A006450 -id:A006450 - cp's OEIS frontend

A058322 Primes for which A049076(p) = 7.

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]]]&]

A049203 Primes for which A049076(p) >= 5.

Original entry on oeis.org

31, 127, 709, 1787, 5381, 8527, 15299, 19577, 27457, 42043, 52711, 72727, 87803, 96797, 112129, 137077, 167449, 173867, 219613, 239489, 250751, 285191, 318211, 352007, 401519, 443419, 464939, 490643, 506683, 527623, 648391, 683873, 718807

Offset: 1

Neil Fernandez

nonn

Union of A049081, A058322, A058324-A058328, A093046, etc. - R. J. Mathar, Jul 07 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)
R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014.
N. Fernandez, An order of primeness.
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.

Cf. A000040, A006450, A038580, A049076, A049090, A049202, A057847, A057849, A057850, A057851, A058332, A093047.

map(ithprime@@4, select(isprime, [$1..137])); # Peter Luschny, Feb 17 2014

Nest[ Prime, Range[35], 5] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(),q,r,s,t); forprime(p=2,lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 16 2017

a(n) = A000040(A049090(n)). - R. J. Mathar, Jul 07 2012

a(n) ~ n (log n)^5. - Charles R Greathouse IV, Feb 16 2017

More terms from Robert G. Wilson v, Nov 10 2000

Name corrected by Sean A. Irvine, Jul 21 2021

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

A058328 Primes for which A049076(p) = 12.

Original entry on oeis.org

9737333, 16123689073, 175650481151, 592821132889, 963726515729, 1765037224331, 3809491708961, 5669795882633, 6947574946087, 9163611272327, 12695664159413, 19638537755027, 20909033866927, 24894639811901

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, A058324, A058325, A058326, A058327, A093046, A006450.

Nest[ Prime, Select[ Range[30], !PrimeQ[ # ] &], 11] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A000040(A058327(n)). - R. J. Mathar, Jul 07 2012

A093046 Primes for which A049076(p) = 13.

Original entry on oeis.org

174440041, 414507281407, 4952019383323, 17461204521323, 28871271685163, 53982894593057, 119543903707171, 180252380737439, 222334565193649, 295872998567819, 414190707114539, 649544694886663, 692919372869953, 829484152743469, 1111923751842437, 1335294947809661, 1532021237514419, 1635795965187779

Offset: 1

Robert G. Wilson v, Mar 15 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, A058324, A058325, A058326, A058327, A058328, A006450.

Nest[ Prime, Select[ Range[30], !PrimeQ[ # ] &], 12]

a(n) = A000040(A058328(n)). - R. J. Mathar, Jul 07 2012

a(7)-a(9) from Robert G. Wilson v, Dec 27 2005

a(10)-a(18) from Robert G. Wilson v, Mar 08 2017 using Kim Walisch's primecount.

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.

A049202 Primes p whose order of primeness A049076(p) is >= 6.

Original entry on oeis.org

127, 709, 5381, 15299, 52711, 87803, 167449, 219613, 318211, 506683, 648391, 919913, 1128889, 1254739, 1471343, 1828669, 2269733, 2364361, 3042161, 3338989, 3509299, 4030889, 4535189, 5054303, 5823667, 6478961, 6816631

Offset: 1

Neil Fernandez

nonn

Union of A058322, A058324-A058328, A093046 etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from Robert G. Wilson v)
R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A000040, A006450, A038580, A049090, A049203, A057849, A057850, A057851, A057847, A058332, A093047.

map(ithprime@@4,select(isprime, [$1..137])); # Peter Luschny, Feb 17 2014

Nest[ Prime, Range[35], 6] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(), q, r, s, t, u); forprime(p=2, lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++) && isprime(u++), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 16 2017

More terms from Robert G. Wilson v, Nov 10 2000

Name corrected by Sean A. Irvine, Jul 21 2021

A102615 Nonprime numbers of order 2.

Original entry on oeis.org

1, 8, 10, 14, 15, 16, 20, 22, 24, 25, 27, 30, 32, 33, 35, 36, 38, 39, 40, 44, 46, 48, 49, 50, 51, 54, 55, 56, 58, 62, 63, 64, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 82, 85, 86, 87, 88, 90, 92, 93, 94, 96, 99, 100, 102, 104, 105, 108, 110, 111, 114, 115, 116, 117, 118, 120

Offset: 1

Cino Hilliard, Jan 31 2005

nonn

nps(n,0) -> list nonprime(n) or the sequence of nonprime numbers. nps(n,1) -> list nonprime(nonprime(n)) or nps of order 1 nps(n,2) -> list nonprime(nonprime(nonprime(n))) or nps of order 2 ..... The order is the number of nestings - 1. We avoid the nestings in the script with a loop.

Nonprimes (A018252) with nonprime (A018252) subscripts. a(n) U A078782(n) = A018252(n), a(n+1) U A175250(n) = A018252(n) for n >= 1. a(n) = nonprime(nonprime(n)) = A018252(A018252(n)). a(4) = 14 because a(4) = b(b(4)) = b(8) = 14, b = nonprime. a(1) = 1, a(n) = nonprimes (A018252) with composite (A002808) subscripts for n >=2. [Jaroslav Krizek, Mar 13 2010]

			Nonprime(2) = 4.
Nonprime(4) = 8 the second entry.

Cf. A018252.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622.  - N. J. A. Sloane, Mar 30 2016

nonPrime[n_] := FixedPoint[n + PrimePi[ # ] &, n]; Nest[nonPrime, Range[66], 2] (* Robert G. Wilson v, Feb 04 2005 *)

\We perform nesting(s) with a loop. cics(n,m) = { local(x,y,z); for(x=1,n, z=x; for(y=1,m+1, z=composite(z); ); print1(z",") ) } composite(n) = \ The n-th composite number. 1 is defined as a composite number. { local(c,x); c=1; x=0; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }

Edited by Robert G. Wilson v, Feb 04 2005

cp's OEIS Frontend

A058322 Primes for which A049076(p) = 7.

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

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A339113 Products of primes of squarefree semiprime index (A322551).

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A049203 Primes for which A049076(p) >= 5.

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A058324 Primes for which A049076(p) = 8.

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A058328 Primes for which A049076(p) = 12.

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A093046 Primes for which A049076(p) = 13.

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A331386 Numbers with at least one prime prime index.

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