A035306 -id:A035306 - cp's OEIS frontend

A348477 Drop all 1 but the first 1 in A035306.

1, 2, 3, 2, 2, 5, 2, 3, 7, 2, 3, 3, 2, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 4, 17, 2, 3, 2, 19, 2, 2, 5, 3, 7, 2, 11, 23, 2, 3, 3, 5, 2, 2, 13, 3, 3, 2, 2, 7, 29, 2, 3, 5, 31, 2, 5, 3, 11, 2, 17, 5, 7, 2, 2, 3, 2, 37, 2, 19, 3, 13, 2, 3, 5, 41, 2, 3, 7, 43, 2, 2, 11, 3, 2, 5, 2, 23, 47, 2, 4, 3, 7, 2, 2, 5, 2, 3, 17, 2, 2, 13, 53, 2, 3, 3, 5, 11, 2, 3, 7, 3, 19, 2, 29

Offset: 1

Seiichi Manyama, Oct 20 2021

List of prime divisors of n and their exponents, ignoring the exponent 1. - Michael De Vlieger, Oct 20 2021

			   n   prime factorization  triangle
   1 = 1.                 ->  1;
   2 = 2.                 ->  2;
   3 = 3.                 ->  3;
   4 = 2^2.               ->  2, 2;
   5 = 5.                 ->  5;
   6 = 2*3.               ->  2, 3;
   7 = 7.                 ->  7;
   8 = 2^3.               ->  2, 3;
   9 = 3^2.               ->  3, 2;
  10 = 2*5.               ->  2, 5;
  11 = 11.                -> 11;
  12 = 2^2*3.             ->  2, 2, 3;
  13 = 13.                -> 13;
  14 = 2*7                ->  2, 7;
  15 = 3*5.               ->  3, 5;
  16 = 2^4.               ->  2, 4;

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.

Column 1 is A020639.
Row lengths are A238949(n) for n > 1.
Cf. A027746, A035306.

Array[DeleteCases[Flatten@ FactorInteger[#], 1] &, 58] /. {} -> {1} // Flatten (* Michael De Vlieger, Oct 20 2021 *)

tabf(nn) = if(nn==1, print1(1, ", "), my(f=factor(nn)); for(i=1, #f~, for(j=1, 2, if((k=f[i, j])>j-1, print1(k, ", ")))));

require 'prime'
def A348477(n)
  ary = (2..n).map{|i| i.prime_division}.flatten
  ary.delete(1)
  [1] + ary
end
p A348477(60)

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]

A381403 a(n) is the mode of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n (using smallest mode if multimodal).

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Feb 27 2025

			The prime factorization of 132 is 2^2*3^1*11^1, the multiset of these bases and exponents is {1, 1, 2, 2, 3, 11} and its smallest most frequent element is 1.

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

Cf. A035306, A381178, A381404.

A381403[n_] := Min[Commonest[Flatten[FactorInteger[n]]]];
Array[A381403, 100, 2]

a(p) = 1, for p prime.

A381404 a(n) is the mode of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n (using largest mode if multimodal).

Original entry on oeis.org

2, 3, 2, 5, 1, 7, 3, 3, 1, 11, 2, 13, 1, 1, 4, 17, 2, 19, 2, 1, 1, 23, 3, 5, 1, 3, 2, 29, 1, 31, 5, 1, 1, 1, 2, 37, 1, 1, 5, 41, 1, 43, 2, 5, 1, 47, 4, 7, 2, 1, 2, 53, 3, 1, 7, 1, 1, 59, 2, 61, 1, 7, 6, 1, 1, 67, 2, 1, 1, 71, 3, 73, 1, 5, 2, 1, 1, 79, 5, 4, 1, 83, 2, 1, 1

Offset: 2

Paolo Xausa, Feb 27 2025

			The prime factorization of 132 is 2^2*3^1*11^1, the multiset of these bases and exponents is {1, 1, 2, 2, 3, 11} and its largest most frequent element is 2.

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

Cf. A035306, A381178, A381403.

Cf. A000040 (fixed points).

A381404[n_] := Max[Commonest[Flatten[FactorInteger[n]]]];
Array[A381404, 100, 2]

a(p) = p, for p prime.

A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 4, 17, 12, 19, 10, 21, 22, 23, 18, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 12, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 12, 14, 20, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 6, 65, 66, 67, 34, 69, 70, 71

Offset: 1

Paolo Xausa, Feb 28 2025

			a(18) = 12 because 18 = 2^1*3^2, lcm(2,1) = 2, lcm(3,2) = 6 and 2*6 = 12.
a(300) = 30 because 300 = 2^2*3^1*5^2, lcm(2,2) = 2, lcm(3,1) = 3, lcm(5,2) = 10 and 2*3*10 = 60.

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

Cf. A008473, A008477, A035306, A144338 (fixed points), A369008 (analogous for gcd).

A381588[n_] := Times @@ LCM @@@ FactorInteger[n];
Array[A381588, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, lcm(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

a(p) = p, for p prime.

A381613 If n = Product (p_j^k_j) then a(n) = Product (min(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Mar 01 2025

First differs from A323308 at n = 27.

			a(18) = 2 because 18 = 2^1*3^2, min(2,1) = 1, min(3,2) = 2 and 1*2 = 2.
a(300) = 4 because 300 = 2^2*3^1*5^2, min(2,2) = 2, min(3,1) = 1, min(5,2) = 2 and 2*1*2 = 4.

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

Cf. A008473, A008477, A035306, A323308, A369008, A381588, A381614.

A381613[n_] := Times @@ Min @@@ FactorInteger[n];
Array[A381613, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, min(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1/p - 1/p^p)/(p-1)) = 1.59383299054679951264... . - Amiram Eldar, Mar 07 2025

A381614 If n = Product (p_j^k_j) then a(n) = Product (max(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 3, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 9, 5, 26, 3, 14, 29, 30, 31, 5, 33, 34, 35, 6, 37, 38, 39, 15, 41, 42, 43, 22, 15, 46, 47, 12, 7, 10, 51, 26, 53, 6, 55, 21, 57, 58, 59, 30, 61, 62, 21, 6, 65, 66, 67, 34, 69, 70, 71, 9, 73

Offset: 1

Paolo Xausa, Mar 01 2025

			a(18) = 6 because 18 = 2^1*3^2, max(2,1) = 2, max(3,2) = 3 and 2*3 = 6.
a(300) = 30 because 300 = 2^2*3^1*5^2, max(2,2) = 2, max(3,1) = 3, max(5,2) = 5 and 2*3*5 = 30.

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

Cf. A008473, A008477, A035306, A065463, A369008, A381588, A381613.

A381614[n_] := Times @@ Max @@@ FactorInteger[n];
Array[A381614, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, max(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

a(p) = p, for p prime.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065463 * Product_{p prime} (1 + 1/((p^2-1)*(p^2+p-1)*p^(2*p-2))) = 0.71628338157754073004... . - Amiram Eldar, Mar 07 2025

cp's OEIS Frontend

A348477 Drop all 1 but the first 1 in A035306.

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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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A381403 a(n) is the mode of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n (using smallest mode if multimodal).

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A381404 a(n) is the mode of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n (using largest mode if multimodal).

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A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

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A381613 If n = Product (p_j^k_j) then a(n) = Product (min(p_j, k_j)), with a(1) = 1.

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A381614 If n = Product (p_j^k_j) then a(n) = Product (max(p_j, k_j)), with a(1) = 1.

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