A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.

3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Author

Gus Wiseman, Jan 16 2025

Keywords

• nonn

Comments

Contains all odd primes.

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 terms together with their prime indices begin:
     2: {1}         53: {16}           109: {29}
     3: {2}         59: {17}           113: {30}
     5: {3}         61: {18}           125: {3,3,3}
     7: {4}         63: {2,2,4}        127: {31}
     9: {2,2}       65: {3,6}          131: {32}
    11: {5}         67: {19}           137: {33}
    13: {6}         71: {20}           139: {34}
    17: {7}         73: {21}           149: {35}
    19: {8}         79: {22}           150: {1,2,3,3}
    23: {9}         81: {2,2,2,2}      151: {36}
    29: {10}        83: {23}           154: {1,4,5}
    30: {1,2,3}     84: {1,1,2,4}      157: {37}
    31: {11}        89: {24}           163: {38}
    37: {12}        97: {25}           165: {2,3,5}
    41: {13}       101: {26}           167: {39}
    43: {14}       103: {27}           169: {6,6}
    47: {15}       107: {28}           173: {40}
    49: {4,4}      108: {1,1,2,2,2}    179: {41}
		

Crossrefs

Including evens gives A326149, counted by A057568.

For nonprime instead of odd we get A326150.

For even instead of odd we get A379319, counted by A379320.

Partitions of this type are counted by A379734, strict A379735, see A379733.

For squarefree instead of odd we get A379844, even case A379845.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- 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, A319916, A324851, A325041, A326152, A379671, A379678.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[#]&&Divisible[Times@@prix[#],Total[prix[#]]]&]