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 A300383 In the ranked poset of integer partitions ordered by refinement, a(n) is the size of the lower ideal generated by the partition with Heinz number n. 26
 1, 1, 2, 1, 3, 2, 5, 1, 3, 3, 7, 2, 11, 5, 5, 1, 15, 3, 22, 3, 8, 7, 30, 2, 6, 11, 4, 5, 42, 5, 56, 1, 11, 15, 11, 3, 77, 22, 17, 3, 101, 8, 135, 7, 7, 30, 176, 2, 14, 6, 23, 11, 231, 4, 15, 5, 33, 42, 297, 5, 385, 56, 11, 1, 23, 11, 490, 15, 45, 11, 627, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The size of the corresponding upper ideal is A317141(n). Chains are A213427(n) and maximal chains are A002846(n). LINKS Table of n, a(n) for n=1..72. FORMULA a(prime(n)) = A000041(n). a(x * y) <= a(x) * a(y). EXAMPLE The a(30) = 5 partitions are (321), (2211), (3111), (21111), (111111), with corresponding Heinz numbers: 30, 36, 40, 48, 64. MATHEMATICA primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@primeMS[n]]]], {n, 50}] CROSSREFS Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273. Cf. A317141, A317142, A317143. Sequence in context: A182167 A117364 A193615 * A120250 A280689 A352961 Adjacent sequences: A300380 A300381 A300382 * A300384 A300385 A300386 KEYWORD nonn AUTHOR Gus Wiseman, Mar 04 2018 STATUS approved

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Last modified December 3 22:01 EST 2023. Contains 367540 sequences. (Running on oeis4.)