A381203 -id:A381203 - cp's OEIS frontend

A381213 Numbers k such that A381201(k) != A381203(k).

16, 48, 64, 80, 112, 144, 162, 176, 192, 208, 240, 256, 272, 304, 320, 324, 336, 368, 400, 432, 448, 464, 496, 528, 560, 576, 592, 624, 648, 656, 688, 704, 720, 729, 752, 768, 784, 810, 816, 832, 848, 880, 912, 944, 960, 976, 1008, 1024, 1040, 1072, 1088, 1104

Offset: 1

Paolo Xausa, Feb 17 2025

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

Cf. A381201, A381203.

A381213Q[n_] := Times @@ # != LCM @@ # & [Union[Flatten[FactorInteger[n]]]];
Select[Range[2000], A381213Q]

A381201 a(n) is the product of the elements of the set of bases and exponents in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 6, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 6, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 24, 14, 10, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 12, 65, 66, 67, 34, 69, 70

Offset: 1

Paolo Xausa, Feb 16 2025

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

			a(12) = 6 because 12 = 2^2*3^1, the set of these bases and exponents is {1, 2, 3} and 1*2*3 = 6.
a(31500) = 210 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and 1*2*3*5*7 = 210.

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

Cf. A000026, A336965, A381202, A381203, A381204, A381205.

A381201[n_] := Times @@ Union[Flatten[FactorInteger[n]]];
Array[A381201, 100]

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

A381204 a(n) is the gcd of the elements of the set of bases and exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Feb 17 2025

The positions of terms > 1 are given by A368107.

			a(16) = 2 because 16 = 2^4, the set of these bases and exponents is {2, 4} and gcd(2, 4) = 2.
a(19683) = 3 because 19683 = 3^9, the set of these bases and exponents is {3, 9} and gcd(3, 9) = 3.

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

Cf. A368107, A381201, A381202, A381203, A381205.

A381204[n_] := GCD @@ Flatten[FactorInteger[n]];
Array[A381204, 100 ,2]

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

A381202 a(n) is the sum of the elements of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

0, 3, 4, 2, 6, 6, 8, 5, 5, 8, 12, 6, 14, 10, 9, 6, 18, 6, 20, 8, 11, 14, 24, 6, 7, 16, 3, 10, 30, 11, 32, 7, 15, 20, 13, 5, 38, 22, 17, 11, 42, 13, 44, 14, 11, 26, 48, 10, 9, 8, 21, 16, 54, 6, 17, 13, 23, 32, 60, 11, 62, 34, 13, 8, 19, 17, 68, 20, 27, 15, 72, 5

Offset: 1

Paolo Xausa, Feb 16 2025

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

			a(12) = 6 because 12 = 2^2*3^1, the set of these bases and exponents is {1, 2, 3} and 1 + 2 + 3 = 6.
a(31500) = 18 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and 1 + 2 + 3 + 5 + 7 = 18.

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

Cf. A008474, A338038, A381201, A381203, A381204, A381205.

A381202[n_] := If[n == 1, 0, Total[Union[Flatten[FactorInteger[n]]]]];
Array[A381202, 100]

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

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

Original entry on oeis.org

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

A381212 a(n) is the smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Feb 19 2025

The corresponding largest elements are given by A081812.
The positions of terms > 1 are given by A001694.
Records of a(n) = 2, 3, 4, 5,.. appear at n=4=2^2, 27=3^3, 625=5^4, 3125=5^5, 117649=7^6, 823543=7^7 ,... (subsequence A051647).- R. J. Mathar, Mar 05 2025

			a(36) = 2 because 36 = 2^2*3^2, the set of these bases and exponents is {2, 3} and its smallest element is 2.
a(31500) = 1 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 smallest element is 1.

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

Cf. A001694, A081812, A288636, A331048, A381201, A381202, A381203, A381204, A381205, A381214.

A381212 := proc(n)
    local a,pe;
    a := n ;
    for pe in ifactors(n)[2] do
        a := min(a,op(1,pe),op(2,pe)) ;
    end do:
    a ;
end proc:
seq(A381212(n),n=2..100) ; # R. J. Mathar, Mar 05 2025

A381212[n_] := Min[Flatten[FactorInteger[n]]];
Array[A381212, 100, 2]

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

cp's OEIS Frontend

A381213 Numbers k such that A381201(k) != A381203(k).

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A381201 a(n) is the product of the elements of the set of bases and exponents in the prime factorization of n.

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A381204 a(n) is the gcd of the elements of the set of bases and exponents in the prime factorization of n.

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A381202 a(n) is the sum of the elements of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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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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A381212 a(n) is the smallest element 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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