A330944 -id:A330944 - cp's OEIS frontend

A320628 Products of primes of nonprime index.

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

Gus Wiseman, Oct 18 2018

nonn

The index of a prime number n is the number m such that n is the m-th prime.

The asymptotic density of this sequence is Product_{p in A006450} (1 - 1/p) = 1/(Sum_{n>=1} 1/A076610(n)) < 1/3. - Amiram Eldar, Feb 02 2021

			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

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

Cf. A018252, A056239, A290103, A302242, A302478, A320533, A320629, A320630, A320631, A320633.
Complement of A331386.
Positions of zeros in A257994.
Primes of prime index are A006450.
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.
Cf. A000040, A000720, A001222, A112798, A295665.

Select[Range[100],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]&]

A257994 Number of prime parts in the partition having Heinz number n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 2, 0, 1, 1, 1, 0, 1, 2, 0, 3, 0, 0, 2, 1, 0, 2, 1, 1, 2, 0, 0, 1, 1, 1, 1, 0, 1, 3, 0, 0, 1, 0, 2, 2, 0, 0, 3, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 1, 1, 1, 1, 0, 2, 0, 0, 3, 0, 1, 1, 0, 1, 4, 1, 1, 1, 2, 0, 1, 1, 0, 3

Offset: 1

Emeric Deutsch, May 20 2015

We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
The number of nonprime parts is given by A330944, so A001222(n) = a(n) + A330944(n). - Gus Wiseman, Jan 17 2020

			a(30) = 2 because the partition with Heinz number 30 = 2*3*5 is [1,2,3], having 2 prime parts.

George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Positions of positive terms are A331386.
Primes of prime index are A006450.
Products of primes of prime index are A076610.
The number of nonprime prime indices is A330944.
Cf. A000040, A000720, A001222, A007821, A018252, A056239, A112798, A215366, A279952, A302242, A320628, A330945.

with(numtheory): a := proc (n) local B, ct, s: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for s to nops(B(n)) do if isprime(B(n)[s]) = true then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 130);

B[n_] := Module[{nn, j, m}, nn = FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[Table[Table[PrimePi[  m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
a[n_] := Module[{ct, s}, ct = 0; For[s = 1, s <= Length[B[n]], s++, If[ PrimeQ[B[n][[s]]], ct++]]; ct];
Table[a[n], {n, 1, 130}] (* Jean-François Alcover, Apr 25 2017, translated from Maple *)
Table[Total[Cases[FactorInteger[n],{p_,k_}/;PrimeQ[PrimePi[p]]:>k]],{n,30}] (* Gus Wiseman, Jan 17 2020 *)

a(n) = my(f = factor(n)); sum(i=1, #f~, if(isprime(primepi(f[i, 1])), f[i, 2], 0)); \\ Amiram Eldar, Nov 03 2023

Additive with a(p^e) = e if primepi(p) is prime, and 0 otherwise. - Amiram Eldar, Nov 03 2023

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

Gus Wiseman, Jan 13 2020

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 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 prime index are A006450.
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.

Select[Range[100],!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

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

Gus Wiseman, Feb 08 2020

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 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.
Prime-indexed primes are A006450, with products A076610.
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

Gus Wiseman, Jan 17 2020

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 asymptotic density of this sequence is 1 - Product_{p in A006450} (1 - 1/p) = 1 - 1/(Sum_{n>=1} 1/A076610(n)) > 2/3. - Amiram Eldar, Feb 02 2021

			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}

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

Complement of A320628.
Positions of terms > 0 in A257994.
Positions of terms > 1 in A295665.
Primes of prime index are A006450.
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.

Select[Range[100],MemberQ[FactorInteger[#],{?(PrimeQ@*PrimePi),}]&]

A257994(a(n)) > 0.

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

Gus Wiseman, Dec 25 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 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.

For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of one in A379300.
Partitions of this type are counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A113646, A376680.

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

Gus Wiseman, Dec 27 2024

nonn

			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)

For all prime parts we have A000607 (strict A000586), ranks A076610.
For no prime parts we have A002095 (strict A096258), ranks A320628.
Ranked by A331915 = positions of one in A257994.
For a unique composite part we have A379302 (strict A379303), ranks A379301.
The strict case is A379305.
For squarefree instead of prime we have A379308 (strict A379309), ranks A379316.
Considering 1 prime gives A379314 (strict A379315), ranks A379312.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A204389, A302540, A330944.

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

Gus Wiseman, Dec 27 2024

nonn

			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)

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.
For a composite instead of prime we have A379303, non-strict A379302 (ranks A379301).
The non-strict version is A379304.
For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).
Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

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

Gus Wiseman, Dec 28 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:
    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.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A340019 MM-numbers of labeled graphs with half-loops, without isolated vertices.

Original entry on oeis.org

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

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 (A006881).

			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}}

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.

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[#])]&]

cp's OEIS Frontend

A320628 Products of primes of nonprime index.

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A257994 Number of prime parts in the partition having Heinz number n.

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Maple

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PARI

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A330945 Numbers whose prime indices are not all prime numbers.

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A331915 Numbers with exactly one prime prime index, counted with multiplicity.

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Mathematica

A331386 Numbers with at least one prime prime index.

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Mathematica

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A379301 Positive integers whose prime indices include a unique composite number.

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Mathematica

A379304 Number of integer partitions of n with a unique prime part.

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Mathematica

A379305 Number of strict integer partitions of n with a unique prime part.

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Mathematica

A379312 Positive integers whose prime indices include a unique 1 or prime number.