A320628
Products of primes of nonprime index.
Original entry on oeis.org
1, 2, 4, 7, 8, 13, 14, 16, 19, 23, 26, 28, 29, 32, 37, 38, 43, 46, 47, 49, 52, 53, 56, 58, 61, 64, 71, 73, 74, 76, 79, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 112, 113, 116, 122, 128, 131, 133, 137, 139, 142, 146, 148, 149, 151, 152, 158, 161, 163
Offset: 1
The sequence of terms begins:
1 = 1
2 = prime(1)
4 = prime(1)^2
7 = prime(4)
8 = prime(1)^3
13 = prime(6)
14 = prime(1)*prime(4)
16 = prime(1)^4
19 = prime(8)
23 = prime(9)
26 = prime(1)*prime(6)
28 = prime(1)^2*prime(4)
29 = prime(10)
32 = prime(1)^5
37 = prime(12)
38 = prime(1)*prime(8)
43 = prime(14)
46 = prime(1)*prime(9)
47 = prime(15)
49 = prime(4)^2
52 = prime(1)^2*prime(6)
53 = prime(16)
56 = prime(1)^3*prime(4)
58 = prime(1)*prime(10)
61 = prime(18)
64 = prime(1)^6
71 = prime(20)
73 = prime(21)
74 = prime(1)*prime(12)
76 = prime(1)^2*prime(8)
79 = prime(22)
86 = prime(1)*prime(14)
89 = prime(24)
91 = prime(4)*prime(6)
92 = prime(1)^2*prime(9)
94 = prime(1)*prime(15)
97 = prime(25)
98 = prime(1)*prime(4)^2
Primes of nonprime index are
A007821.
Products of primes of prime index are
A076610.
Products of primes of nonprime index are this sequence.
The number of prime prime indices is given by
A257994.
The number of nonprime prime indices is given by
A330944.
A330944
Number of nonprime prime indices of n.
Original entry on oeis.org
0, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 1, 2, 1, 1, 1, 3, 0, 2, 0, 3, 1, 1, 0, 5, 0, 1, 1, 2, 1, 2, 1, 3, 0, 2, 1, 2, 0, 2, 1, 4, 2, 1, 0, 3, 1, 1, 0, 4, 1, 2, 0, 2, 1, 1, 1, 6, 1, 1, 0, 2, 1, 2, 1, 3, 1, 2, 0, 3, 1, 2, 1, 4, 0, 1, 0, 3, 0, 2, 1
Offset: 1
24 has prime indices {1,1,1,2}, of which {1,1,1} are nonprime, so a(24) = 3.
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.
Numbers whose prime indices are not all prime are
A330945.
Cf.
A000040,
A000720,
A001222,
A007097,
A018252,
A056239,
A112798,
A302242,
A320629,
A320633,
A330946,
A330947.
-
Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}/;!PrimeQ[PrimePi[p]]:>k]],{n,30}]
-
a(n) = my(f=factor(n)); sum(k=1, #f~, if(!isprime(primepi(f[k,1])), f[k,2], 0)); \\ Daniel Suteu, Jan 14 2020
A330945
Numbers whose prime indices are not all prime numbers.
Original entry on oeis.org
2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84, 86, 87
Offset: 1
The sequence of terms together with their prime indices of prime indices begins:
2: {{}}
4: {{},{}}
6: {{},{1}}
7: {{1,1}}
8: {{},{},{}}
10: {{},{2}}
12: {{},{},{1}}
13: {{1,2}}
14: {{},{1,1}}
16: {{},{},{},{}}
18: {{},{1},{1}}
19: {{1,1,1}}
20: {{},{},{2}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}
26: {{},{1,2}}
28: {{},{},{1,1}}
29: {{1,3}}
Complement of
A076610 (products of primes of prime index).
Numbers n such that
A330944(n) > 0.
The restriction to odd terms is
A330946.
The restriction to nonprimes is
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 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,
A320633,
A330943,
A330947,
A330949.
A331915
Numbers with exactly one prime prime index, counted with multiplicity.
Original entry on oeis.org
3, 5, 6, 10, 11, 12, 17, 20, 21, 22, 24, 31, 34, 35, 39, 40, 41, 42, 44, 48, 57, 59, 62, 65, 67, 68, 69, 70, 77, 78, 80, 82, 83, 84, 87, 88, 95, 96, 109, 111, 114, 115, 118, 119, 124, 127, 129, 130, 134, 136, 138, 140, 141, 143, 145, 147, 154, 156, 157, 159
Offset: 1
The sequence of terms together with their prime indices begins:
3: {2} 57: {2,8} 114: {1,2,8}
5: {3} 59: {17} 115: {3,9}
6: {1,2} 62: {1,11} 118: {1,17}
10: {1,3} 65: {3,6} 119: {4,7}
11: {5} 67: {19} 124: {1,1,11}
12: {1,1,2} 68: {1,1,7} 127: {31}
17: {7} 69: {2,9} 129: {2,14}
20: {1,1,3} 70: {1,3,4} 130: {1,3,6}
21: {2,4} 77: {4,5} 134: {1,19}
22: {1,5} 78: {1,2,6} 136: {1,1,1,7}
24: {1,1,1,2} 80: {1,1,1,1,3} 138: {1,2,9}
31: {11} 82: {1,13} 140: {1,1,3,4}
34: {1,7} 83: {23} 141: {2,15}
35: {3,4} 84: {1,1,2,4} 143: {5,6}
39: {2,6} 87: {2,10} 145: {3,10}
40: {1,1,1,3} 88: {1,1,1,5} 147: {2,4,4}
41: {13} 95: {3,8} 154: {1,4,5}
42: {1,2,4} 96: {1,1,1,1,1,2} 156: {1,1,2,6}
44: {1,1,5} 109: {29} 157: {37}
48: {1,1,1,1,2} 111: {2,12} 159: {2,16}
These are numbers n such that
A257994(n) = 1.
The number of distinct prime prime indices is
A279952.
Numbers with at least one prime prime index are
A331386.
The set S of numbers with exactly one prime index in S are
A331785.
The set S of numbers with exactly one distinct prime index in S are
A331913.
Numbers with at most one prime prime index are
A331914.
Numbers with exactly one distinct prime prime index are
A331916.
Numbers with at most one distinct prime prime index are
A331995.
Cf.
A000040,
A000720,
A007097,
A007821,
A018252,
A112798,
A289509,
A320628,
A330944,
A330945,
A331784.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],_?PrimeQ]==1&]
A331386
Numbers with at least one prime prime index.
Original entry on oeis.org
3, 5, 6, 9, 10, 11, 12, 15, 17, 18, 20, 21, 22, 24, 25, 27, 30, 31, 33, 34, 35, 36, 39, 40, 41, 42, 44, 45, 48, 50, 51, 54, 55, 57, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 93, 95, 96, 99, 100, 102, 105, 108
Offset: 1
The sequence of terms together with their prime indices begins:
3: {2}
5: {3}
6: {1,2}
9: {2,2}
10: {1,3}
11: {5}
12: {1,1,2}
15: {2,3}
17: {7}
18: {1,2,2}
20: {1,1,3}
21: {2,4}
22: {1,5}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
30: {1,2,3}
31: {11}
33: {2,5}
34: {1,7}
Primes of nonprime index are
A007821.
Products of primes of prime index are
A076610.
Products of primes of nonprime index are
A320628.
The number of nonprime prime indices is given by
A330944.
Cf.
A000040,
A000720,
A001222,
A018252,
A056239,
A076610,
A112798,
A302242,
A320633,
A330943,
A330944,
A330947,
A330949.
A379315
Number of strict integer partitions of n with a unique 1 or prime part.
Original entry on oeis.org
0, 1, 1, 1, 0, 2, 1, 3, 1, 3, 2, 7, 3, 7, 4, 10, 7, 15, 7, 17, 13, 23, 16, 31, 20, 37, 31, 48, 38, 62, 48, 76, 68, 93, 80, 119, 105, 147, 137, 175, 166, 226, 208, 267, 263, 326, 322, 407, 391, 481, 492, 586, 591, 714, 714, 849, 884, 1020, 1050, 1232, 1263
Offset: 0
The a(10) = 2 through a(15) = 10 partitions:
(8,2) (11) (9,3) (13) (9,5) (8,7)
(9,1) (6,5) (10,2) (7,6) (12,2) (10,5)
(7,4) (6,4,2) (8,5) (8,4,2) (11,4)
(8,3) (10,3) (9,4,1) (12,3)
(9,2) (12,1) (14,1)
(10,1) (6,4,3) (6,5,4)
(6,4,1) (8,4,1) (8,4,3)
(8,6,1)
(9,4,2)
(10,4,1)
A376682 gives k-th differences of old primes.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]
-
seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k)) * prod(k=4, n, 1 + if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ Andrew Howroyd, Dec 28 2024
A379301
Positive integers whose prime indices include a unique composite number.
Original entry on oeis.org
7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131
Offset: 1
The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.
A066247 is the characteristic function for the composite numbers.
Other counts of prime indices:
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],CompositeQ]]==1&]
A379304
Number of integer partitions of n with a unique prime part.
Original entry on oeis.org
0, 0, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 26, 31, 41, 47, 62, 72, 93, 108, 136, 156, 199, 226, 279, 321, 398, 452, 555, 630, 767, 873, 1051, 1188, 1433, 1618, 1930, 2185, 2595, 2921, 3458, 3891, 4580, 5155, 6036, 6776, 7926, 8883, 10324, 11577, 13421, 15014
Offset: 0
The a(2) = 1 through a(9) = 9 partitions:
(2) (3) (31) (5) (42) (7) (62) (54)
(21) (211) (311) (51) (43) (71) (63)
(2111) (3111) (421) (431) (621)
(21111) (511) (4211) (711)
(31111) (5111) (4311)
(211111) (311111) (42111)
(2111111) (51111)
(3111111)
(21111111)
A095195 gives k-th differences of prime numbers.
-
Table[Length[Select[IntegerPartitions[n],Count[#,_?PrimeQ]==1&]],{n,0,30}]
A379305
Number of strict integer partitions of n with a unique prime part.
Original entry on oeis.org
0, 0, 1, 2, 1, 1, 2, 3, 3, 3, 3, 6, 8, 8, 8, 10, 12, 17, 18, 18, 22, 28, 30, 36, 40, 44, 52, 62, 67, 78, 87, 97, 113, 129, 137, 156, 177, 200, 227, 251, 271, 312, 350, 382, 425, 475, 521, 588, 648, 705, 785, 876, 957, 1061, 1164, 1272, 1411, 1558, 1693, 1866
Offset: 0
The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):
(2) (3) (31) (5) (42) (7) (62) (54) (82) (B) (93)
(21) (51) (43) (71) (63) (541) (65) (A2)
(421) (431) (621) (631) (74) (B1)
(83) (642)
(92) (651)
(821) (741)
(831)
(921)
A095195 gives k-th differences of prime numbers.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?PrimeQ]==1&]],{n,0,30}]
A379312
Positive integers whose prime indices include a unique 1 or prime number.
Original entry on oeis.org
2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214
Offset: 1
The terms together with their prime indices begin:
2: {1}
3: {2}
5: {3}
11: {5}
14: {1,4}
17: {7}
21: {2,4}
26: {1,6}
31: {11}
35: {3,4}
38: {1,8}
39: {2,6}
41: {13}
46: {1,9}
57: {2,8}
58: {1,10}
59: {17}
65: {3,6}
67: {19}
69: {2,9}
74: {1,12}
77: {4,5}
These "old" primes are listed by
A008578.
A080339 is the characteristic function for the old prime numbers.
Other counts of prime indices:
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]
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