A006450 -id:A006450 - cp's OEIS frontend

A057850 Primes p whose order of primeness A078442(p) is at least 8.

5381, 52711, 648391, 2269733, 9737333, 17624813, 37139213, 50728129, 77557187, 131807699, 174440041, 259336153, 326851121, 368345293, 440817757, 563167303, 718064159, 751783477, 997525853, 1107276647, 1170710369, 1367161723

Offset: 1

Robert G. Wilson v, Nov 10 2000

nonn

Union of A058325-A058328, A093046 etc. - R. J. Mathar, Jul 07 2012

Robert G. Wilson v, Table of n, a(n) for n = 1..11457
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. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057851, A057847, A058332, A093047.

Nest[ Prime, Range[35], 8] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(), q, r, s, t, u, vv, w); forprime(p=2, lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++) && isprime(u++) && isprime(vv++) && isprime(w++), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 19 2017

a(n) = A049090(A049090(n)). - James G. Merickel, Feb 14 2010

a(n) = A000040(A057849(n)). - R. J. Mathar, Jul 07 2012

Name clarified by Andrew Howroyd, Nov 17 2024

A057851 Primes p whose order of primeness A078442(p) is at least 9.

Original entry on oeis.org

52711, 648391, 9737333, 37139213, 174440041, 326851121, 718064159, 997525853, 1559861749, 2724711961, 3657500101, 5545806481, 7069067389, 8012791231, 9672485827, 12501968177, 16123689073, 16917026909, 22742734291

Offset: 1

Robert G. Wilson v, Nov 10 2000

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1381 terms from Robert G. Wilson v)
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057847, A058332, A093047.

Nest[ Prime, Range[35], 9] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(), q, r, s, t, u, vv, w, x); forprime(p=2, lim, if(isprime(q++) && isprime(r++) && isprime(s++) && isprime(t++) && isprime(u++) && isprime(vv++) && isprime(w++) && isprime(x++), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Feb 19 2017

Name clarified by Andrew Howroyd, Nov 17 2024

A058332 Primes p whose order of primeness A078442(p) is at least 11.

Original entry on oeis.org

9737333, 174440041, 3657500101, 16123689073, 88362852307, 175650481151, 414507281407, 592821132889, 963726515729, 1765037224331, 2428095424619, 3809491708961, 4952019383323, 5669795882633, 6947574946087, 9163611272327

Offset: 1

Robert G. Wilson v, Dec 12 2000

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..47.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A093047.

Nest[ Prime, Range[35], 11] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = prime(A057847(n)). - Andrew Howroyd, Nov 17 2024

Name clarified by Andrew Howroyd, Nov 17 2024

A093047 Primes p whose order of primeness A078442(p) is at least 12.

Original entry on oeis.org

174440041, 3657500101, 88362852307, 414507281407, 2428095424619, 4952019383323, 12055296811267, 17461204521323, 28871271685163, 53982894593057, 75063692618249, 119543903707171, 156740126985437, 180252380737439, 222334565193649

Offset: 1

Robert G. Wilson v, Mar 15 2000

Primes p whose primeness is > 12: 3657500101, 88362852307, 2428095424619, 12055296811267, 75063692618249, 156740126985437, ..., . - Robert G. Wilson v, Mar 15 2000

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

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A058332, A093047.

Nest[ Prime, Range[35], 12] (* Robert G. Wilson v, Mar 15 2004 *)

a(n) = A058332(prime(n)). - Andrew Howroyd, Nov 17 2024

Name clarified by Andrew Howroyd, Nov 17 2024

A102617 Primes p(n) such that n is a second-order nonprime number.

Original entry on oeis.org

2, 19, 29, 43, 47, 53, 71, 79, 89, 97, 103, 113, 131, 137, 149, 151, 163, 167, 173, 193, 199, 223, 227, 229, 233, 251, 257, 263, 271, 293, 307, 311, 317, 337, 347, 349, 359, 379, 383, 389, 397, 409, 421, 439, 443, 449, 457, 463, 479, 487, 491, 503, 523, 541

Offset: 1

Cino Hilliard, Jan 31 2005

nonn

The prime/nonprime compound sequence ABB. - N. J. A. Sloane, Apr 06 2016

			Nonprime(4) = 8.
The 8th prime is 19, the second entry.

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_Integer] := FixedPoint[n + PrimePi[ # ] &, n]; Prime /@ nonPrime /@ nonPrime /@ Range[54] (* Robert G. Wilson v, Feb 04 2005 *)

\We perform nesting(s) with a loop. cips(n,m) = { local(x,y,z); for(x=1,n, z=x; for(y=1,m+1, z=composite(z); ); print1(prime(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

A270792 The prime/nonprime compound sequence ABA.

Original entry on oeis.org

7, 13, 23, 37, 61, 73, 101, 107, 139, 181, 197, 239, 269, 281, 313, 373, 419, 433, 467, 499, 521, 577, 613, 653, 719, 751, 761, 811, 823, 853, 977, 1013, 1051, 1069, 1163, 1187, 1237, 1289, 1307, 1373, 1439, 1453, 1549, 1559, 1583

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

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

A270794 The prime/nonprime compound sequence BAA.

Original entry on oeis.org

6, 9, 18, 26, 45, 57, 81, 91, 112, 143, 165, 203, 228, 244, 267, 303, 345, 354, 411, 437, 454, 495, 530, 564, 623, 668, 687, 714, 728, 749, 856, 893, 931, 959, 1032, 1054, 1104, 1158, 1185, 1233, 1268, 1298, 1372, 1392, 1425, 1445, 1539, 1672, 1698, 1714, 1742, 1773, 1802, 1886, 1914, 1966, 2031, 2050, 2104

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

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

A270796 The prime/nonprime compound sequence BBA.

Original entry on oeis.org

8, 10, 15, 20, 27, 32, 38, 40, 49, 58, 63, 72, 78, 82, 88, 99, 110, 114, 121, 125, 129, 140, 146, 155, 166, 172, 175, 183, 185, 189, 212, 217, 225, 230, 245, 248, 258, 265, 272, 279, 289, 292, 306, 309, 315, 319, 334, 355, 360, 362, 368, 375, 377, 393, 402, 408, 416, 420, 427, 435, 438, 452, 473, 478, 482, 486, 507

Offset: 1

N. J. A. Sloane, Mar 30 2016

nonn

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

A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 553, 559, 577, 607, 611, 631, 647, 653, 661, 667, 673, 677

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a loop is an edge with two equal vertices, distinguished from a half-loop, which has only one vertex.
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 semiprimes, where a semiprime (A001358) is a product of any two prime numbers.

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
      1: {}              161: {{1,1},{2,2}}    347: {{2,9}}
      7: {{1,1}}         163: {{1,8}}          373: {{1,12}}
     13: {{1,2}}         167: {{2,6}}          377: {{1,2},{1,3}}
     23: {{2,2}}         199: {{1,9}}          389: {{4,5}}
     29: {{1,3}}         203: {{1,1},{1,3}}    421: {{1,13}}
     43: {{1,4}}         227: {{4,4}}          439: {{3,7}}
     47: {{2,3}}         233: {{2,7}}          443: {{1,14}}
     73: {{2,4}}         257: {{3,5}}          449: {{2,10}}
     79: {{1,5}}         269: {{2,8}}          467: {{4,6}}
     91: {{1,1},{1,2}}   271: {{1,10}}         487: {{2,11}}
     97: {{3,3}}         293: {{1,11}}         491: {{1,15}}
    101: {{1,6}}         299: {{1,2},{2,2}}    499: {{3,8}}
    137: {{2,5}}         301: {{1,1},{1,4}}    511: {{1,1},{2,4}}
    139: {{1,7}}         313: {{3,6}}          553: {{1,1},{1,5}}
    149: {{3,4}}         329: {{1,1},{2,3}}    559: {{1,2},{1,4}}

The case with only one edge is A106349.
The case covering an initial interval is A320461.
The version allowing multiple edges is A339112.
The half-loop version covering an initial interval is A340018.
The half-loop version is A340019.
A006450 lists primes of prime index.
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.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

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

A320633 Composite numbers whose prime indices are also composite.

Original entry on oeis.org

49, 91, 133, 161, 169, 203, 247, 259, 299, 301, 329, 343, 361, 371, 377, 427, 437, 481, 497, 511, 529, 551, 553, 559, 611, 623, 637, 667, 679, 689, 703, 707, 721, 749, 791, 793, 817, 841, 851, 893, 917, 923, 931, 949, 959, 973, 989, 1007, 1027, 1043, 1057

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of terms begins:
   49 = prime(4)^2
   91 = prime(4)*prime(6)
  133 = prime(4)*prime(8)
  161 = prime(4)*prime(9)
  169 = prime(6)^2
  203 = prime(4)*prime(10)
  247 = prime(6)*prime(8)
  259 = prime(4)*prime(12)
  299 = prime(6)*prime(9)
  301 = prime(4)*prime(14)
  329 = prime(4)*prime(15)
  343 = prime(4)^3
  361 = prime(8)^2
  371 = prime(4)*prime(16)
  377 = prime(6)*prime(10)
  427 = prime(4)*prime(18)
  437 = prime(8)*prime(9)
  481 = prime(6)*prime(12)
  497 = prime(4)*prime(20)
  511 = prime(4)*prime(21)
  529 = prime(9)^2
  551 = prime(8)*prime(10)
  553 = prime(4)*prime(22)
  559 = prime(6)*prime(14)
  611 = prime(6)*prime(15)
  623 = prime(4)*prime(24)
  637 = prime(4)^2*prime(6)

Cf. A000040, A006450, A007821, A018252, A050370, A056239, A076610, A112798, A302242, A302478, A320533, A320628, A320629.

Select[Range[2,1000],And[OddQ[#],!PrimeQ[#],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]]&]

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A057850 Primes p whose order of primeness A078442(p) is at least 8.

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A057851 Primes p whose order of primeness A078442(p) is at least 9.

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A058332 Primes p whose order of primeness A078442(p) is at least 11.

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A093047 Primes p whose order of primeness A078442(p) is at least 12.

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A102617 Primes p(n) such that n is a second-order nonprime number.

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A270792 The prime/nonprime compound sequence ABA.

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A270794 The prime/nonprime compound sequence BAA.

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A270796 The prime/nonprime compound sequence BBA.

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A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

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