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Numbers with at least as many divisors as the sum of their prime indices.
8

%I #5 May 23 2019 14:51:35

%S 1,2,3,4,6,8,10,12,16,18,20,24,28,30,32,36,40,42,48,54,56,60,64,66,70,

%T 72,80,84,88,90,96,100,108,112,120,126,128,132,140,144,150,156,160,

%U 162,168,176,180,192,198,200,204,208,210,216,220,224,228,234,240

%N Numbers with at least as many divisors as the sum of their prime indices.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n).

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 2: {1}

%e 3: {2}

%e 4: {1,1}

%e 6: {1,2}

%e 8: {1,1,1}

%e 10: {1,3}

%e 12: {1,1,2}

%e 16: {1,1,1,1}

%e 18: {1,2,2}

%e 20: {1,1,3}

%e 24: {1,1,1,2}

%e 28: {1,1,4}

%e 30: {1,2,3}

%e 32: {1,1,1,1,1}

%e 36: {1,1,2,2}

%e 40: {1,1,1,3}

%e 42: {1,2,4}

%e 48: {1,1,1,1,2}

%e 54: {1,2,2,2}

%t Select[Range[100],DivisorSigma[0,#]>=Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

%Y Positions of nonnegative terms in A325794.

%Y Heinz numbers of the partitions counted by A325832.

%Y Cf. A000005, A002033, A056239, A112798, A299702.

%Y Cf. A325694, A325780, A325781, A325792, A325793, A325795, A325797, A325798.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 23 2019