Revision History for A371179
(Underlined text is an addition;
strikethrough text is a deletion.)
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#5 by Michael De Vlieger at Wed Mar 20 21:53:35 EDT 2024
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#4 by Gus Wiseman at Wed Mar 20 18:45:57 EDT 2024
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#3 by Gus Wiseman at Wed Mar 20 18:45:26 EDT 2024
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| NAME
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Positive integers with morefewer distinct prime factors (A001221) than distinct divisors of prime indices (A370820).
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| FORMULA
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A001221(a(n)) > )) < A370820(a(n)).
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| CROSSREFS
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The complement counting all prime factors on the LHS is A370802, counted by A371130.
Partitions of this type are counted by A371132, ranks A371179, strict A371180.
Counting all prime indices (not just distinct) on the LHS gives A371168, counted by A371173.
A305148 counts partitions containing no proper divisors ofpairwise partsindivisible (stable) partitions, ranks A316476.
Cf. A003963, ~, A239312, A285573, ~A319055, A355529, `, A370802, A370803, `A370808, ~A370813, A371130, A371165, `, A371171, `, A371172.
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#2 by Gus Wiseman at Tue Mar 19 16:56:22 EDT 2024
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| NAME
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allocatedPositive integers with more distinct prime factors (A001221) than distinct divisors forof Gusprime Wisemanindices (A370820).
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| DATA
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3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101
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| OFFSET
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1,1
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| COMMENTS
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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.
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| FORMULA
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A001221(a(n)) > A370820(a(n)).
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| EXAMPLE
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The terms together with their prime indices begin:
3: {2} 28: {1,1,4} 52: {1,1,6} 74: {1,12}
5: {3} 29: {10} 53: {16} 75: {2,3,3}
7: {4} 31: {11} 55: {3,5} 76: {1,1,8}
9: {2,2} 33: {2,5} 56: {1,1,1,4} 77: {4,5}
11: {5} 35: {3,4} 57: {2,8} 78: {1,2,6}
13: {6} 37: {12} 58: {1,10} 79: {22}
14: {1,4} 38: {1,8} 59: {17} 81: {2,2,2,2}
15: {2,3} 39: {2,6} 61: {18} 83: {23}
17: {7} 41: {13} 63: {2,2,4} 85: {3,7}
19: {8} 43: {14} 65: {3,6} 86: {1,14}
21: {2,4} 45: {2,2,3} 67: {19} 87: {2,10}
23: {9} 46: {1,9} 69: {2,9} 89: {24}
25: {3,3} 47: {15} 70: {1,3,4} 91: {4,6}
26: {1,6} 49: {4,4} 71: {20} 92: {1,1,9}
27: {2,2,2} 51: {2,7} 73: {21} 93: {2,11}
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| MATHEMATICA
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Select[Range[100], PrimeNu[#]<Length[Union @@ Divisors/@PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]&]
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| CROSSREFS
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The LHS is A001221, distinct case of A001222.
The complement counting all prime factors on the LHS is A370802, counted by A371130.
The RHS is A370820, for prime factors A303975.
Partitions of this type are counted by A371132, ranks A371179, strict A371180.
Counting all prime indices (not just distinct) on the LHS gives A371168, counted by A371173.
The complement is A371177, counted by A371178, strict A371128.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions containing no proper divisors of parts, ranks A316476.
Cf. A003963, ~A239312, A285573, ~A319055, A355529, `A370803, `A370808, ~A370813, A371165, `A371171, `A371172.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Mar 19 2024
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Thu Mar 14 04:14:09 EDT 2024
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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