Displaying 1-10 of 24 results found.
Number of graphs with loops spanning n labeled vertices.
+10
68
1, 1, 5, 45, 809, 28217, 1914733, 254409765, 66628946641, 34575388318705, 35680013894626133, 73392583417010454429, 301348381381966079690489, 2471956814761996896091805993, 40530184362443281653842556898237, 1328619783326799871943604598592805525
COMMENTS
The span of a graph is the union of its edges.
FORMULA
Inverse binomial transform of A006125(n+1) = 2^binomial(n+1,2).
EXAMPLE
The a(2) = 5 edge-sets:
{{1,2}}
{{1,1},{1,2}}
{{1,1},{2,2}}
{{1,2},{2,2}}
{{1,1},{1,2},{2,2}}
MATHEMATICA
Table[Sum[(-1)^(n-k)*Binomial[n, k]*2^Binomial[k+1, 2], {k, 0, n}], {n, 10}]
(* second program *)
Table[Select[Expand[Product[1+x[i]*x[j], {j, n}, {i, j}]], And@@Table[!FreeQ[#, x[i]], {i, n}]&]/.x[_]->1, {n, 7}]
PROG
(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k+1, 2)) \\ Andrew Howroyd, Jan 06 2024
Numbers whose multiset multisystem spans an initial interval of positive integers.
+10
40
1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117
COMMENTS
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. The n-th multiset multisystem is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the 78th multiset multisystem is {{},{1},{1,2}}.
EXAMPLE
The sequence of terms together with their multiset multisystems begins:
1: {}
2: {{}}
3: {{1}}
4: {{},{}}
6: {{},{1}}
7: {{1,1}}
8: {{},{},{}}
9: {{1},{1}}
12: {{},{},{1}}
13: {{1,2}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
18: {{},{1},{1}}
19: {{1,1,1}}
21: {{1},{1,1}}
24: {{},{},{},{1}}
26: {{},{1,2}}
27: {{1},{1},{1}}
28: {{},{},{1,1}}
30: {{},{1},{2}}
32: {{},{},{},{},{}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];
Select[Range[100], normQ[primeMS/@primeMS[#]]&]
CROSSREFS
Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.
Number of unlabeled graphs with loops spanning n vertices.
+10
37
1, 1, 4, 14, 70, 454, 4552, 74168, 2129348, 111535148, 10812483376, 1945437208224, 650378721156736, 404749938336404704, 470163239887698967104, 1022592854829028311090816, 4177826139658552046627175072, 32163829440870460348768023969632
COMMENTS
The span of a graph is the union of its edges. The not necessarily spanning case is A000666.
MATHEMATICA
Table[Sum[2^PermutationCycles[Ordering[Map[Sort, Select[Tuples[Range[n], 2], OrderedQ]/.Rule@@@Table[{i, prm[[i]]}, {i, n}], {1}]], Length], {prm, Permutations[Range[n]]}]/n!, {n, 0, 8}]//Differences (* Mathematica 8.0+ *)
PROG
(Python)
from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A322700(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum(((q>>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))-sum(Fraction(1<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum(((q>>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()), prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n-1))) if n else 1 # Chai Wah Wu, Jul 14 2024
Number of non-loop-graphical integer partitions of 2n.
+10
20
0, 0, 1, 3, 7, 14, 28, 51, 91, 156, 260, 425, 680, 1068, 1654, 2524, 3802, 5668, 8350, 12190, 17634, 25306, 36011, 50902, 71441, 99642
COMMENTS
An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with equal source and target. See A339657 for the Heinz numbers, and A339656 for the complement. The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.
EXAMPLE
The a(2) = 1 through a(5) = 14 partitions (A = 10):
(4) (6) (8) (A)
(4,2) (4,4) (5,5)
(5,1) (5,3) (6,4)
(6,2) (7,3)
(7,1) (8,2)
(5,2,1) (9,1)
(6,1,1) (5,3,2)
(5,4,1)
(6,2,2)
(6,3,1)
(7,2,1)
(8,1,1)
(6,2,1,1)
(7,1,1,1)
For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:
{{1,1},{1,1},{1,2},{2,2},{3,3}}
{{1,1},{1,1},{1,2},{2,3},{2,3}}
{{1,1},{1,1},{1,3},{2,2},{2,3}}
{{1,1},{1,2},{1,2},{1,2},{3,3}}
{{1,1},{1,2},{1,2},{1,3},{2,3}}
{{1,1},{1,2},{1,3},{1,3},{2,2}}
{{1,2},{1,2},{1,2},{1,3},{1,3}},
but since none of these is a loop-graph (because they are not strict), y is counted under a(5).
MATHEMATICA
spsbin[{}]:={{}}; spsbin[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsbin[Complement[set, s]]]/@Cases[Subsets[set], {i, _}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n], Select[mpsbin[#], UnsameQ@@#&]=={}&]], {n, 0, 5}]
CROSSREFS
A062740 counts labeled connected loop-graphs.
A101048 counts partitions into semiprimes.
A322661 counts covering loop-graphs.
A320655 counts factorizations into semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A339655 (this sequence) counts non-loop-graphical partitions of 2n ( A339657). The following count partitions of even length and give their Heinz numbers:
Cf. A007717, A025065, A147878, A320732, A320921, A338898, A338902, A338912, A338913, A339659, A339660, A339662.
Number of loop-graphical integer partitions of 2n.
+10
20
1, 2, 4, 8, 15, 28, 49, 84, 140, 229, 367, 577, 895, 1368, 2064, 3080, 4547, 6642, 9627, 13825, 19704, 27868, 39164, 54656, 75832, 104584
COMMENTS
An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement. The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the unordered prime signature of n is loop-graphical.
EXAMPLE
The a(0) = 1 through a(4) = 15 partitions:
() (2) (2,2) (3,3) (3,3,2)
(1,1) (3,1) (2,2,2) (4,2,2)
(2,1,1) (3,2,1) (4,3,1)
(1,1,1,1) (4,1,1) (2,2,2,2)
(2,2,1,1) (3,2,2,1)
(3,1,1,1) (3,3,1,1)
(2,1,1,1,1) (4,2,1,1)
(1,1,1,1,1,1) (5,1,1,1)
(2,2,2,1,1)
(3,2,1,1,1)
(4,1,1,1,1)
(2,2,1,1,1,1)
(3,1,1,1,1,1)
(2,1,1,1,1,1,1)
(1,1,1,1,1,1,1,1)
For example, there are four possible loop-graphs with degrees y = (2,2,1,1), namely
{{1,1},{2,2},{3,4}}
{{1,1},{2,3},{2,4}}
{{1,2},{1,3},{2,4}}
{{1,2},{1,4},{2,3}}
{{1,3},{1,4},{2,2}},
so y is counted under a(3). On the other hand, there are two possible loop-multigraphs with degrees z = (4,2), namely
{{1,1},{1,1},{2,2}}
{{1,1},{1,2},{1,2}},
but neither of these is a loop-graph, so z is not counted under a(3).
MATHEMATICA
spsbin[{}]:={{}}; spsbin[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsbin[Complement[set, s]]]/@Cases[Subsets[set], {i, _}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n], Select[mpsbin[#], UnsameQ@@#&]!={}&]], {n, 0, 5}]
CROSSREFS
A062740 counts labeled connected loop-graphs.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A322661 counts covering loop-graphs.
The following count vertex-degree partitions and give their Heinz numbers:
- A321728 is conjectured to count non-half-loop-graphical partitions of n. The following count partitions of even length and give their Heinz numbers:
Cf. A001055, A001222, A025065, A095268, A101048, A320656, A320921, A338902, A338912, A338913, A339659.
Products of primes of squarefree semiprime index ( A322551).
+10
19
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
COMMENTS
A squarefree semiprime ( A006881) is a product of any two distinct prime numbers. Also MM-numbers of labeled multigraphs (without uncovered vertices). 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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
EXAMPLE
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}}
MATHEMATICA
sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000], FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; !sqfsemiQ[PrimePi[p]]]&]
CROSSREFS
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).
Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.
+10
15
1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0
COMMENTS
A semiprime ( A001358) is a product of any two prime numbers. In the even case, these factorizations have A001222(n)/2 factors. - Gus Wiseman, Dec 31 2020 Records 1, 2, 3, 4, 5, 9, 13, 15, 17, ... occur at 1, 60, 210, 840, 1260, 4620, 27720, 30030, 69300, ...
EXAMPLE
a(4) = 1, as there is just one way to factor 4 into distinct semiprimes, namely as {4}.
The a(n) factorizations for n = 60, 210, 840, 1260, 4620, 12600, 18480:
4*15 6*35 4*6*35 4*9*35 4*15*77 4*6*15*35 4*6*10*77
6*10 10*21 4*10*21 4*15*21 4*21*55 4*6*21*25 4*6*14*55
14*15 4*14*15 6*10*21 4*33*35 4*9*10*35 4*6*22*35
6*10*14 6*14*15 6*10*77 4*9*14*25 4*10*14*33
9*10*14 6*14*55 4*10*15*21 4*10*21*22
6*22*35 6*10*14*15 4*14*15*22
10*14*33 6*10*14*22
10*21*22
14*15*22
(End)
MATHEMATICA
Table[Count[Subsets[Select[Divisors[n], PrimeOmega[#] == 2 &]], _?(Times @@ # == n &)], {n, 105}] (* Michael De Vlieger, Dec 11 2020 *)
PROG
(PARI) A322353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A322353(n/d, d-1))); (s)); \\ Antti Karttunen, Dec 10 2020
CROSSREFS
Unlabeled multiset partitions of this type are counted by A007717. The version for partitions is A112020, or A101048 without distinctness. Positions of zeros include A320892. Positions of nonzero terms are A320912. The case of squarefree factors is A339661, or A320656 without distinctness. Allowing prime factors gives A339839, or A320732 without distinctness. A037143 lists primes and semiprimes.
A339846 counts even-length factorizations, with ordered version A174725.
Products of primes of semiprime index ( A106349).
+10
15
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
COMMENTS
A semiprime ( A001358) is a product of any two prime numbers. Also MM-numbers of labeled multigraphs with loops (without uncovered vertices). 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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
EXAMPLE
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)
MAPLE
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:
MATHEMATICA
semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100], FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; !semiQ[PrimePi[p]]]&]
CROSSREFS
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).
Primes indexed by squarefree semiprimes.
+10
14
13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 199, 233, 257, 269, 271, 293, 313, 347, 373, 389, 421, 439, 443, 449, 467, 487, 491, 499, 577, 607, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947, 983, 1051, 1061, 1093
COMMENTS
A squarefree semiprime is a product of two distinct prime numbers.
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. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of non-loop edges.
EXAMPLE
The sequence of edges whose MM-numbers belong to the sequence begins: {{1,2}}, {{1,3}}, {{1,4}}, {{2,3}}, {{2,4}}, {{1,5}}, {{1,6}}, {{2,5}}, {{1,7}}, {{3,4}}, {{1,8}}, {{2,6}}, {{1,9}}, {{2,7}}, {{3,5}}, {{2,8}}.
MATHEMATICA
Select[Range[100], PrimeOmega[#]==1&&PrimeOmega[PrimePi[#]]==2&&SquareFreeQ[PrimePi[#]]&]
PROG
(PARI) isok(p) = isprime(p) && (ip=primepi(p)) && (omega(ip)==2) && (bigomega(ip) == 2); \\ Michel Marcus, Dec 16 2018
CROSSREFS
Cf. A001358, A003963, A006881, A056239, A085156, A106349, A112798, A302242, A302491, A320458, A320459, A320461.
MM-numbers of labeled graphs with half-loops, without isolated vertices.
+10
14
1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 39, 41, 43, 47, 51, 55, 59, 65, 67, 73, 79, 83, 85, 87, 93, 101, 109, 123, 127, 129, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 187, 191, 195, 199, 201, 205, 211, 215, 219, 221, 233, 235, 237, 241, 249
COMMENTS
Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal vertices.
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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}. Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime ( A006881).
EXAMPLE
The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
1: {} 55: {{2},{3}} 137: {{2,5}}
3: {{1}} 59: {{7}} 139: {{1,7}}
5: {{2}} 65: {{2},{1,2}} 141: {{1},{2,3}}
11: {{3}} 67: {{8}} 143: {{3},{1,2}}
13: {{1,2}} 73: {{2,4}} 145: {{2},{1,3}}
15: {{1},{2}} 79: {{1,5}} 149: {{3,4}}
17: {{4}} 83: {{9}} 155: {{2},{5}}
29: {{1,3}} 85: {{2},{4}} 157: {{12}}
31: {{5}} 87: {{1},{1,3}} 163: {{1,8}}
33: {{1},{3}} 93: {{1},{5}} 165: {{1},{2},{3}}
39: {{1},{1,2}} 101: {{1,6}} 167: {{2,6}}
41: {{6}} 109: {{10}} 177: {{1},{7}}
43: {{1,4}} 123: {{1},{6}} 179: {{13}}
47: {{2,3}} 127: {{11}} 187: {{3},{4}}
51: {{1},{4}} 129: {{1},{1,4}} 191: {{14}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], And[SquareFreeQ[#], And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]
CROSSREFS
The version with full loops covering an initial interval is A320461. The case covering an initial interval is A340018. The version with full loops is A340020. A006450 lists primes of prime index.
A106349 lists primes of semiprime index.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A322551 lists primes of squarefree semiprime index.
A330944 counts nonprime prime indices.
A339112 lists MM-numbers of multigraphs with loops.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326788.
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