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

1,2

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.

Links

Table of n, a(n) for n=1..53.

Formula

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

Example

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}

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[#]]]&]

Prog

(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

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.

Sequence in context: A371108 A371050 A070007 * A325041 A154267 A173351

Adjacent sequences: A380214 A380215 A380216 * A380218 A380219 A380220

Keyword

nonn

Author

Gus Wiseman, Jan 18 2025

Status

approved