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A111907
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A number n is included if the same number of primes, among primes <= the largest prime dividing n, divide n as do not.
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29
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1, 3, 9, 14, 21, 27, 28, 35, 56, 63, 78, 81, 98, 112, 130, 147, 156, 175, 182, 189, 195, 196, 224, 234, 243, 245, 260, 273, 286, 312, 364, 392, 429, 441, 448, 455, 468, 520, 567, 570, 572, 585, 624, 650, 686, 702, 715, 728, 729, 784, 798, 819, 875, 896, 936
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OFFSET
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1,2
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COMMENTS
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Also numbers whose greatest prime index (A061395) is twice their number of distinct prime factors (A001221). - Gus Wiseman, Mar 19 2023
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LINKS
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EXAMPLE
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28 is included because 7 is the largest prime dividing 28. And of the primes <= 7 (2,3,5,7), 2 and 7 (2 primes) divide 28 and 3 and 5 (also 2 primes) do not divide 28.
The terms together with their prime indices begin:
1: {}
3: {2}
9: {2,2}
14: {1,4}
21: {2,4}
27: {2,2,2}
28: {1,1,4}
35: {3,4}
56: {1,1,1,4}
63: {2,2,4}
78: {1,2,6}
81: {2,2,2,2}
98: {1,4,4}
112: {1,1,1,1,4}
130: {1,3,6}
147: {2,4,4}
156: {1,1,2,6}
For example, 156 is included because it has prime indices {1,1,2,6}, with distinct parts {1,2,6} and distinct non-parts {3,4,5}, both of length 3. Alternatively, 156 has greatest prime index 6 and omega 3, and 6 = 2*3.
(End)
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MATHEMATICA
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Select[Range[100], 2*PrimeNu[#]==PrimePi[FactorInteger[#][[-1, 1]]]&] (* Gus Wiseman, Mar 19 2023 *)
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PROG
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(PARI) {m=950; v=vector(m); for(n=1, m, f=factor(n)[, 1]~; c=0; pc=0; forprime(p=2, vecmax(f), j=1; s=length(f); while(j<=s&&p!=f[j], j++); if(j<=s, c++); pc++); v[n]=sign(pc-2*c)); for(n=1, m, if(v[n]==0, print1(n, ", ")))} (Klaus Brockhaus)
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CROSSREFS
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For length instead of maximum we have A067801.
These partitions are counted by A239959.
A061395 gives greatest prime index.
Comparing twice the number of distinct parts to greatest part:
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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