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 A325397 Heinz numbers of integer partitions whose k-th differences are weakly decreasing for all k >= 0. 8
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS First differs from A325361 in lacking 150. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences. The enumeration of these partitions by sum is given by A325353. LINKS EXAMPLE Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:    12: {1,1,2}    20: {1,1,3}    24: {1,1,1,2}    28: {1,1,4}    36: {1,1,2,2}    40: {1,1,1,3}    42: {1,2,4}    44: {1,1,5}    45: {2,2,3}    48: {1,1,1,1,2}    52: {1,1,6}    56: {1,1,1,4}    60: {1,1,2,3}    63: {2,2,4}    66: {1,2,5}    68: {1,1,7}    72: {1,1,1,2,2}    76: {1,1,8}    78: {1,2,6}    80: {1,1,1,1,3} The first partition that has weakly decreasing differences (A320466, A325361) but is not represented in this sequence is (3,3,2,1), which has Heinz number 150 and whose first and second differences are (0,-1,-1) and (-1,0) respectively. MATHEMATICA primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]; Select[Range[100], And@@Table[GreaterEqual@@Differences[primeptn[#], k], {k, 0, PrimeOmega[#]}]&] CROSSREFS Cf. A056239, A112798, A320466, A320509, A325353, A325361, A325364, A325389, A325398, A325399, A325400, A325405, A325467. Sequence in context: A329138 A065200 A325361 * A289812 A206551 A247762 Adjacent sequences:  A325394 A325395 A325396 * A325398 A325399 A325400 KEYWORD nonn AUTHOR Gus Wiseman, May 02 2019 STATUS approved

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Last modified December 14 05:36 EST 2019. Contains 329978 sequences. (Running on oeis4.)