%I #6 Mar 16 2024 21:41:30

%S 3,5,7,11,13,14,15,17,19,21,23,26,29,31,33,35,37,38,39,41,43,46,47,49,

%T 51,52,53,55,57,58,59,61,65,67,69,70,71,73,74,76,77,78,79,83,85,86,87,

%U 89,91,93,94,95,97,101,103,105,106,107,109,111,113,114,115

%N Positive integers with fewer prime factors (A001222) than distinct divisors of prime indices (A370820).

%C 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.

%F A001222(a(n)) < A370820(a(n)).

%e The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.

%e The terms together with their prime indices begin:

%e 3: {2} 35: {3,4} 59: {17} 86: {1,14}

%e 5: {3} 37: {12} 61: {18} 87: {2,10}

%e 7: {4} 38: {1,8} 65: {3,6} 89: {24}

%e 11: {5} 39: {2,6} 67: {19} 91: {4,6}

%e 13: {6} 41: {13} 69: {2,9} 93: {2,11}

%e 14: {1,4} 43: {14} 70: {1,3,4} 94: {1,15}

%e 15: {2,3} 46: {1,9} 71: {20} 95: {3,8}

%e 17: {7} 47: {15} 73: {21} 97: {25}

%e 19: {8} 49: {4,4} 74: {1,12} 101: {26}

%e 21: {2,4} 51: {2,7} 76: {1,1,8} 103: {27}

%e 23: {9} 52: {1,1,6} 77: {4,5} 105: {2,3,4}

%e 26: {1,6} 53: {16} 78: {1,2,6} 106: {1,16}

%e 29: {10} 55: {3,5} 79: {22} 107: {28}

%e 31: {11} 57: {2,8} 83: {23} 109: {29}

%e 33: {2,5} 58: {1,10} 85: {3,7} 111: {2,12}

%t Select[Range[100],PrimeOmega[#]<Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

%Y The opposite version is A370348 counted by A371171.

%Y The version for equality is A370802, counted by A371130, strict A371128.

%Y The RHS is A370820, for prime factors instead of divisors A303975.

%Y For divisors instead of prime factors on the LHS we get A371166.

%Y The complement is counted by A371169.

%Y The weak version is A371170.

%Y Partitions of this type are counted by A371173.

%Y Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).

%Y A000005 counts divisors.

%Y A001221 counts distinct prime factors.

%Y A027746 lists prime factors, indices A112798, length A001222.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y Cf. A003963, A319899, A355737, A355739, A355741, A370808, A370814, A371127.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 16 2024