A381212 -id:A381212 - cp's OEIS frontend

A381205 a(n) is the cardinality of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

0, 2, 2, 1, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 1, 3, 2, 4, 2, 2, 3, 3, 3, 2, 2, 3, 3, 4, 2, 4, 2, 3, 4, 3, 2, 4, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 2, 4, 2, 3, 4, 2, 3, 4, 2, 3, 3, 4, 2, 2, 2, 3, 4, 3, 3, 4, 2, 4, 2, 3, 2, 4, 3, 3, 3, 4, 2, 4

Offset: 1

Paolo Xausa, Feb 17 2025

The prime factorization of 1 is the empty set, so a(1) = 0 by convention.

			a(16) = 2 because 12 = 2^3, the set of these bases and exponents is {2, 3} and its size is 2.
a(31500) = 5 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and its size is 5.

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

Cf. A051674 (positions of ones), A381201, A381202, A381203, A381204, A381212.

a:= n-> nops({map(i-> i[], ifactors(n)[2])[]}):
seq(a(n), n=1..90);  # Alois P. Heinz, Feb 18 2025

A381205[n_] := If[n == 1, 0, Length[Union[Flatten[FactorInteger[n]]]]];
Array[A381205, 100]

a(n) = my(f=factor(n)); #setunion(Set(f[,1]), Set(f[,2])); \\ Michel Marcus, Feb 18 2025

from sympy import factorint
def a(n): return len(set().union(*(set(pe) for pe in factorint(n).items())))
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Feb 18 2025

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

Paolo Xausa, Feb 22 2025

			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}.

Paolo Xausa, Table of n, a(n) for n = 2..10326 (rows 2..3000 of triangle, flattened).

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.

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

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]

A381214 a(n) is the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

1, 2, 0, 4, 2, 6, 1, 1, 4, 10, 2, 12, 6, 4, 2, 16, 2, 18, 4, 6, 10, 22, 2, 3, 12, 0, 6, 28, 4, 30, 3, 10, 16, 6, 1, 36, 18, 12, 4, 40, 6, 42, 10, 4, 22, 46, 3, 5, 4, 16, 12, 52, 2, 10, 6, 18, 28, 58, 4, 60, 30, 6, 4, 12, 10, 66, 16, 22, 6, 70, 1, 72, 36, 4, 18

Offset: 2

Paolo Xausa, Feb 19 2025

			a(36) = 1 because 36 = 2^2*3^2, the set of these bases and exponents is {2, 3} and 3 - 2 = 1.
a(31500) = 6 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and 7 - 1 = 6.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A046665, A081812, A381212.

Cf. A051674 (positions of zeros), A381215 (positions of ones).

A381214[n_] := Max[#] - Min[#] & [Flatten[FactorInteger[n]]];
Array[A381214, 100, 2]

a(n) = my(f=factor(n), s=setunion(Set(f[,1]), Set(f[,2]))); vecmax(s) - vecmin(s); \\ Michel Marcus, Feb 20 2025

a(n) = A081812(n) - A381212(n).

A381215 Numbers k such that the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of k is 1.

Original entry on oeis.org

2, 8, 9, 36, 72, 81, 108, 216, 625, 15625, 117649, 5764801, 25937424601, 3138428376721, 23298085122481, 3937376385699289, 48661191875666868481, 14063084452067724991009, 104127350297911241532841, 37589973457545958193355601, 907846434775996175406740561329

Offset: 1

Paolo Xausa, Feb 19 2025

			72 is a term because 72 = 2^3*3^2, the set of these bases and exponents is {2, 3} and 3 - 2 = 1.

Paolo Xausa, Table of n, a(n) for n = 1..155

Positions of ones in A381214.

Cf. A036878, A081812, A104126, A381212, A381317.

Join[{2, 8, 9, 36, 72, 81, 108, 216}, Flatten[Map[#^{# - 1, # + 1} &, Prime[Range[3, 10]]]]]

For n >= 9, a(n) = A381317(n-4).

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A381205 a(n) is the cardinality of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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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.

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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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A381214 a(n) is the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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A381215 Numbers k such that the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of k is 1.

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