A325792 - OEIS

%I #7 May 23 2019 14:52:44

%S 1,2,4,6,8,16,18,20,32,42,54,56,64,100,128,162,176,204,234,256,260,

%T 294,308,315,350,392,416,486,500,512,690,696,798,920,1024,1026,1064,

%U 1088,1116,1122,1190,1365,1430,1458,1496,1755,1936,1968,2025,2048,2058,2079

%N Positive integers with as many proper divisors as the sum of their prime indices.

%C First differs from A325780 in having 204.

%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 term 42 is in the sequence because it has 7 proper divisors (1, 2, 3, 6, 7, 14, 21) and its sum of prime indices is also 1 + 2 + 4 = 7.

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

%e      1: {}

%e      2: {1}

%e      4: {1,1}

%e      6: {1,2}

%e      8: {1,1,1}

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

%e     18: {1,2,2}

%e     20: {1,1,3}

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

%e     42: {1,2,4}

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

%e     56: {1,1,1,4}

%e     64: {1,1,1,1,1,1}

%e    100: {1,1,3,3}

%e    128: {1,1,1,1,1,1,1}

%e    162: {1,2,2,2,2}

%e    176: {1,1,1,1,5}

%e    204: {1,1,2,7}

%e    234: {1,2,2,6}

%e    256: {1,1,1,1,1,1,1,1}

%t Select[Range[100],DivisorSigma[0,#]-1==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

%Y Positions of 1's in A325794.

%Y Heinz numbers of the partitions counted by A325828.

%Y Cf. A000005, A002033, A056239, A112798, A299702, A304793.

%Y Cf. A325694, A325780, A325781, A325793, A325795, A325796, A325797, A325798.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 23 2019