A241914 -id:A241914 - cp's OEIS frontend

A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 1

Antti Karttunen, May 03 2014, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

a(1) = 0 by convention (stands for an empty partition).

For n >= 2, A203623(n-1)+2 gives the index to the beginning of row n and for n>=1, A203623(n)+1 is the index to the end of row n.

			Table begins:
Row     Partition
[ 1]    0;         (stands for empty partition)
[ 2]    1;         (as 2 = 2^1)
[ 3]    1,1;       (as 3 = 2^0 * 3^1)
[ 4]    2;         (as 4 = 2^2)
[ 5]    1,1,1;     (as 5 = 2^0 * 3^0 * 5^1)
[ 6]    2,2;       (as 6 = 2^1 * 3^1)
[ 7]    1,1,1,1;   (as 7 = 2^0 * 3^0 * 5^0 * 7^1)
[ 8]    3;         (as 8 = 2^3)
[ 9]    1,2;       (as 9 = 2^0 * 3^2)
[10]    2,2,2;     (as 10 = 2^1 * 3^0 * 5^1)
[11]    1,1,1,1,1;
[12]    3,3;
[13]    1,1,1,1,1,1;
[14]    2,2,2,2;
[15]    1,2,2;     (as 15 = 2^0 * 3^1 * 5^1)
[16]    4;
[17]    1,1,1,1,1,1,1;
[18]    2,3;       (as 18 = 2^1 * 3^2)
etc.
If n is 2^k (k>=1), then the partition is a singleton {k}, otherwise, add one to the exponent of 2 (= A007814(n)), and subtract one from the exponent of the greatest prime dividing n (= A071178(n)), leaving the intermediate exponents as they are, and then take partial sums of all, thus resulting for e.g. 15 = 2^0 * 3^1 * 5^1 the modified sequence of exponents {0+1, 1, 1-1} -> {1,1,0}, whose partial sums {1,1+1,1+1+0} -> {1,2,2} give the corresponding partition at row 15.

Antti Karttunen, Table of n, a(n) for n = 1..10081; rows 1..521 flattened.
Marc LeBrun's original "crazy order" mapping for partitions (Copy of Marc's Jan 11 2006 message in OEIS Wiki)

For n>=2, the length of row n is given by A061395(n).
Cf. also A067255, A203623, A241914.
Other tables of partitions: A112798 (also based on prime factorization), A227739, A242628 (encoded in the binary representation of n), and A036036-A036037, A080576-A080577, A193073 for various lexicographical orderings.
Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.
Permutation A122111 is induced when partitions in this list are conjugated.
A241912 gives the row numbers for which the corresponding rows in A112798 and here are the conjugate partitions of each other.

Table[If[n == 1, {0}, Function[s, Function[t, Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, s]]]]@ ConstantArray[0, Transpose[s][[1, -1]]]][FactorInteger[n] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]], {n, 31}] // Flatten (* Michael De Vlieger, May 12 2017 *)

If A241914(n)=0 and A241914(n+1)=0, a(n) = A067255(n); otherwise, if A241914(n)=0 and A241914(n+1)>0, a(n) = A067255(n)+1; otherwise, if A241914(n)>0 and A241914(n+1)=0, a(n) = a(n-1) + A067255(n) - 1, otherwise, when A241914(n)>0 and A241914(n+1)>0, a(n) = a(n-1) + A067255(n).

A241915 After a(1)=1, numbers 1 .. A061395(n), followed by numbers 1 .. A061395(n+1), etc.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 3, 1, 1, 2, 3, 4, 5, 6, 7, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1

Offset: 1

Antti Karttunen, May 01 2014

			Viewed as an irregular table, the sequence is constructed as:
"Row"
  [1] 1; (by convention, a(1)=1)
  [2] 1; (because A061395(2)=1 (the index of the largest prime factor), we have here terms from 1 to 1)
  [3] 1, 2; (because A061395(3)=2, we have terms from 1 to 2)
  [4] 1;
  [5] 1, 2, 3; (because A061395(5)=3, we have terms from 1 to 3)
  [6] 1, 2;    (because A061395(6)=2, we have terms from 1 to 2)
  [7] 1, 2, 3, 4; (because A061395(7)=4, we have terms from 1 to 4)
etc.

Antti Karttunen, Table of n, a(n) for n = 1..10082

One more than A241914.

Cf. A203623, A241920, A241911.

(define (A241915 n) (if (= n 1) 1 (- n (A203623 (- (A241920 n) 1)) 1)))

a(1)=1, a(n) = n - A203623(A241920(n)-1) - 1.

A241910 After a(1)=0, numbers 0 .. bigomega(n)-1, followed by numbers 0 .. bigomega(n+1)-1, etc., where bigomega(n)=A001222(n) is the number of prime factors of n (with repetition).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 01 2014

			Viewed as an irregular table, the sequence is constructed as:
"Row"
  [1] 0; (by convention, a(1)=0)
  [2] 0; (because bigomega(2)=1, we have here terms from 0 to 0)
  [3] 0; (same with 3, bigomega(3)=1)
  [4] 0, 1; (as bigomega(4)=2, we have terms from 0 to 2-1)
  [5] 0;
  [6] 0, 1;
  [7] 0;
  [8] 0, 1, 2; (as bigomega(8)=3, we have terms from 0 to 3-1).
etc.

Antti Karttunen, Table of n, a(n) for n = 1..10000

One less than A241911.

Cf. A022559, A082288, A112798, A241914.

a(1)=0, a(n) = n - A022559(A082288(n)-1) - 2.

A241920 After a(1)=1, each n appears A061395(n) times, where A061395 gives the index of the largest prime factor of n.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 22

Offset: 1

Antti Karttunen, May 01 2014

nonn

Only numbers that occur just once are the powers of two (A000079).

Antti Karttunen, Table of n, a(n) for n = 1..10081

Cf. A082288, A061395, A203623, A067255, A241918, A241914, A241915.

cp's OEIS Frontend

A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

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A241915 After a(1)=1, numbers 1 .. A061395(n), followed by numbers 1 .. A061395(n+1), etc.

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A241910 After a(1)=0, numbers 0 .. bigomega(n)-1, followed by numbers 0 .. bigomega(n+1)-1, etc., where bigomega(n)=A001222(n) is the number of prime factors of n (with repetition).

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A241920 After a(1)=1, each n appears A061395(n) times, where A061395 gives the index of the largest prime factor of n.

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