 1352, 2809, 972, 3025
Offset: 1

Views

Author

Ilya Gutkovskiy, May 09 2018

Keywords

Examples

			a(12) = a(2^2*3^1) = 2^prime(2)*3^prime(1) = 2^3*3^2 = 72.

Links

Crossrefs

Cf. A000040, A001248, A002110, A003963, A008477, A030078, A048767, A050997, A056166 (records), A061742, A181819.
Cf. A064988 (apply prime to p), A321874 (apply prime to both p & e).

Programs

  • Maple
    a:= n-> mul(i[1]^ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 20 2021
  • Mathematica
    a[n_] := Times @@ (#[[1]]^Prime[#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 55}]
  • PARI
    a(n) = my(f=factor(n)); prod(k=1, #f~, f[k,1]^prime(f[k,2])); \\ Michel Marcus, May 09 2018
  • PARI
    apply( A304203(n)=factorback((n=factor(n))[,1],apply(prime,n[,2])), [1..50]) \\ M. F. Hasler, Nov 20 2018

Formula

a(prime(i)^k) = prime(i)^prime(k).
a(A000040(k)) = A001248(k).
a(A001248(k)) = A030078(k).
a(A030078(k)) = A050997(k).
a(A002110(k)) = A061742(k).
Multiplicative with a(p^e) = p^prime(e). - M. F. Hasler, Nov 20 2018
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^prime(k)) = 1.80728269690724154161... . - Amiram Eldar, Jan 20 2024

A328878 If n = Product (p_j^k_j) then a(n) = Product (prime(p_j)).

Original entry on oeis.org

1, 3, 5, 3, 11, 15, 17, 3, 5, 33, 31, 15, 41, 51, 55, 3, 59, 15, 67, 33, 85, 93, 83, 15, 11, 123, 5, 51, 109, 165, 127, 3, 155, 177, 187, 15, 157, 201, 205, 33, 179, 255, 191, 93, 55, 249, 211, 15, 17, 33, 295, 123, 241, 15, 341, 51, 335, 327, 277, 165, 283, 381, 85, 3, 451
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 29 2019

Keywords

Examples

			a(54) = 15 because 54 = 2 * 3^3 = prime(1) * prime(2)^3 and prime(prime(1)) * prime(prime(2)) = 3 * 5 = 15.

Links

Crossrefs

Cf. A006450, A007947, A064988, A156061, A181819, A321874.

Programs

  • Maple
    a:= n-> mul(ithprime(i[1]), i=ifactors(n)[2]):
seq(a(n), n=1..65);  # Alois P. Heinz, Nov 26 2024
  • Mathematica
    a[n_] := Times @@ (Prime[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]
  • PARI
    a(n)={my(f=factor(n)[,1]); prod(i=1, #f, prime(f[i]))} \\ Andrew Howroyd, Oct 29 2019

A322177 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^prime(k_j)).

Original entry on oeis.org

0, 9, 25, 27, 121, 34, 289, 243, 125, 130, 961, 52, 1681, 298, 146, 2187, 3481, 134, 4489, 148, 314, 970, 6889, 268, 1331, 1690, 3125, 316, 11881, 155, 16129, 177147, 986, 3490, 410, 152, 24649, 4498, 1706, 364, 32041, 323, 36481, 988, 246, 6898, 44521, 2212, 4913, 1340
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2018

Keywords

Examples

			a(12) = a(2^2 * 3^1) = prime(2)^prime(2) + prime(3)^prime(1) = 3^3 + 5^2 = 52.

Crossrefs

Cf. A000040, A008475, A222416, A304251, A321874.

Programs

  • Mathematica
    a[n_] := Plus @@ (Prime[#[[1]]]^Prime[#[[2]]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 50}]
  • PARI
    a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f[k,1])^prime(f[k,2]);); vecsum(f[,1]); \\ Michel Marcus, Nov 30 2018
Showing 1-4 of 4 results.

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