A381398 -id:A381398 - cp's OEIS frontend

A381399 a(n) is the number of prime elements in the set of bases and exponents in the prime factorization of n.

0, 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, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3

Offset: 1

Paolo Xausa, Feb 22 2025

Differs from A115588 at n = 64, 81, 256, 320, 405, 448, 512... = A381400.

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A115588, A381398, A381400, A381401.

A381399[n_] := Count[Union[Flatten[FactorInteger[n]]], _?PrimeQ];
Array[A381399, 100]

A381178 Irregular triangle read by rows, where row n lists the elements of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Feb 27 2025

Terms in each row are sorted; cf. A035306, where they are given in (base, exponent) groups.

			Triangle begins:
   [2]  1, 2;
   [3]  1, 3;
   [4]  2, 2;
   [5]  1, 5;
   [6]  1, 1, 2, 3;
   [7]  1, 7;
   [8]  2, 3;
   [9]  2, 3;
  [10]  1, 1, 2, 5;
  ...
The prime factorization of 10 is 2^1*5^1 and the multiset of these bases and exponents is {1, 1, 2, 5}.
The prime factorization of 132 is 2^2*3^1*11^1 and the multiset of these bases and exponents is {1, 1, 2, 2, 3, 11}.

Paolo Xausa, Table of n, a(n) for n = 2..11293 (rows 2..2500 of triangle, flattened).

Cf. A000026 (row products), A001221 (row lengths, divided by 2), A008474 (row sums).
Cf. A081812 (right border), A381212 (first column), A381576 (second column).
Cf. A035306, A381203, A381204, A381398, A381401, A381403, A381404.

A381178row[n_] := Sort[Flatten[FactorInteger[n]]];
Array[A381178row, 30, 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

Paolo Xausa, Feb 24 2025

			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.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A106490, A349281, A381398, A381399.

A381401[n_] := Count[FactorInteger[n], _?PrimeQ, {2}];
Array[A381401, 100]

A381402 Numbers k such that the set P of bases and exponents in the prime factorization of k (including exponents = 1) contains all numbers from min(P) to max(P).

Original entry on oeis.org

2, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 108, 144, 162, 216, 240, 324, 432, 625, 648, 720, 810, 1200, 1296, 1620, 2000, 2025, 2160, 2592, 3125, 3240, 3600, 3750, 3888, 4050, 5000, 5625, 6000, 6480, 7500, 8100, 10125, 10800, 11250, 12960, 15000, 15625

Offset: 1

Paolo Xausa, Feb 24 2025

nonn

			48 is a term because 48 = 2^4*3^1, the set of these bases and exponents is {1, 2, 3, 4} and this set contains all numbers from 1 to 4.
2000 is a term because 2000 = 2^4*5^3, the set of these bases and exponents is {2, 3, 4, 5} and this set contains all numbers from 2 to 5.

Cf. A381398.

A381402Q[k_] := Range @@ MinMax[#] == # & [Union[Flatten[FactorInteger[k]]]];
Select[Range[2, 20000], A381402Q]

cp's OEIS Frontend

A381399 a(n) is the number of prime elements in the set of bases and exponents in the prime factorization of n.

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A381178 Irregular triangle read by rows, where row n lists the elements of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n.

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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.

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Mathematica

A381402 Numbers k such that the set P of bases and exponents in the prime factorization of k (including exponents = 1) contains all numbers from min(P) to max(P).

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Mathematica