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 A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum. 16
 1, 15, 21, 25, 27, 33, 35, 39, 42, 45, 49, 50, 51, 54, 55, 57, 63, 65, 66, 69, 70, 75, 77, 78, 81, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 110, 111, 114, 115, 117, 119, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is greater than their sum of prime indices (A056239). The enumeration of these partitions by sum is given by A114324. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 FORMULA A003963(a(n)) > A056239(a(n)). EXAMPLE The sequence of terms together with their prime indices begins:    1: {}   15: {2,3}   21: {2,4}   25: {3,3}   27: {2,2,2}   33: {2,5}   35: {3,4}   39: {2,6}   42: {1,2,4}   45: {2,2,3}   49: {4,4}   50: {1,3,3}   51: {2,7}   54: {1,2,2,2}   55: {3,5}   57: {2,8}   63: {2,2,4}   65: {3,6}   66: {1,2,5}   69: {2,9}   70: {1,3,4}   75: {2,3,3}   77: {4,5}   78: {1,2,6}   81: {2,2,2,2} MAPLE q:= n-> (l-> mul(i, i=l)>add(i, i=l))(map(i->     numtheory[pi](i[1])\$i[2], ifactors(n)[2])): select(q, [\$1..200])[];  # Alois P. Heinz, Mar 27 2019 MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; Select[Range[100], Times@@primeMS[#]>Plus@@primeMS[#]&] CROSSREFS Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000. Cf. A325032, A325033, A325036, A325038, A325041, A325042, A325044. Sequence in context: A294171 A306102 A177024 * A154545 A156063 A181780 Adjacent sequences:  A325034 A325035 A325036 * A325038 A325039 A325040 KEYWORD nonn AUTHOR Gus Wiseman, Mar 25 2019 STATUS approved

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Last modified June 1 13:11 EDT 2020. Contains 334762 sequences. (Running on oeis4.)