%I #35 Jun 30 2022 08:40:39
%S 1,2,6,9,20,30,56,75,84,125,176,210,264,350,416,441,624,660,735,1088,
%T 1100,1386,1560,1632,1715,2310,2401,2432,2600,3267,3276,3648,4080,
%U 5390,5445,5460,5888,6800,7546,7722,8568,8832,9120,12705,12740,12870,13689
%N 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.
%C 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
%H Alois P. Heinz, <a href="/A088902/b088902.txt">Table of n, a(n) for n = 1..400</a>
%e 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.
%e From _Gus Wiseman_, Jun 28 2022: (Start)
%e The terms together with their prime indices begin:
%e 1: ()
%e 2: (1)
%e 6: (2,1)
%e 9: (2,2)
%e 20: (3,1,1)
%e 30: (3,2,1)
%e 56: (4,1,1,1)
%e 75: (3,3,2)
%e 84: (4,2,1,1)
%e 125: (3,3,3)
%e 176: (5,1,1,1,1)
%e 210: (4,3,2,1)
%e 264: (5,2,1,1,1)
%e (End)
%p 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
%t 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 *)
%o (Scheme, with _Antti Karttunen_'s Intseq-library)
%o (define A088902 (FIXED-POINTS 1 1 A122111))
%Y Fixed points of A122111.
%Y Cf. A242422, A215366.
%Y A002110 (primorial numbers) is a subsequence.
%Y After a(1) and a(2), a subsequence of A241913.
%Y These partitions are counted by A000700.
%Y The same count comes from A258116.
%Y The complement is A352486, counted by A330644.
%Y These are the positions of zeros in A352491.
%Y A000041 counts integer partitions, strict A000009.
%Y A325039 counts partitions w/ product = conjugate product, ranked by A325040.
%Y Heinz number (rank) and partition:
%Y - A003963 = product of partition, conjugate A329382.
%Y - A008480 = number of permutations of partition, conjugate A321648.
%Y - A056239 = sum of partition.
%Y - A296150 = parts of partition, reverse A112798, conjugate A321649.
%Y - A352487 = less than conjugate, counted by A000701.
%Y - A352488 = greater than or equal to conjugate, counted by A046682.
%Y - A352489 = less than or equal to conjugate, counted by A046682.
%Y - A352490 = greater than conjugate, counted by A000701.
%Y Cf. A000720, A098825, A195017, A238351, A238745, A347450, A350841.
%K easy,nonn
%O 1,2
%A _Naohiro Nomoto_, Nov 28 2003
%E More terms from _David Wasserman_, Aug 26 2005