A325793 Positive integers whose number of divisors is equal to their sum of prime indices.

3, 10, 28, 66, 70, 88, 208, 228, 306, 340, 364, 490, 495, 525, 544, 550, 675, 744, 870, 966, 1160, 1216, 1242, 1254, 1288, 1326, 1330, 1332, 1672, 1768, 1785, 1870, 2002, 2064, 2145, 2295, 2457, 2900, 2944, 3250, 3280, 3430, 3468, 3540, 3724, 4125, 4144, 4248

Offset

1,1

Comments

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

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The term 70 is in the sequence because it has 8 divisors {1, 2, 5, 7, 10, 14, 35, 70} and its sum of prime indices is also 1 + 3 + 4 = 8.
The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    28: {1,1,4}
    66: {1,2,5}
    70: {1,3,4}
    88: {1,1,1,5}
   208: {1,1,1,1,6}
   228: {1,1,2,8}
   306: {1,2,2,7}
   340: {1,1,3,7}
   364: {1,1,4,6}
   490: {1,3,4,4}
   495: {2,2,3,5}
   525: {2,3,3,4}
   544: {1,1,1,1,1,7}
   550: {1,3,3,5}
   675: {2,2,2,3,3}
   744: {1,1,1,2,11}
   870: {1,2,3,10}
   966: {1,2,4,9}

Maple

filter:= proc(n) local F, t;
  F:= ifactors(n)[2];
  add(numtheory:-pi(t[1])*t[2], t=F) = mul(t[2]+1, t=F)
end proc:
select(filter, [$1..10000]); # Robert Israel, Oct 16 2023

Mathematica

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

Crossrefs

Positions of 0's in A325794.

Contains A239885 except for 1.

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

Cf. A325694, A325780, A325781, A325792, A325795, A325796, A325797, A325798.

Sequence in context: A262724 A059193 A226863 * A325800 A037167 A267707

Adjacent sequences: A325790 A325791 A325792 * A325794 A325795 A325796

Keyword

nonn

Author

Gus Wiseman, May 23 2019

Status

approved