A007821 -id:A007821 - cp's OEIS frontend

A330944 Number of nonprime prime indices of n.

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

Gus Wiseman, Jan 13 2020

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.

			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 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.
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

a(n) + A257994(n) = A001222(n).

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

A049081 Primes prime(k) for which A049076(k) = 5.

Original entry on oeis.org

31, 1787, 8527, 19577, 27457, 42043, 72727, 96797, 112129, 137077, 173867, 239489, 250751, 285191, 352007, 401519, 443419, 464939, 490643, 527623, 683873, 718807, 755387, 839483, 864013, 985151, 1021271, 1080923, 1159901, 1278779

Offset: 1

Neil Fernandez

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049080, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049081 := proc(n)
        ithprime(A049080(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[44], !PrimeQ[ # ] &], 5] (* Robert G. Wilson v, Mar 15 2004 *)

More terms from Robert G. Wilson v, Dec 12 2000

A320629 Products of odd primes of nonprime index.

Original entry on oeis.org

1, 7, 13, 19, 23, 29, 37, 43, 47, 49, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 169, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311

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 (1/2) * Product_{p in A006450} (1 - 1/p) = 1/(2*Sum_{n>=1} 1/A076610(n)) < 1/6. - Amiram Eldar, Feb 02 2021

			The sequence of terms begins:
    1 = 1
    7 = prime(4)
   13 = prime(6)
   19 = prime(8)
   23 = prime(9)
   29 = prime(10)
   37 = prime(12)
   43 = prime(14)
   47 = prime(15)
   49 = prime(4)^2
   53 = prime(16)
   61 = prime(18)
   71 = prime(20)
   73 = prime(21)
   79 = prime(22)
   89 = prime(24)
   91 = prime(4)*prime(6)
   97 = prime(25)
  101 = prime(26)
  103 = prime(27)
  107 = prime(28)
  113 = prime(30)
  131 = prime(32)
  133 = prime(4)*prime(8)
  137 = prime(33)
  139 = prime(34)
  149 = prime(35)
  151 = prime(36)
  161 = prime(4)*prime(9)

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

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A320533, A320628, A320630, A320631, A320633.

Select[Range[1,100,2],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]&]

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&]

A049079 Primes prime(k) for which A049076(k) = 3.

Original entry on oeis.org

5, 59, 179, 331, 431, 599, 919, 1153, 1297, 1523, 1847, 2381, 2477, 2749, 3259, 3637, 3943, 4091, 4273, 4549, 5623, 5869, 6113, 6661, 6823, 7607, 7841, 8221, 8719, 9461, 9739, 9859, 11743, 11953, 12097, 12301, 12547, 13469, 13709, 14177, 14723, 14867

Offset: 1

Neil Fernandez

nonn

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049079 := proc(n)
        ithprime(A049078(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[60], !PrimeQ[ # ] &], 3] (* Robert G. Wilson v, Mar 15 2004 *)

Definition edited by Zak Seidov, Sep 15 2013

A058322 Primes for which A049076(p) = 7.

Original entry on oeis.org

127, 15299, 87803, 219613, 318211, 506683, 919913, 1254739, 1471343, 1828669, 2364361, 3338989, 3509299, 4030889, 5054303, 5823667, 6478961, 6816631, 7220981, 7807321, 10311439, 10875143, 11469013, 12838937, 13243033, 15239333, 15837299, 16827557, 18143603

Offset: 1

Robert G. Wilson v, Dec 12 2000

nonn

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049080, A049081, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A058322 := proc(n)
        ithprime(A049081(n)) ;
end proc: # R. J. Mathar, Jul 07 2012
# second Maple program:
map(ithprime@@6, remove(isprime, [$1..42]))[];  # Alois P. Heinz, Mar 15 2020

Nest[ Prime, Select[ Range[40], !PrimeQ[ # ] &], 6] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A049081(n)).

More terms from Alois P. Heinz, Mar 15 2020

A049080 Primes prime(k) for which A049076(k) = 4.

Original entry on oeis.org

11, 277, 1063, 2221, 3001, 4397, 7193, 9319, 10631, 12763, 15823, 21179, 22093, 24859, 30133, 33967, 37217, 38833, 40819, 43651, 55351, 57943, 60647, 66851, 68639, 77431, 80071, 84347, 90023, 98519, 101701, 103069, 125113, 127643

Offset: 1

Neil Fernandez

nonn

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049080 := proc(n)
        ithprime(A049079(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[50], !PrimeQ[ # ] &], 4] (* Robert G. Wilson v, Mar 15 2004 *)

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

Gus Wiseman, Mar 12 2021

nonn

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}}.

			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:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of squarefree semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The semiprime instead of squarefree semiprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A002100 counts partitions into squarefree semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices, which are listed by A112798.
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).
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
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).
Cf. A000040, A000720, A001055, A001222, A003963, A289509, A320461.

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]

A058324 Primes for which A049076(p) = 8.

Original entry on oeis.org

709, 167449, 1128889, 3042161, 4535189, 7474967, 14161729, 19734581, 23391799, 29499439, 38790341, 56011909, 59053067, 68425619, 87019979, 101146501, 113256643, 119535373, 127065427, 138034009, 185350441, 196100297, 207460717, 233784751, 241568891, 280256489

Offset: 1

Robert G. Wilson v, Dec 12 2000

nonn

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049080, A049081, A058322, A058325, A058326, A058327, A058328, A093046, A006450.

map(ithprime@@7, remove(isprime, [$1..38]))[];  # Alois P. Heinz, Mar 15 2020

Nest[ Prime, Select[ Range[34], !PrimeQ[ # ] &], 7] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A058322(n)). - R. J. Mathar, Jul 07 2012

More terms from Alois P. Heinz, Mar 15 2020

cp's OEIS Frontend

A330944 Number of nonprime prime indices of n.

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A049081 Primes prime(k) for which A049076(k) = 5.

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A320629 Products of odd primes of nonprime index.

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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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A049079 Primes prime(k) for which A049076(k) = 3.

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A058322 Primes for which A049076(p) = 7.

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A049080 Primes prime(k) for which A049076(k) = 4.

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