A380217 Numbers whose product of prime indices is a multiple of their sum of prime indices plus one.

1, 15, 42, 54, 75, 77, 95, 99, 100, 132, 182, 195, 221, 234, 245, 253, 290, 312, 315, 329, 350, 357, 405, 420, 423, 437, 450, 459, 476, 494, 510, 540, 555, 559, 560, 612, 627, 665, 715, 720, 740, 798, 816, 833, 854, 855, 858, 893, 897, 899, 979, 1026, 1064

Offset: 1

Author

Gus Wiseman, Jan 18 2025

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. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The prime indices of 75 are {2,3,3}, with product 18 and sum 8, and since 18 is a multiple of 8+1, 75 is in the sequence.
The terms together with their prime indices begin:
     1: {}
    15: {2,3}
    42: {1,2,4}
    54: {1,2,2,2}
    75: {2,3,3}
    77: {4,5}
    95: {3,8}
    99: {2,2,5}
   100: {1,1,3,3}
   132: {1,1,2,5}
   182: {1,4,6}
   195: {2,3,6}
   221: {6,7}
   234: {1,2,2,6}
   245: {3,4,4}
		

Crossrefs

The case of equality is A325041, counted by A380218 = A028422 except n=3.

Without "plus one" we get A326149, counted by A057568, see A379733, A379734, A379735.

Double all terms to get A379319.

Partitions of this type are counted by A379320.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- multiple: A057567, ranks A326155

- divisor: A057568, ranks A326149

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029, A379720

- less or equal: A319005, ranks A379721, see A025147

- different: A379736, ranks A379722, see A111133

Cf. A069016, A301988, A318950, A319000, A324851, A379318, A379671, A379844, A379845.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Divisible[Times@@prix[#],1+Total[prix[#]]]&]

PARI

vpind(n)=my(v=List(), f=factor(n)); for(i=1, #f~, for(j=1, f[i, 2], listput(v, primepi(f[i, 1])))); Vec(v); \\ A112798
isok(k) = my(vind = vpind(k)); (vecprod(vind) % (vecsum(vind)+1)) == 0; \\ Michel Marcus, Jan 21 2025

Formula

a(n) = A379319(n)/2.