A331785
Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.
Original entry on oeis.org
1, 2, 3, 5, 11, 14, 21, 26, 31, 34, 35, 38, 39, 43, 46, 51, 57, 58, 65, 69, 73, 74, 77, 82, 85, 87, 94, 95, 98, 101, 106, 111, 115, 118, 122, 123, 127, 134, 139, 141, 142, 143, 145, 147, 149, 158, 159, 163, 166, 167, 177, 178, 182, 183, 185, 187, 191, 194, 199
Offset: 1
The sequence of terms together with their prime indices begins:
1: {} 73: {21} 142: {1,20} 205: {3,13}
2: {1} 74: {1,12} 143: {5,6} 206: {1,27}
3: {2} 77: {4,5} 145: {3,10} 209: {5,8}
5: {3} 82: {1,13} 147: {2,4,4} 213: {2,20}
11: {5} 85: {3,7} 149: {35} 214: {1,28}
14: {1,4} 87: {2,10} 158: {1,22} 217: {4,11}
21: {2,4} 94: {1,15} 159: {2,16} 218: {1,29}
26: {1,6} 95: {3,8} 163: {38} 226: {1,30}
31: {11} 98: {1,4,4} 166: {1,23} 233: {51}
34: {1,7} 101: {26} 167: {39} 235: {3,15}
35: {3,4} 106: {1,16} 177: {2,17} 237: {2,22}
38: {1,8} 111: {2,12} 178: {1,24} 238: {1,4,7}
39: {2,6} 115: {3,9} 182: {1,4,6} 245: {3,4,4}
43: {14} 118: {1,17} 183: {2,18} 249: {2,23}
46: {1,9} 122: {1,18} 185: {3,12} 253: {5,9}
51: {2,7} 123: {2,13} 187: {5,7} 262: {1,32}
57: {2,8} 127: {31} 191: {43} 265: {3,16}
58: {1,10} 134: {1,19} 194: {1,25} 266: {1,4,8}
65: {3,6} 139: {34} 199: {46} 267: {2,24}
69: {2,9} 141: {2,15} 201: {2,19} 269: {57}
For example, the prime indices of 77 are {4,5}, of which only 5 is in the sequence, so 77 is in the sequence.
Numbers S without all prime indices in S are
A324694.
Numbers S without any prime indices in S are
A324695.
Numbers S with at most one prime index in S are
A331784.
Numbers S with at most one distinct prime index in S are
A331912.
Numbers S with exactly one distinct prime index in S are
A331913.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aQ[n_]:=n==1||Length[Select[primeMS[n],aQ]]==1;
Select[Range[100],aQ]
A330946
Odd numbers whose prime indices are not all prime numbers.
Original entry on oeis.org
7, 13, 19, 21, 23, 29, 35, 37, 39, 43, 47, 49, 53, 57, 61, 63, 65, 69, 71, 73, 77, 79, 87, 89, 91, 95, 97, 101, 103, 105, 107, 111, 113, 115, 117, 119, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 159, 161, 163, 167, 169, 171, 173, 175, 181, 183, 185
Offset: 1
The sequence of terms together with their prime indices of prime indices begins:
7: {{1,1}}
13: {{1,2}}
19: {{1,1,1}}
21: {{1},{1,1}}
23: {{2,2}}
29: {{1,3}}
35: {{2},{1,1}}
37: {{1,1,2}}
39: {{1},{1,2}}
43: {{1,4}}
47: {{2,3}}
49: {{1,1},{1,1}}
53: {{1,1,1,1}}
57: {{1},{1,1,1}}
61: {{1,2,2}}
63: {{1},{1},{1,1}}
65: {{2},{1,2}}
69: {{1},{2,2}}
71: {{1,1,3}}
73: {{2,4}}
Odd numbers n such that
A330944(n) > 0.
Including even numbers gives
A330945.
The restriction to nonprimes is
A330949.
Taking nonprimes instead of odds gives
A330947.
The number of prime prime indices is given by
A257994.
Primes of nonprime index are
A007821.
Products of primes of prime index are
A076610.
Products of primes of nonprime index are
A320628.
The set S of numbers whose prime indices do not all belong to S is
A324694.
Cf.
A000040,
A000720,
A001222,
A018252,
A056239,
A112798,
A302242,
A320629,
A330943,
A330947,
A330948.
-
Select[Range[1,100,2],!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]
A324739
Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.
Original entry on oeis.org
1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 312, 624, 936, 1872, 3744, 7488, 12480, 24960, 37440, 74880, 142848, 285696, 456192, 912384, 1548288, 3096576, 5308416, 10616832, 15925248, 31850496, 51978240, 103956480, 200835072, 401670144, 771489792, 1542979584, 2314469376
Offset: 1
The a(1) = 1 through a(6) = 20 subsets:
{} {} {} {} {} {}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{4} {4} {4}
{2,4} {5} {5}
{3,4} {2,4} {6}
{2,5} {2,4}
{3,4} {2,5}
{4,5} {2,6}
{2,4,5} {3,4}
{3,6}
{4,5}
{4,6}
{5,6}
{2,4,5}
{2,4,6}
{2,5,6}
{3,4,6}
{4,5,6}
{2,4,5,6}
The maximal case is
A324762. The case of subsets of {1...n} is
A324738. The strict integer partition version is
A324750. The integer partition version is
A324755. The Heinz number version is
A324760. An infinite version is
A324694.
Cf.
A000720,
A001221,
A001462,
A007097,
A084422,
A085945,
A112798,
A276625,
A279861,
A290689,
A290822,
A304360,
A306844.
-
Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,10}]
-
pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019
A324749
Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 4, 3, 4, 6, 6, 8, 11, 10, 14, 14, 19, 21, 26, 28, 35, 38, 44, 50, 60, 65, 79, 88, 98, 113, 131, 144, 165, 185, 211, 234, 268, 297, 334, 374, 420, 470, 525, 584, 649, 727, 801, 902, 998, 1100, 1220, 1357, 1500, 1657, 1833, 2029, 2220, 2462
Offset: 0
The a(0) = 1 through a(10) = 6 strict integer partitions:
() (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
(3,1) (4,2) (4,3) (6,2) (5,4) (6,4)
(5,1) (5,2) (7,1) (6,3) (7,3)
(6,1) (7,2) (8,2)
(9,1)
(6,3,1)
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]
A331913
Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 26, 27, 31, 32, 39, 49, 52, 53, 58, 59, 64, 65, 67, 74, 81, 82, 83, 86, 87, 91, 94, 97, 101, 103, 104, 111, 116, 117, 121, 122, 123, 125, 127, 128, 129, 131, 141, 142, 143, 145, 146, 148, 158, 164, 167, 172, 178
Offset: 1
The sequence of terms together with their prime indices begins:
1: {} 52: {1,1,6} 116: {1,1,10}
2: {1} 53: {16} 117: {2,2,6}
3: {2} 58: {1,10} 121: {5,5}
4: {1,1} 59: {17} 122: {1,18}
5: {3} 64: {1,1,1,1,1,1} 123: {2,13}
7: {4} 65: {3,6} 125: {3,3,3}
8: {1,1,1} 67: {19} 127: {31}
9: {2,2} 74: {1,12} 128: {1,1,1,1,1,1,1}
11: {5} 81: {2,2,2,2} 129: {2,14}
16: {1,1,1,1} 82: {1,13} 131: {32}
17: {7} 83: {23} 141: {2,15}
19: {8} 86: {1,14} 142: {1,20}
23: {9} 87: {2,10} 143: {5,6}
25: {3,3} 91: {4,6} 145: {3,10}
26: {1,6} 94: {1,15} 146: {1,21}
27: {2,2,2} 97: {25} 148: {1,1,12}
31: {11} 101: {26} 158: {1,22}
32: {1,1,1,1,1} 103: {27} 164: {1,1,13}
39: {2,6} 104: {1,1,1,6} 167: {39}
49: {4,4} 111: {2,12} 172: {1,1,14}
Numbers S without all prime indices in S are
A324694.
Numbers S without any prime indices in S are
A324695.
Numbers S with at most one prime index in S are
A331784.
Numbers S with exactly one prime index in S are
A331785.
Numbers S with at most one distinct prime index in S are
A331912.
A330949
Odd nonprime numbers whose prime indices are not all prime numbers.
Original entry on oeis.org
21, 35, 39, 49, 57, 63, 65, 69, 77, 87, 91, 95, 105, 111, 115, 117, 119, 129, 133, 141, 143, 145, 147, 159, 161, 169, 171, 175, 183, 185, 189, 195, 203, 207, 209, 213, 215, 217, 219, 221, 231, 235, 237, 245, 247, 253, 259, 261, 265, 267, 273, 285, 287, 291
Offset: 1
The sequence of terms together with their prime indices of prime indices begins:
21: {{1},{1,1}}
35: {{2},{1,1}}
39: {{1},{1,2}}
49: {{1,1},{1,1}}
57: {{1},{1,1,1}}
63: {{1},{1},{1,1}}
65: {{2},{1,2}}
69: {{1},{2,2}}
77: {{1,1},{3}}
87: {{1},{1,3}}
91: {{1,1},{1,2}}
95: {{2},{1,1,1}}
105: {{1},{2},{1,1}}
111: {{1},{1,1,2}}
115: {{2},{2,2}}
117: {{1},{1},{1,2}}
119: {{1,1},{4}}
129: {{1},{1,4}}
133: {{1,1},{1,1,1}}
141: {{1},{2,3}}
Including even numbers gives
A330948.
The number of prime prime indices is given by
A257994.
The number of nonprime prime indices is given by
A330944.
Primes of nonprime index are
A007821.
Products of primes of prime index are
A076610.
Products of primes of nonprime index are
A320628.
The set S of numbers whose prime indices do not all belong to S is
A324694.
Cf.
A000040,
A000720,
A001222,
A018252,
A056239,
A112798,
A302242,
A320629,
A320633,
A330943,
A330947.
-
Select[Range[1,100,2],!PrimeQ[#]&&!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]
A324839
Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.
Original entry on oeis.org
1, 0, 1, 1, 2, 3, 8, 16, 35, 74, 166, 367, 831, 1878, 4299, 9857, 22775, 52777, 122957, 287337
Offset: 1
The a(1) = 1 through a(8) = 16 rooted identity trees:
o ((o)) (((o))) ((o(o))) (((o(o)))) ((o)(o(o))) (((o))(o(o)))
((((o)))) ((o((o)))) ((o(o(o)))) (((o)(o(o))))
(((((o))))) ((((o(o))))) (((o(o(o)))))
(((o)((o)))) ((o)((o(o))))
(((o((o))))) ((o)(o((o))))
((o)(((o)))) ((o((o(o)))))
((o(((o))))) ((o(o)((o))))
((((((o)))))) ((o(o((o)))))
(((((o(o))))))
((((o)((o)))))
((((o((o))))))
(((o)(((o)))))
(((o(((o))))))
((o)((((o)))))
((o((((o))))))
(((((((o)))))))
Cf.
A324694,
A324696,
A324704,
A324738,
A324744,
A324758,
A324759,
A324767,
A324770,
A324771,
A324838,
A324840,
A324844,
A324846.
-
idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],And@@Table[!SubsetQ[#,b],{b,#}]&]],{n,10}]
A324845
Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 38, 40, 43, 44, 46, 49, 50, 51, 53, 57, 58, 59, 62, 63, 64, 67, 68, 69, 70, 71, 73, 76, 77, 79, 80, 81, 83, 85, 86, 87, 88, 92, 93, 95, 97, 98, 99, 100, 103, 106
Offset: 1
The sequence of terms together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
7: ((oo))
8: (ooo)
9: ((o)(o))
10: (o((o)))
11: ((((o))))
14: (o(oo))
16: (oooo)
17: (((oo)))
19: ((ooo))
20: (oo((o)))
21: ((o)(oo))
22: (o(((o))))
23: (((o)(o)))
25: (((o))((o)))
27: ((o)(o)(o))
Cf.
A324694,
A324738,
A324744,
A324749,
A324754,
A324759,
A324765,
A324768,
A324838,
A324842,
A324844,
A324846,
A324847,
A324849.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[!Divisible[n,x],{x,DeleteCases[primeMS[n],1]}],And@@qaQ/@primeMS[n]];
Select[Range[100],qaQ]
A306719
Lexicographically earliest sequence containing 2 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.
Original entry on oeis.org
2, 4, 8, 10, 20, 22, 28, 30, 50, 58, 64, 72, 80, 82, 88, 108, 114, 134, 148, 172, 190, 204, 214, 230, 238, 244, 262, 272, 312, 322, 340, 344, 360, 362, 400, 410, 422, 442, 458, 498, 514, 552, 554, 568, 594, 610, 620, 640, 688, 712, 730, 750, 758, 784, 792, 814
Offset: 1
Cf.
A000002,
A000720,
A001222,
A001462,
A007097,
A055396,
A061395,
A079000,
A079254,
A109298,
A112798,
A276625,
A277098,
A304360.
Cf.
A324694,
A324695,
A324696,
A324697,
A324698,
A324700,
A324701,
A324702,
A324703,
A324704,
A324705.
-
aQ[n_]:=Switch[n,0,False,2,True,,And@@Cases[FactorInteger[n-1],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[0,100],aQ]
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