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A367225
Numbers m without a divisor whose prime indices sum to bigomega(m).
25
3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 55, 58, 59, 61, 62, 63, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 106, 107, 109, 113
OFFSET
1,1
COMMENTS
Also numbers m whose prime indices do not have a submultiset summing to bigomega(m).
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.
These are the Heinz numbers of the partitions counted by A367213.
EXAMPLE
The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
3: {2} 29: {10} 58: {1,10}
5: {3} 31: {11} 59: {17}
7: {4} 34: {1,7} 61: {18}
10: {1,3} 35: {3,4} 62: {1,11}
11: {5} 37: {12} 63: {2,2,4}
13: {6} 38: {1,8} 65: {3,6}
14: {1,4} 41: {13} 67: {19}
17: {7} 43: {14} 68: {1,1,7}
19: {8} 44: {1,1,5} 71: {20}
22: {1,5} 46: {1,9} 73: {21}
23: {9} 47: {15} 74: {1,12}
25: {3,3} 49: {4,4} 76: {1,1,8}
26: {1,6} 52: {1,1,6} 77: {4,5}
27: {2,2,2} 53: {16} 79: {22}
28: {1,1,4} 55: {3,5} 82: {1,13}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], FreeQ[Total/@prix/@Divisors[#], PrimeOmega[#]]&]
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free
-------------------------------------------
A000700 counts self-conjugate partitions, ranks A088902.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A229816 counts partitions whose length is not a part, ranks A367107.
A237667 counts sum-free partitions, ranks A364531.
A365924 counts incomplete partitions, ranks A365830.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Sequence in context: A241561 A241008 A335126 * A173708 A239929 A246955
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 15 2023
STATUS
approved