A241913 -id:A241913 - cp's OEIS frontend

A088902 Numbers n such that n = product (p_k)^(c_k) and set of its (c_k k's)'s is a self-conjugate partition, where p_k is k-th prime and c_k > 0.

1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 350, 416, 441, 624, 660, 735, 1088, 1100, 1386, 1560, 1632, 1715, 2310, 2401, 2432, 2600, 3267, 3276, 3648, 4080, 5390, 5445, 5460, 5888, 6800, 7546, 7722, 8568, 8832, 9120, 12705, 12740, 12870, 13689

Offset: 1

Naohiro Nomoto, Nov 28 2003

The Heinz numbers of the self-conjugate partitions. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] to be Product(p_j-th prime, j=1..r) (a concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56. It is in the sequence since [1,1,1,4] is self-conjugate. - Emeric Deutsch, Jun 05 2015

			20 is in the sequence because 20 = 2^2 * 5^1 = (p_1)^2 *(p_3)^1, (two 1's, one 3's) = (1,1,3) is a self-conjugate partition of 5.
From _Gus Wiseman_, Jun 28 2022: (Start)
The terms together with their prime indices begin:
    1: ()
    2: (1)
    6: (2,1)
    9: (2,2)
   20: (3,1,1)
   30: (3,2,1)
   56: (4,1,1,1)
   75: (3,3,2)
   84: (4,2,1,1)
  125: (3,3,3)
  176: (5,1,1,1,1)
  210: (4,3,2,1)
  264: (5,2,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..400

Fixed points of A122111.
Cf. A242422, A215366.
A002110 (primorial numbers) is a subsequence.
After a(1) and a(2), a subsequence of A241913.
These partitions are counted by A000700.
The same count comes from A258116.
The complement is A352486, counted by A330644.
These are the positions of zeros in A352491.
A000041 counts integer partitions, strict A000009.
A325039 counts partitions w/ product = conjugate product, ranked by A325040.
Heinz number (rank) and partition:
- A003963 = product of partition, conjugate A329382.
- A008480 = number of permutations of partition, conjugate A321648.
- A056239 = sum of partition.
- A296150 = parts of partition, reverse A112798, conjugate A321649.
- A352487 = less than conjugate, counted by A000701.
- A352488 = greater than or equal to conjugate, counted by A046682.
- A352489 = less than or equal to conjugate, counted by A046682.
- A352490 = greater than conjugate, counted by A000701.
Cf. A000720, A098825, A195017, A238351, A238745, A347450, A350841.

with(numtheory): c := proc (n) local B, C: B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: C := proc (P) local a: a := proc (j) local c, i: c := 0: for i to nops(P) do if j <= P[i] then c := c+1 else end if end do: c end proc: [seq(a(k), k = 1 .. max(P))] end proc: mul(ithprime(C(B(n))[q]), q = 1 .. nops(C(B(n)))) end proc: SC := {}: for i to 14000 do if c(i) = i then SC := `union`(SC, {i}) else end if end do: SC; # Emeric Deutsch, May 09 2015

Select[Range[14000], Function[n, n == If[n == 1, 1, Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@ Length@ l, m = Min@ l}, Length@ Union@ l]] &@ Catenate[ConstantArray[PrimePi@ #1, #2] & @@@ FactorInteger@ n]]]] (* Michael De Vlieger, Aug 27 2016, after JungHwan Min at A122111 *)

More terms from David Wasserman, Aug 26 2005

A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 8, 6, 25, 11, 27, 13, 49, 15, 16, 17, 18, 19, 125, 35, 121, 23, 81, 10, 169, 12, 343, 29, 75, 31, 32, 77, 289, 21, 54, 37, 361, 143, 625, 41, 245, 43, 1331, 45, 529, 47, 243, 14, 50, 221, 2197, 53, 36, 55, 2401, 323, 841, 59, 375, 61, 961, 175, 64

Offset: 1

Antti Karttunen, May 03 2014

nonn

For other numbers than the powers of 2 (that are fixed), this permutation reverses the sequence of exponents in the prime factorization of n from the exponent of 2 to that of the largest prime factor, except that the exponents of 2 and the greatest prime factor present are adjusted by one. Note that some of the exponents might be zeros.
Self-inverse permutation of natural numbers, composition of A122111 & A241909 in either order: a(n) = A122111(A241909(n)) = A241909(A122111(n)).
This permutation preserves both bigomega and the (index of) largest prime factor: for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
From the above it follows, that this fixes both primes (A000040) and powers of two (A000079), among other numbers.
Even positions from n=4 onward contain only terms of A070003, and the odd positions only the terms of A102750, apart from 1 which is at a(1), and 2 which is at a(2).

Antti Karttunen, Table of n, a(n) for n = 1..8192
A. Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

A241912 gives the fixed points; A241913 their complement.
Cf. A006530, A137502, A070003, A102750, A278525.
{A000027, A122111, A241909, A241916} form a 4-group.
The sum of prime indices of a(n) is A243503(n).
Even bisection of A358195 = Heinz numbers of rows of A358172.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A019565, A161511, A246277, A253565, A326844, A356958, A358137.

nn = 65; f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[If[IntegerQ@ #, n/4, g@ Reverse@(# - Join[{1}, ConstantArray[0, Length@ # - 2], {1}] &@ f@ n)] &@ Log2@ n, {n, 4, 4 nn, 4}] (* Michael De Vlieger, Aug 27 2016 *)

A209229(n) = (n && !bitand(n,n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf,1]), g = f); for(i=1,nbf,g[i,1] = prime(1+igp-primepi(f[i,1]))); factorback(g)/2); \\ Antti Karttunen, Jul 02 2018

(define (A241916 n) (A122111 (A241909 n)))

a(1)=1, and for n>1, a(n) = A006530(n) * A137502(n)/2.
a(n) = A122111(A241909(n)) = A241909(A122111(n)).
If 2n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=1..k-1} prime(x_k-x_i+1). The opposite version is A000027, even bisection of A246277. - Gus Wiseman, Dec 28 2022

Description clarified by Antti Karttunen, Jul 02 2018

A241912 Fixed points of A241916.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 18, 19, 23, 29, 31, 32, 37, 41, 43, 45, 47, 50, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 98, 101, 103, 105, 107, 108, 109, 113, 119, 127, 128, 131, 135, 137, 139, 149, 150, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Antti Karttunen, May 03 2014

nonn

A natural number n occurs here if and only if it is either a power of 2, or satisfies A001511(n) = A071178(n) [the exponent of highest power of 2 dividing n is one less than the exponent of the largest prime factor of n], and all the intermediate exponents form a palindrome. [Please see the definition of A241916.]

Numbers for which the corresponding rows in A112798 and A241918 are the conjugate partitions of each other.

			98 = 2*7*7 = p_1^1 * p_2^0 * p_3^0 * p_4^2 is included because 2 occurs once, the highest prime factor 7 occurs twice (thus A001511(150) = A071178(150) = 2), and the intermediate exponents (in this case {0,0}) form a palindrome.
150 = 2*3*5*5 = p_1^1 * p_2^1 * p_3^2 is included because 2 occurs once, the highest prime factor 5 occurs twice (thus A001511(150) = A071178(150) = 2), and the intermediate exponents (in this case 1) form a palindrome.

Antti Karttunen, Table of n, a(n) for n = 1..4931

Complement: A241913.
A079704 is a subsequence.
Cf. A088902, A241916, A242418.

f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[#[[n + 1]]/2, {n, Length@ # - 1}] &@ Select[Range@ 400, g@ f@ # == g@ Reverse@ f@ # &] (* Michael De Vlieger, Aug 27 2016 *)

a(n) = A242418(n+1)/2.

cp's OEIS Frontend

A088902 Numbers n such that n = product (p_k)^(c_k) and set of its (c_k k's)'s is a self-conjugate partition, where p_k is k-th prime and c_k > 0.

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A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

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A241912 Fixed points of A241916.

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