A325792 Positive integers with as many proper divisors as the sum of their prime indices.

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 204, 234, 256, 260, 294, 308, 315, 350, 392, 416, 486, 500, 512, 690, 696, 798, 920, 1024, 1026, 1064, 1088, 1116, 1122, 1190, 1365, 1430, 1458, 1496, 1755, 1936, 1968, 2025, 2048, 2058, 2079

Offset

1,2

Comments

First differs from A325780 in having 204.

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).

Links

Table of n, a(n) for n=1..52.

Example

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.
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    42: {1,2,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   176: {1,1,1,1,5}
   204: {1,1,2,7}
   234: {1,2,2,6}
   256: {1,1,1,1,1,1,1,1}

Mathematica

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

Crossrefs

Positions of 1's in A325794.

Heinz numbers of the partitions counted by A325828.

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

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

Sequence in context: A077569 A270140 A333020 * A325780 A283423 A073935

Adjacent sequences: A325789 A325790 A325791 * A325793 A325794 A325795

Keyword

nonn

Author

Gus Wiseman, May 23 2019

Status

approved