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.
A339113
Products of primes of squarefree semiprime index (A322551).
Original entry on oeis.org
1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 169, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 841, 907, 929, 937
Offset: 1
The sequence of terms together with the corresponding multigraphs begins:
1: {} 233: {{2,7}} 487: {{2,11}}
13: {{1,2}} 257: {{3,5}} 491: {{1,15}}
29: {{1,3}} 269: {{2,8}} 499: {{3,8}}
43: {{1,4}} 271: {{1,10}} 559: {{1,2},{1,4}}
47: {{2,3}} 293: {{1,11}} 577: {{1,16}}
73: {{2,4}} 313: {{3,6}} 607: {{2,12}}
79: {{1,5}} 347: {{2,9}} 611: {{1,2},{2,3}}
101: {{1,6}} 373: {{1,12}} 631: {{3,9}}
137: {{2,5}} 377: {{1,2},{1,3}} 647: {{1,17}}
139: {{1,7}} 389: {{4,5}} 653: {{4,7}}
149: {{3,4}} 421: {{1,13}} 673: {{1,18}}
163: {{1,8}} 439: {{3,7}} 677: {{2,13}}
167: {{2,6}} 443: {{1,14}} 727: {{2,14}}
169: {{1,2},{1,2}} 449: {{2,10}} 751: {{4,8}}
199: {{1,9}} 467: {{4,6}} 757: {{1,19}}
These primes (of squarefree semiprime index) are listed by
A322551.
The strict (squarefree) case is
A309356.
The prime instead of squarefree semiprime version:
The nonprime instead of squarefree semiprime version:
The semiprime instead of squarefree semiprime version:
A002100 counts partitions into squarefree semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320911 lists products of squarefree semiprimes (Heinz numbers of
A338914).
A339561 lists products of distinct squarefree semiprimes (ranking:
A339560).
MM-numbers:
A255397 (normal),
A302478 (set multisystems),
A320630 (set multipartitions),
A302494 (sets of sets),
A305078 (connected),
A316476 (antichains),
A318991 (chains),
A320456 (covers),
A328514 (connected sets of sets),
A329559 (clutters),
A340019 (half-loop graphs).
-
sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]
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&]
A379308
Number of integer partitions of n with a unique squarefree part.
Original entry on oeis.org
0, 1, 1, 1, 0, 2, 2, 2, 0, 3, 5, 5, 1, 6, 9, 9, 2, 10, 14, 18, 6, 18, 24, 30, 11, 28, 39, 47, 24, 48, 63, 76, 41, 74, 95, 118, 65, 120, 149, 181, 107, 181, 221, 266, 169, 266, 335, 398, 262, 394, 487, 578, 391, 578, 697, 844, 592, 834, 997, 1198, 867
Offset: 0
The a(1) = 1 through a(11) = 5 partitions:
(1) (2) (3) . (5) (6) (7) . (5,4) (10) (11)
(4,1) (4,2) (4,3) (8,1) (6,4) (7,4)
(4,4,1) (8,2) (8,3)
(9,1) (9,2)
(4,4,2) (4,4,3)
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf.
A000586,
A000607,
A002095,
A013928,
A023895,
A034891,
A072284,
A073247,
A120327,
A175804,
A376657,
A377430.
-
Table[Length[Select[IntegerPartitions[n],Count[#,_?SquareFreeQ]==1&]],{n,0,30}]
A379309
Number of strict integer partitions of n with a unique squarefree part.
Original entry on oeis.org
0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 4, 4, 1, 4, 7, 7, 2, 6, 8, 11, 4, 9, 13, 17, 7, 13, 20, 22, 13, 20, 29, 33, 21, 29, 40, 47, 27, 41, 56, 64, 42, 59, 77, 85, 60, 74, 104, 115, 83, 101, 141, 155, 113, 138, 179, 206, 156, 183, 236, 272, 212, 239, 309, 343, 282, 315
Offset: 0
The a(9) = 2 through a(15) = 7 partitions:
(5,4) (10) (11) (9,3) (13) (14) (15)
(8,1) (6,4) (7,4) (8,5) (8,6) (8,7)
(8,2) (8,3) (12,1) (9,5) (9,6)
(9,1) (9,2) (8,4,1) (10,4) (11,4)
(12,2) (12,3)
(8,4,2) (8,4,3)
(9,4,1) (9,4,2)
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf.
A000586,
A000607,
A002095,
A023895,
A034891,
A036497,
A072284,
A073247,
A096258,
A204389,
A377430.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]
-
lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ Jinyuan Wang, Feb 21 2025
A339112
Products of primes of semiprime index (A106349).
Original entry on oeis.org
1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647
Offset: 1
The sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):
1: 149: (34) 313: (36)
7: (11) 161: (11)(22) 329: (11)(23)
13: (12) 163: (18) 343: (11)(11)(11)
23: (22) 167: (26) 347: (29)
29: (13) 169: (12)(12) 373: (1C)
43: (14) 199: (19) 377: (12)(13)
47: (23) 203: (11)(13) 389: (45)
49: (11)(11) 227: (44) 421: (1D)
73: (24) 233: (27) 439: (37)
79: (15) 257: (35) 443: (1E)
91: (11)(12) 269: (28) 449: (2A)
97: (33) 271: (1A) 467: (46)
101: (16) 293: (1B) 487: (2B)
137: (25) 299: (12)(22) 491: (1F)
139: (17) 301: (11)(14) 499: (38)
These primes (of semiprime index) are listed by
A106349.
The strict (squarefree) case is
A340020.
The prime instead of semiprime version:
The nonprime instead of semiprime version:
The squarefree semiprime instead of semiprime version:
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A101048 counts partitions into semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320892 lists even-omega non-products of distinct semiprimes.
A320911 lists products of squarefree semiprimes (Heinz numbers of
A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of
A338916).
MM-numbers:
A255397 (normal),
A302478 (set multisystems),
A320630 (set multipartitions),
A302494 (sets of sets),
A305078 (connected),
A316476 (antichains),
A318991 (chains),
A320456 (covers),
A328514 (connected sets of sets),
A329559 (clutters),
A340019 (half-loop graphs).
-
N:= 1000: # for terms up to N
SP:= {}: p:= 1:
for i from 1 do
p:= nextprime(p);
if 2*p > N then break fi;
Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
SP:= SP union Q;
od:
SP:= sort(convert(SP,list)):
PSP:= map(ithprime,SP):
R:= {1}:
for p in PSP do
Rp:= {}:
for k from 1 while p^k <= N do
Rpk:= select(`<=`,R, N/p^k);
Rp:= Rp union map(`*`,Rpk, p^k);
od;
R:= R union Rp;
od:
sort(convert(R,list)); # Robert Israel, Nov 03 2024
-
semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]
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&]
A302491
Prime numbers of squarefree index.
Original entry on oeis.org
2, 3, 5, 11, 13, 17, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 101, 109, 113, 127, 137, 139, 149, 157, 163, 167, 179, 181, 191, 199, 211, 233, 241, 257, 269, 271, 277, 283, 293, 313, 317, 331, 347, 349, 353, 367, 373, 389, 397, 401, 421, 431, 439, 443, 449, 461
Offset: 1
Cf.
A000040,
A000961,
A001222,
A005117,
A007097,
A056239,
A063834,
A275024,
A281113,
A302242,
A302478.
-
map(ithprime, select(numtheory:-issqrfree, [$1..500])); # Robert Israel, Nov 06 2023
-
Prime/@Select[Range[100],SquareFreeQ]
-
forprime(p=1, 500, if(issquarefree(primepi(p)), print1(p, ", "))) \\ Felix Fröhlich, Apr 10 2018
-
list(lim)=my(v=List(),k); forprime(p=2,lim\1, if(issquarefree(k++), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 03 2023
A379316
Positive integers whose prime indices include a unique squarefree number.
Original entry on oeis.org
2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211
Offset: 1
The terms together with their prime indices begin:
2: {1}
3: {2}
5: {3}
11: {5}
13: {6}
14: {1,4}
17: {7}
21: {2,4}
29: {10}
31: {11}
35: {3,4}
38: {1,8}
41: {13}
43: {14}
46: {1,9}
A008966 is the characteristic function for the squarefree 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[#],SquareFreeQ]]==1&]
A382201
MM-numbers of sets of sets with distinct sums.
Original entry on oeis.org
1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 39, 41, 43, 47, 51, 55, 58, 59, 62, 65, 66, 67, 73, 78, 79, 82, 83, 85, 86, 87, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 129, 130, 134, 137, 139, 141, 145, 146, 149, 155, 157, 158, 163, 165
Offset: 1
The terms together with their prime indices of prime indices begin:
1: {}
2: {{}}
3: {{1}}
5: {{2}}
6: {{},{1}}
10: {{},{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
22: {{},{3}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
39: {{1},{1,2}}
Set partitions of this type are counted by
A275780.
Twice-partitions of this type are counted by
A279785.
For just sets of sets we have
A302478.
For distinct blocks instead of block-sums we have
A302494.
For equal instead of distinct sums we have
A302497.
For just distinct sums we have
A326535.
Cf.
A000720,
A003963,
A005117,
A007716,
A293511,
A302242,
A319899,
A326534,
A368100,
A368101,
A381635,
A382215.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@SquareFreeQ/@prix[#]&&UnsameQ@@Total/@prix/@prix[#]&]
Showing 1-10 of 53 results.
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