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.

A381400 Numbers k >= 2 such that A115588(k) != A381399(k).

Original entry on oeis.org

64, 81, 256, 320, 405, 448, 512, 567, 625, 704, 729, 832, 891, 1024, 1053, 1088, 1216, 1280, 1377, 1472, 1539, 1600, 1792, 1856, 1863, 1875, 1984, 2240, 2349, 2368, 2401, 2511, 2560, 2624, 2752, 2816, 2835, 2997, 3008, 3072, 3136, 3321, 3328, 3392, 3483, 3520
Offset: 1

Views

Author

Paolo Xausa, Feb 22 2025

Keywords

Links

Crossrefs

Cf. A115588, A381399.

Programs

  • Mathematica
    A115588[n_] := Length[Union[NestWhile[DeleteCases[Flatten[FactorInteger[#]], 1] &, {n}, AnyTrue[#, CompositeQ] &]]];
A381399[n_] := Count[Union[Flatten[FactorInteger[n]]], _?PrimeQ];
A381400Q[k_] := A115588[k] != A381399[k];
Select[Range[2, 5000], A381400Q]

A115588 Number of distinct prime numbers necessary to represent a natural number n > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 3, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 1, 3, 1, 3, 3, 2
Offset: 2

Views

Author

Lucas B. Vieira, Mar 09 2006

Keywords

Comments

The sequence gives the number of distinct prime numbers needed to represent a given natural number greater than or equal to 2 upon recursive factorization of powers. In order to do this, we must factor any subsequent composite number that may appear on the exponents of the next factorizations (e.g., 4 in 48=2^4*3), until only prime numbers are used. - Lucas B. Vieira, Mar 15 2006
In this sequence, a(n)=1 if n is prime, or a power tower (tetration or iterated exponentiation) of a prime base (e.g., 2^2, 3^3^3^3, 7^7). The sequence reaches a new boundary whenever n is a primorial number (factorial of primes). - Lucas B. Vieira, Mar 15 2006

Examples

			a(4)=1, since 4=2^2 and the only prime used was 2.
a(30)=3 because 30=2*3*5 and three primes were necessary.
a(65536)=1, since 65536=2^16=2^(2^4)=2^(2^(2^2)) and, again, only one prime was needed.
a(1) would be undefined, so it is not included.

Links

Crossrefs

Cf. A000040, A001221, A002110, A381399, A381400.

Programs

  • Maple
    b:= n-> `if`(n=1, {}, {seq([i[1], b(i[2])[]][], i=ifactors(n)[2])}):
a:= n-> nops(b(n)):
seq(a(n), n=1..200);  # Alois P. Heinz, Apr 27 2012
  • Mathematica
    A115588[n_] := Length[Union[NestWhile[DeleteCases[Flatten[FactorInteger[#]], 1] &, {n}, AnyTrue[#, CompositeQ] &]]];
Array[A115588, 100, 2] (* Paolo Xausa, Feb 24 2025 *)
  • PARI
    listf(f, list) = {for (k=1, #f~, listput(list, f[k,1]); if (isprime(f[k,2]), listput(list, f[k,2]), if (f[k,2] > 1, my(vexp = Vec(listf(factor(f[k,2]), list))); for (i=1, #vexp, listput(list, vexp[i]););););); list;}
a(n) = {my(f=factor(n), list=List()); #select(isprime, Set(Vec(listf(f, list))));} \\ Michel Marcus, Dec 02 2020

A381398 Irregular triangle read by rows, where row n lists the elements of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 1, 2, 3, 1, 7, 2, 3, 2, 3, 1, 2, 5, 1, 11, 1, 2, 3, 1, 13, 1, 2, 7, 1, 3, 5, 2, 4, 1, 17, 1, 2, 3, 1, 19, 1, 2, 5, 1, 3, 7, 1, 2, 11, 1, 23, 1, 2, 3, 2, 5, 1, 2, 13, 3, 1, 2, 7, 1, 29, 1, 2, 3, 5, 1, 31, 2, 5, 1, 3, 11, 1, 2, 17, 1, 5, 7, 2, 3
Offset: 2

Views

Author

Paolo Xausa, Feb 22 2025

Keywords

Examples

			Triangle begins:
   [2]  1, 2;
   [3]  1, 3;
   [4]  2;
   [5]  1, 5;
   [6]  1, 2, 3;
   [7]  1, 7;
   [8]  2, 3;
   [9]  2, 3;
  [10]  1, 2, 5;
  ...
The prime factorization of 10 is 2^1*5^1 and the set of these bases and exponents is {1, 2, 5}.

Links

Crossrefs

Cf. A381201 (row products), A381202 (row sums), A381205 (row lengths).
Cf. A381203 (row lcms), A381204 (row gcds).
Cf. A081812 (row largest elements), A381212 (row smallest elements).
Cf. A035306, A381399, A381402.

Programs

  • Mathematica
    A381398row[n_] := Union[Flatten[FactorInteger[n]]];
Array[A381398row, 50, 2]

A381401 a(n) is the number of (possibly non-distinct) prime elements in the multiset of bases and exponents in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2, 3, 1, 4
Offset: 1

Views

Author

Paolo Xausa, Feb 24 2025

Keywords

Examples

			a(144) = 3 because the prime factorization of 144 is 2^4*3^2 and the multiset of these bases and exponents is {2, 2, 3, 4}, containing 3 primes.

Links

Crossrefs

Cf. A106490, A349281, A381398, A381399.

Programs

  • Mathematica
    A381401[n_] := Count[FactorInteger[n], _?PrimeQ, {2}];
Array[A381401, 100]
Showing 1-4 of 4 results.

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