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

Author

Gus Wiseman, May 23 2019

Keywords

• nonn

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).

Examples

			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}
		

Links

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

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.

Programs

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]]&]