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%I #11 Mar 23 2024 22:12:57
%S 7,13,19,23,29,37,43,47,53,61,71,73,74,79,89,91,95,97,101,103,106,107,
%T 111,113,122,131,137,139,141,142,143,145,149,151,159,161,163,167,169,
%U 173,178,181,183,185,193,197,199,203,209,213,214,215,219,221,223,226
%N Positive integers with fewer divisors (A000005) 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 A000005(a(n)) < A370820(a(n)).
%e The terms together with their prime indices begin:
%e 7: {4} 101: {26} 163: {38} 223: {48}
%e 13: {6} 103: {27} 167: {39} 226: {1,30}
%e 19: {8} 106: {1,16} 169: {6,6} 227: {49}
%e 23: {9} 107: {28} 173: {40} 229: {50}
%e 29: {10} 111: {2,12} 178: {1,24} 233: {51}
%e 37: {12} 113: {30} 181: {42} 239: {52}
%e 43: {14} 122: {1,18} 183: {2,18} 247: {6,8}
%e 47: {15} 131: {32} 185: {3,12} 251: {54}
%e 53: {16} 137: {33} 193: {44} 257: {55}
%e 61: {18} 139: {34} 197: {45} 259: {4,12}
%e 71: {20} 141: {2,15} 199: {46} 262: {1,32}
%e 73: {21} 142: {1,20} 203: {4,10} 263: {56}
%e 74: {1,12} 143: {5,6} 209: {5,8} 265: {3,16}
%e 79: {22} 145: {3,10} 213: {2,20} 267: {2,24}
%e 89: {24} 149: {35} 214: {1,28} 269: {57}
%e 91: {4,6} 151: {36} 215: {3,14} 271: {58}
%e 95: {3,8} 159: {2,16} 219: {2,21} 281: {60}
%e 97: {25} 161: {4,9} 221: {6,7} 293: {62}
%t Select[Range[100],Length[Divisors[#]] < Length[Union@@Divisors/@PrimePi/@First/@FactorInteger[#]]&]
%Y The RHS is A370820, for prime factors instead of divisors A303975.
%Y For (equal to) instead of (less than) we have A371165, counted by A371172.
%Y For (greater than) instead of (less than) we have A371167.
%Y For prime factors on the LHS we get A371168, counted by A371173.
%Y Other equalities: A319899, A370802 (A371130), A371128, A371177 (A371178).
%Y Other inequalities: A370348 (A371171), A371169, A371170.
%Y A000005 counts divisors.
%Y A001221 counts distinct prime factors.
%Y A027746 lists prime factors, A112798 indices, length A001222.
%Y A239312 counts divisor-choosable partitions, ranks A368110.
%Y A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370320 counts non-divisor-choosable partitions, ranks A355740.
%Y A370814 counts divisor-choosable factorizations, complement A370813.
%Y Cf. A000792, A003963, A355737, A355739, A355741, A368100, A370808, A371127.
%K nonn
%O 1,1
%A _Gus Wiseman_, Mar 14 2024
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