A302590 -id:A302590 - cp's OEIS frontend

A340018 MM-numbers of labeled graphs with half-loops covering an initial interval of positive integers, without isolated vertices.

1, 3, 13, 15, 39, 65, 141, 143, 145, 165, 195, 377, 429, 435, 611, 705, 715, 1131, 1363, 1551, 1595, 1833, 1885, 1937, 2021, 2117, 2145, 2235, 2365, 2397, 2409, 2431, 2465, 2805, 3055, 4089, 4147, 4785, 5655, 5811, 6063, 6149, 6235, 6351, 6409, 6721, 6815

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

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, and whose prime indices together cover an initial interval of positive integers. A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}
     3: {{1}}
    13: {{1,2}}
    15: {{1},{2}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
   141: {{1},{2,3}}
   143: {{3},{1,2}}
   145: {{2},{1,3}}
   165: {{1},{2},{3}}
   195: {{1},{2},{1,2}}
   377: {{1,2},{1,3}}
   429: {{1},{3},{1,2}}
   435: {{1},{2},{1,3}}
   611: {{1,2},{2,3}}
   705: {{1},{2},{2,3}}
   715: {{2},{3},{1,2}}
  1131: {{1},{1,2},{1,3}}

The version with full loops is A320461.
The version not necessarily covering an initial interval is A340019.
MM-numbers of graphs with loops are 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.
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, A326754, A326788.

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[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

A316154 Number of integer partitions of prime(n) into a prime number of prime parts.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 12, 19, 39, 50, 93, 136, 166, 239, 409, 682, 814, 1314, 1774, 2081, 3231, 4272, 6475, 11077, 14270, 16265, 20810, 23621, 30031, 68251, 85326, 118917, 132815, 226097, 251301, 342448, 463940, 565844, 759873, 1015302, 1117708, 1787452, 1961624

Offset: 1

Gus Wiseman, Jun 25 2018

nonn

			The a(7) = 9 partitions of 17 into a prime number of prime parts: (13,2,2), (11,3,3), (7,7,3), (7,5,5), (7,3,3,2,2), (5,5,3,2,2), (5,3,3,3,3), (5,2,2,2,2,2,2), (3,3,3,2,2,2,2).

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from Andrew Howroyd)

Cf. A000040, A000586, A000607, A038499, A056768, A064688, A070215, A085755, A302590, A316092, A316153, A316185, A344782.

b:= proc(n, p, c) option remember; `if`(n=0 or p=2,
      `if`(n::even and isprime(c+n/2), 1, 0),
      `if`(p>n, 0, b(n-p, p, c+1))+b(n, prevprime(p), c))
    end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 26 2018

Table[Length[Select[IntegerPartitions[Prime[n]],And[PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,20}]
(* Second program: *)
b[n_, p_, c_] := b[n, p, c] = If[n == 0 || p == 2, If[EvenQ[n] && PrimeQ[c + n/2], 1, 0], If[p>n, 0, b[n - p, p, c + 1]] + b[n, NextPrime[p, -1], c]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 50] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

seq(n)={my(p=vector(n,k,prime(k))); my(v=Vec(1/prod(k=1, n, 1 - x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018

a(n) = A085755(A000040(n)). - Alois P. Heinz, Jun 26 2018

Terms a(21) and beyond from Andrew Howroyd, Jun 26 2018

A316185 Number of strict integer partitions of the n-th prime into a prime number of prime parts.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 2, 2, 3, 5, 5, 6, 8, 10, 13, 18, 20, 26, 32, 34, 45, 54, 66, 90, 106, 117, 135, 142, 165, 269, 311, 375, 398, 546, 579, 689, 823, 938, 1107, 1301, 1352, 1790, 1850, 2078, 2153, 2878, 3811, 4241, 4338, 4828, 5495, 5637, 7076, 8000, 9032

Offset: 1

Gus Wiseman, Jun 25 2018

nonn

			The a(14) = 8 partitions of 43 into a prime number of distinct prime parts: (41,2), (31,7,5), (29,11,3), (23,17,3), (23,13,7), (19,17,7), (19,13,11), (17,11,7,5,3).

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000586, A000607, A038499, A045450, A056768, A064688, A070215, A085755, A302590, A316092, A316153, A316154.

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(isprime(n), n, h(n-1)))
    end:
b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(isprime(c), 1, 0), `if`(i<2, 0, b(n, h(i-1), c)+
      `if`(i>n, 0, b(n-i, h(min(n-i, i-1)), c+1))))
    end:
a:= n-> b(ithprime(n)$2, 0):
seq(a(n), n=1..56);  # Alois P. Heinz, May 26 2021

Table[Length[Select[IntegerPartitions[Prime[n]],And[UnsameQ@@#,PrimeQ[Length[#]],And@@PrimeQ/@#]&]],{n,10}]
(* Second program: *)
h[n_] := h[n] = If[n == 0, 0, If[PrimeQ[n], n, h[n - 1]]];
b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[PrimeQ[c], 1, 0], If[i < 2, 0, b[n, h[i - 1], c] +
     If[i > n, 0, b[n - i, h[Min[n - i, i - 1]], c + 1]]]];
a[n_] := b[Prime[n], Prime[n], 0];
Array[a, 56] (* Jean-François Alcover, Jun 11 2021, after Alois P. Heinz *)

seq(n)={my(p=vector(n, k, prime(k))); my(v=Vec(prod(k=1, n, 1 + x^p[k]*y + O(x*x^p[n])))); vector(n, k, sum(i=1, k, polcoeff(v[1+p[k]], p[i])))} \\ Andrew Howroyd, Jun 26 2018

a(n) = A045450(A000040(n)).

More terms from Alois P. Heinz, Jun 26 2018

A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 7, 8, 1, 2, 1, 4, 13, 14, 1, 16, 1, 2, 19, 4, 7, 2, 23, 8, 1, 26, 1, 28, 29, 2, 1, 32, 1, 2, 7, 4, 37, 38, 13, 8, 1, 14, 43, 4, 1, 46, 47, 16, 49, 2, 1, 52, 53, 2, 1, 56, 19, 58, 1, 4, 61, 2, 7, 64, 13, 2, 1, 4, 23, 14, 71, 8, 73, 74, 1, 76, 7

Offset: 1

Amiram Eldar, Feb 03 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006450, A007821, A076610, A302590, A320628, A360325, A360330, A360331.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(isprime(primepi(f[i,1])), f[i,1]=1)); factorback(f);}

a(n) = 1 if and only if n is in A076610.
a(n) = n if and only if n is in A320628.
a(n) = n/A360325(n).
Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and p^e otherwise (A007821).
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p in A006450} p/(p+1) < 0.4 (see A302590 for an estimate of 1/c).

A340104 Products of distinct primes of nonprime index (A007821).

Original entry on oeis.org

1, 2, 7, 13, 14, 19, 23, 26, 29, 37, 38, 43, 46, 47, 53, 58, 61, 71, 73, 74, 79, 86, 89, 91, 94, 97, 101, 103, 106, 107, 113, 122, 131, 133, 137, 139, 142, 146, 149, 151, 158, 161, 163, 167, 173, 178, 181, 182, 193, 194, 197, 199, 202, 203, 206, 214, 223, 226

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

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 sequence of terms together with the corresponding prime indices of prime indices begins:
     1: {}              58: {{},{1,3}}        113: {{1,2,3}}
     2: {{}}            61: {{1,2,2}}         122: {{},{1,2,2}}
     7: {{1,1}}         71: {{1,1,3}}         131: {{1,1,1,1,1}}
    13: {{1,2}}         73: {{2,4}}           133: {{1,1},{1,1,1}}
    14: {{},{1,1}}      74: {{},{1,1,2}}      137: {{2,5}}
    19: {{1,1,1}}       79: {{1,5}}           139: {{1,7}}
    23: {{2,2}}         86: {{},{1,4}}        142: {{},{1,1,3}}
    26: {{},{1,2}}      89: {{1,1,1,2}}       146: {{},{2,4}}
    29: {{1,3}}         91: {{1,1},{1,2}}     149: {{3,4}}
    37: {{1,1,2}}       94: {{},{2,3}}        151: {{1,1,2,2}}
    38: {{},{1,1,1}}    97: {{3,3}}           158: {{},{1,5}}
    43: {{1,4}}        101: {{1,6}}           161: {{1,1},{2,2}}
    46: {{},{2,2}}     103: {{2,2,2}}         163: {{1,8}}
    47: {{2,3}}        106: {{},{1,1,1,1}}    167: {{2,6}}
    53: {{1,1,1,1}}    107: {{1,1,4}}         173: {{1,1,1,3}}

These primes (of nonprime index) are listed by A007821.
The non-strict version is A320628, with odd case A320629.
The odd case is A340105.
The prime instead of nonprime version:
  primes: A006450
  products: A076610
  strict: A302590
The semiprime instead of nonprime version:
  primes: A106349
  products: A339112
  strict: A340020
The squarefree semiprime instead of nonprime version:
  strict: A309356
  primes: A322551
  products: A339113
A056239 gives the sum of prime indices, which are listed by A112798.
A257994 counts prime prime indices.
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).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A330944 counts nonprime prime indices.
A330945 lists numbers with a nonprime prime index (nonprime case: A330948).
A339561 lists products of distinct squarefree semiprimes (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).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A018252, A289509, A320461, A320631.

Select[Range[100],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeQ[PrimePi[p]]]&]

Equals A005117 /\ A320628.

A340105 Odd products of distinct primes of nonprime index (A007821).

Original entry on oeis.org

1, 7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311, 313, 317

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

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 sequence of terms together with the corresponding sets of multisets begins:
     1: {}              91: {{1,1},{1,2}}      173: {{1,1,1,3}}
     7: {{1,1}}         97: {{3,3}}            181: {{1,2,4}}
    13: {{1,2}}        101: {{1,6}}            193: {{1,1,5}}
    19: {{1,1,1}}      103: {{2,2,2}}          197: {{2,2,3}}
    23: {{2,2}}        107: {{1,1,4}}          199: {{1,9}}
    29: {{1,3}}        113: {{1,2,3}}          203: {{1,1},{1,3}}
    37: {{1,1,2}}      131: {{1,1,1,1,1}}      223: {{1,1,1,1,2}}
    43: {{1,4}}        133: {{1,1},{1,1,1}}    227: {{4,4}}
    47: {{2,3}}        137: {{2,5}}            229: {{1,3,3}}
    53: {{1,1,1,1}}    139: {{1,7}}            233: {{2,7}}
    61: {{1,2,2}}      149: {{3,4}}            239: {{1,1,6}}
    71: {{1,1,3}}      151: {{1,1,2,2}}        247: {{1,2},{1,1,1}}
    73: {{2,4}}        161: {{1,1},{2,2}}      251: {{1,2,2,2}}
    79: {{1,5}}        163: {{1,8}}            257: {{3,5}}
    89: {{1,1,1,2}}    167: {{2,6}}            259: {{1,1},{1,1,2}}

These primes (of nonprime index) are listed by A007821.
The non-strict version is A320629, with not necessarily odd version A320628.
The not necessarily odd version is A340104.
The prime instead of odd nonprime version:
  primes: A006450
  products: A076610
  strict: A302590
The squarefree semiprime instead of odd nonprime version:
  strict: A309356
  primes: A322551
  products: A339113
The semiprime instead of odd nonprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes.
A056239 gives the sum of prime indices, which are listed by A112798.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A330944 counts nonprime prime indices.
A330945 lists numbers with a nonprime prime index (nonprime case: A330948).
A339561 lists products of distinct squarefree semiprimes.
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).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A018252, A289509, A320461, A320631, A320911, A320912.

Select[Range[1,100,2],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeQ[PrimePi[p]]]&]

Equals A005117 /\ A005408 /\ A320628 = A005117 /\ A320629.

A368728 Numbers whose prime indices are 1, prime, or semiprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 75

Offset: 1

Gus Wiseman, Jan 07 2024

nonn

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.

These are products of primes indexed by elements of A037143.
For just primes we have A076610, strict A302590.
For just semiprimes we have A339112, strict A340020.
For squarefree semiprimes we have A339113, strict A309356.
The odd case is A368729, strict A340019.
The complement is A368833.
A000607 counts partitions into primes, A034891 with ones allowed.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628, A320911, A320912.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Length/@prix/@prix[#]<=2&]

Closed under multiplication.

A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 55, 59, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 97, 99, 101, 105, 109, 115, 117, 119, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 147, 149

Offset: 1

Gus Wiseman, Jan 07 2024

nonn

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}}.

			The terms together with the corresponding multigraphs begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

In the unlabeled case these multigraphs are counted by A320663.
These are products of primes indexed by elements of A037143 greater than 1.
For just primes we have A076610, squarefree A302590.
For just semiprimes we have A339112, squarefree A340020.
For just half-loops we have A340019.
This is the odd case of A368728, complement A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320912, A339113.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&Max@@Length/@prix/@prix[#]<=2&]

A368732 Primes whose index is one, another prime number, or a semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 167, 179, 191, 199, 211, 227, 233, 241, 257, 269, 271, 277, 283, 293, 313, 331, 347, 353, 367, 373, 389, 401, 421, 431, 439, 443, 449, 461, 467, 487

Offset: 1

Gus Wiseman, Jan 08 2024

nonn

For just primes we have A006450, products A076610, strict A302590.
These indices are A037143.
For just semiprimes we have A106349, products A339112, strict A340020.
Products of these primes are A368728, odd A368729, odd strict A340019.
Products of the complementary primes are A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
A322551 lists primes of squarefree semiprime index.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320911, A320912, A339113.

Prime/@Select[Range[100],PrimeOmega[#]<=2&]

A368833 Numbers whose prime indices are not 1, prime, or semiprime.

Original entry on oeis.org

19, 37, 38, 53, 57, 61, 71, 74, 76, 89, 95, 103, 106, 107, 111, 113, 114, 122, 131, 133, 142, 148, 151, 152, 159, 171, 173, 178, 181, 183, 185, 190, 193, 197, 206, 209, 212, 213, 214, 222, 223, 226, 228, 229, 239, 244, 247, 251, 259, 262, 263, 265, 266, 267

Offset: 1

Gus Wiseman, Jan 08 2024

nonn

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 terms together with their prime indices begin:
   19: {8}
   37: {12}
   38: {1,8}
   53: {16}
   57: {2,8}
   61: {18}
   71: {20}
   74: {1,12}
   76: {1,1,8}
   89: {24}
   95: {3,8}
  103: {27}
  106: {1,16}
  107: {28}
  111: {2,12}
  113: {30}
  114: {1,2,8}
  122: {1,18}
  131: {32}
  133: {4,8}
  142: {1,20}
  148: {1,1,12}

These are non-products of primes indexed by elements of A037143.
The complement for just primes is A076610, strict A302590.
The complement for just semiprimes is A339112, strict A340020.
The complement for just squarefree semiprimes is A339113, strict A309356.
The complement is A368728.
The complement for just primes and semiprimes is A368729, strict A340019.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Max@@PrimeOmega/@prix[#]>2&]

cp's OEIS Frontend

A340018 MM-numbers of labeled graphs with half-loops covering an initial interval of positive integers, without isolated vertices.

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A316154 Number of integer partitions of prime(n) into a prime number of prime parts.

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A316185 Number of strict integer partitions of the n-th prime into a prime number of prime parts.

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A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

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A340104 Products of distinct primes of nonprime index (A007821).

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A340105 Odd products of distinct primes of nonprime index (A007821).

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A368728 Numbers whose prime indices are 1, prime, or semiprime.

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A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

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