A203623 -id:A203623 - cp's OEIS frontend

A243503 Sums of parts of partitions (i.e., their sizes) as ordered in the table A241918: a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

0, 1, 2, 2, 3, 4, 4, 3, 3, 6, 5, 6, 6, 8, 5, 4, 7, 5, 8, 9, 7, 10, 9, 8, 4, 12, 4, 12, 10, 8, 11, 5, 9, 14, 6, 7, 12, 16, 11, 12, 13, 11, 14, 15, 7, 18, 15, 10, 5, 7, 13, 18, 16, 6, 8, 16, 15, 20, 17, 11, 18, 22, 10, 6, 10, 14, 19, 21, 17, 10, 20, 9, 21, 24, 6, 24

Offset: 1

Antti Karttunen, Jun 05 2014

nonn

Each n occurs A000041(n) times in total.

Where are the first and the last occurrence of each n located?

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A243504 (the products of parts), A241918, A000041, A227183, A075158, A056239, A241909.
Sum of prime indices of A241916, the even bisection of A358195.
Sums of even-indexed rows of A358172.
A112798 lists prime indices, length A001222, sum A056239, max A061395.
Cf. A005940, A019565, A246277, A326844, A356958, A359362.

Table[If[n==1,0,With[{y=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Last[y]*Length[y]+Last[y]-Total[y]+Length[y]-1]],{n,100}] (* Gus Wiseman, Jan 09 2023 *)

a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).
a(n) = A056239(A241909(n)).
a(n) = A227183(A075158(n-1)).
a(A000040(n)) = a(A000079(n)) = n for all n >= 1.
a(A122111(n)) = a(n) for all n.
a(A243051(n)) = a(n) for all n, and likewise for A243052, A243053 and other rows of A243060.
a(n) = A061395(n) * A001222(n) + A061395(n) - A056239(n) + A001222(n) - 1. - Gus Wiseman, Jan 09 2023
a(n) = A326844(2n) + A001222(n). - Gus Wiseman, Jan 09 2023

A243504 Product of parts of integer partitions as ordered by the table A241918: a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 3, 2, 8, 1, 9, 1, 16, 4, 4, 1, 6, 1, 27, 8, 32, 1, 16, 2, 64, 3, 81, 1, 18, 1, 5, 16, 128, 4, 12, 1, 256, 32, 64, 1, 54, 1, 243, 9, 512, 1, 25, 2, 12, 64, 729, 1, 8, 8, 256, 128, 1024, 1, 48, 1, 2048, 27, 6, 16, 162, 1, 2187, 256, 36, 1, 20, 1, 4096

Offset: 1

Antti Karttunen, Jun 05 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

The positions of ones after a(1)=1 is given by A000040 (primes).

Cf. A243503 (the sum of parts), A241918, A227184, A075158, A003963, A241909.

a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).
a(n) = A003963(A241909(n)).
a(n) = A227184(A075158(n-1)).
a(A000040(n)) = 1 for all n.
a(A000079(n)) = n for all n.

A241909 Self-inverse permutation of natural numbers: a(1)=1, a(p_i) = 2^i, and if n = p_i1 * p_i2 * p_i3 * ... * p_{ik-1} * p_ik, where p's are primes, with their indexes are sorted into nondescending order: i1 <= i2 <= i3 <= ... <= i_{k-1} <= ik, then a(n) = 2^(i1-1) * 3^(i2-i1) * 5^(i3-i2) * ... * p_k^(1+(ik-i_{k-1})). Here k = A001222(n) and ik = A061395(n).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 16, 5, 6, 27, 32, 25, 64, 81, 18, 7, 128, 15, 256, 125, 54, 243, 512, 49, 12, 729, 10, 625, 1024, 75, 2048, 11, 162, 2187, 36, 35, 4096, 6561, 486, 343, 8192, 375, 16384, 3125, 50, 19683, 32768, 121, 24, 45, 1458, 15625, 65536, 21, 108, 2401

Offset: 1

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

nonn

This permutation maps between the partitions as ordered in A112798 and A241918 (the original motivation for this sequence).
For all n > 2, A007814(a(n)) = A055396(n)-1, which implies that this self-inverse permutation maps between primes (A000040) and the powers of two larger than one (A000079(n>=1)), and apart from a(1) & a(2), this also maps each even number to some odd number, and vice versa, which means there are no fixed points after 2.
A122111 commutes with this one, that is, a(n) = A122111(a(A122111(n))).
Conjugates between A243051 and A242424 and other rows of A243060 and A243070.

			For n = 12 = 2 * 2 * 3 = p_1 * p_1 * p_2, we obtain by the first formula 2^(1-1) * 3^(1-1) * 5^(1+(2-1)) = 5^2 = 25. By the second formula, as n = 2^2 * 3^1, we obtain the same result, p_{1+2} * p_{2+1} = p_3 * p_3 = 25, thus a(12) = 25.
Using the product formula over the terms of row n of table A241918, we see, because 9450 = 2*3*3*3*5*5*7 = p_1^1 * p_2^3 * p_3^2 * p_4^1, that the corresponding row in A241918 is {2,5,7,7}, and multiplying p_2 * p_5 * p_7^2 yields 3 * 11 * 17 * 17 = 9537, thus a(9450) = 9537.
Similarly, for 9537, the corresponding row in A241918 is {1,2,2,2,3,3,4}, and multiplying p_1^1 * p_2^3 * p_3^2 * p_4^1, we obtain 9450 back.

Antti Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

Cf. A203623, A241918, A242378, A007814, A064989, A064216, A001222, A061395, A125976, A243060, A243070.
Cf. also A278220 (= A046523(a(n))), A331280 (its rgs_transform), A331299 (= min(n,a(n))).
{A000027, A122111, A241909, A241916} form a 4-group.
Cf. A049084, A027746.

a241909 1 = 1
a241909 n = product $ zipWith (^) a000040_list $ zipWith (-) is (1 : is)
            where is = reverse ((j + 1) : js)
                  (j:js) = reverse $ map a049084 $ a027746_row n
-- Reinhard Zumkeller, Aug 04 2014

Array[If[# == 1, 1, Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[FactorInteger[#] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]] &, 56] (* Michael De Vlieger, Jan 23 2020 *)

A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m)); \\ Antti Karttunen, Jan 17 2020

If n is a prime with index i (p_i), then a(n) = 2^i, otherwise when n = p_i1 * p_i2 * p_i3 * ... p_ik, where p_i1, p_i2, p_i3, ..., p_ik are the primes present (not necessarily all distinct) in the prime factorization of n, sorted into nondescending order, a(n) = 2^(i1-1) * 3^(i2-i1) * 5^(i3-i2) * ... * p_k^(1+(ik-i_{k-1})).
Equally, if n = 2^k, then a(n) = p_k, otherwise, when n = 2^e1 * 3^e2 * 5^e3 * ... * p_k^{e_k}, i.e., where e1 ... e_k are the exponents (some of them possibly zero, except the last) of the primes 2, 3, 5, ... in the prime factorization of n, a(n) = p_{1+e1} * p_{1+e1+e2} * p_{1+e1+e2+e3} * ... * p_{e1+e2+e3+...+e_k}.
From the equivalence of the above two formulas (which are inverses of each other) it follows that a(a(n)) = n, i.e., that this permutation is an involution. For a proof, please see the attached notes.
The first formula corresponds to this recurrence:
a(1) = 1, a(p_k) = 2^k for primes with index k, otherwise a(n) = (A000040(A001222(n))^(A241917(n)+1)) * A052126(a(A052126(n))).
And the latter formula with this recurrence:
a(1) = 1, and for n>1, if n = 2^k, a(n) = A000040(k), otherwise a(n) = A000040(A001511(n)) * A242378(A007814(n), a(A064989(n))).
[Here A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n.]
We also have:
a(1)=1, and for n>1, a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A000040(A241918(i)).
For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).
For all n > 1, a(2n-1) = 2*a(A064216(n)).

Typos in the name corrected by Antti Karttunen, May 31 2014

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.

Original entry on oeis.org

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).

A241914 After a(1)=0, numbers 0 .. A061395(n)-1, followed by numbers 0 .. A061395(n+1)-1, etc.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 2, 0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 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 A061395(2)=1 (the index of the largest prime factor), we have here terms from 0 to 1-1)
  [3] 0, 1; (because A061395(3)=2, we have terms from 0 to 2-1)
  [4] 0;
  [5] 0, 1, 2; (because A061395(5)=3, we have terms from 0 to 3-1)
  [6] 0, 1;    (because A061395(6)=2, we have terms from 0 to 2-1)
  [7] 0, 1, 2, 3; (because A061395(7)=4, we have terms from 0 to 4-1)
etc.

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

One less than A241915.

Cf. A203623, A241920, A241910, A241918.

(define (A241914 n) (if (= n 1) 0 (- n (+ 2 (A203623 (- (A241920 n) 1))))))

a(1)=0, a(n) = n - A203623(A241920(n)-1) - 2.

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.

cp's OEIS Frontend

A243503 Sums of parts of partitions (i.e., their sizes) as ordered in the table A241918: a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

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A243504 Product of parts of integer partitions as ordered by the table A241918: a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

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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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A241914 After a(1)=0, numbers 0 .. A061395(n)-1, followed by numbers 0 .. A061395(n+1)-1, etc.

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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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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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