A088902 - OEIS

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

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OFFSET

1,2

COMMENTS

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

LINKS

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

EXAMPLE

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)

MAPLE

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

MATHEMATICA

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 *)

PROG

(Scheme, with Antti Karttunen's Intseq-library)

(define A088902 (FIXED-POINTS 1 1 A122111))

CROSSREFS

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.

Sequence in context: A106529 A325040 A350949 * A265016 A279897 A336131

Adjacent sequences: A088899 A088900 A088901 * A088903 A088904 A088905

KEYWORD

easy,nonn

AUTHOR

Naohiro Nomoto, Nov 28 2003

EXTENSIONS

More terms from David Wasserman, Aug 26 2005

STATUS

approved