This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379318 #10 Jan 19 2025 09:35:45
%S A379318 3,5,7,9,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,63,65,67,71,73,
%T A379318 79,81,83,89,97,101,103,107,109,113,125,127,131,137,139,149,151,157,
%U A379318 163,165,167,169,173,179,181,191,193,197,199,211,223,227,229,233,239
%N A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.
%C A379318 Contains all odd primes.
%C A379318 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.
%e A379318 The terms together with their prime indices begin:
%e A379318 2: {1} 53: {16} 109: {29}
%e A379318 3: {2} 59: {17} 113: {30}
%e A379318 5: {3} 61: {18} 125: {3,3,3}
%e A379318 7: {4} 63: {2,2,4} 127: {31}
%e A379318 9: {2,2} 65: {3,6} 131: {32}
%e A379318 11: {5} 67: {19} 137: {33}
%e A379318 13: {6} 71: {20} 139: {34}
%e A379318 17: {7} 73: {21} 149: {35}
%e A379318 19: {8} 79: {22} 150: {1,2,3,3}
%e A379318 23: {9} 81: {2,2,2,2} 151: {36}
%e A379318 29: {10} 83: {23} 154: {1,4,5}
%e A379318 30: {1,2,3} 84: {1,1,2,4} 157: {37}
%e A379318 31: {11} 89: {24} 163: {38}
%e A379318 37: {12} 97: {25} 165: {2,3,5}
%e A379318 41: {13} 101: {26} 167: {39}
%e A379318 43: {14} 103: {27} 169: {6,6}
%e A379318 47: {15} 107: {28} 173: {40}
%e A379318 49: {4,4} 108: {1,1,2,2,2} 179: {41}
%t A379318 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A379318 Select[Range[2,100],OddQ[#]&&Divisible[Times@@prix[#],Total[prix[#]]]&]
%Y A379318 Including evens gives A326149, counted by A057568.
%Y A379318 For nonprime instead of odd we get A326150.
%Y A379318 For even instead of odd we get A379319, counted by A379320.
%Y A379318 Partitions of this type are counted by A379734, strict A379735, see A379733.
%Y A379318 For squarefree instead of odd we get A379844, even case A379845.
%Y A379318 Counting and ranking multisets by comparing sum and product:
%Y A379318 - same: A001055, ranks A301987
%Y A379318 - divisible: A057567, ranks A326155
%Y A379318 - greater than: A096276 shifted right, ranks A325038
%Y A379318 - greater or equal: A096276, ranks A325044
%Y A379318 - less than: A114324, ranks A325037, see A318029, A379720
%Y A379318 - less or equal: A319005, ranks A379721, see A025147
%Y A379318 - different: A379736, ranks A379722, see A111133
%Y A379318 Cf. A069016, A301988, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.
%K A379318 nonn
%O A379318 1,1
%A A379318 _Gus Wiseman_, Jan 16 2025