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 A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product. 15
 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 less than their sum of prime indices (A056239). The enumeration of these partitions by sum is given by A096276 shifted once to the right. LINKS Table of n, a(n) for n=1..60. FORMULA A003963(a(n)) < A056239(a(n)). a(n) = 2 * A325044(n). EXAMPLE The sequence of terms together with their prime indices begins: 4: {1,1} 6: {1,2} 8: {1,1,1} 10: {1,3} 12: {1,1,2} 14: {1,4} 16: {1,1,1,1} 18: {1,2,2} 20: {1,1,3} 22: {1,5} 24: {1,1,1,2} 26: {1,6} 28: {1,1,4} 32: {1,1,1,1,1} 34: {1,7} 36: {1,1,2,2} 38: {1,8} 40: {1,1,1,3} 44: {1,1,5} 46: {1,9} 48: {1,1,1,1,2} MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; Select[Range[100], Times@@primeMS[#]

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Last modified October 4 02:45 EDT 2023. Contains 365872 sequences. (Running on oeis4.)