Displaying 51-60 of 269 results found.
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Primes p whose order of primeness A078442(p) is at least 8.
+10
12
5381, 52711, 648391, 2269733, 9737333, 17624813, 37139213, 50728129, 77557187, 131807699, 174440041, 259336153, 326851121, 368345293, 440817757, 563167303, 718064159, 751783477, 997525853, 1107276647, 1170710369, 1367161723
PROG
(PARI) 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
CROSSREFS
Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057851, A057847, A058332, A093047.
Primes p whose order of primeness A078442(p) is at least 9.
+10
12
52711, 648391, 9737333, 37139213, 174440041, 326851121, 718064159, 997525853, 1559861749, 2724711961, 3657500101, 5545806481, 7069067389, 8012791231, 9672485827, 12501968177, 16123689073, 16917026909, 22742734291
PROG
(PARI) 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
CROSSREFS
Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057847, A058332, A093047.
Primes p whose order of primeness A078442(p) is at least 11.
+10
12
9737333, 174440041, 3657500101, 16123689073, 88362852307, 175650481151, 414507281407, 592821132889, 963726515729, 1765037224331, 2428095424619, 3809491708961, 4952019383323, 5669795882633, 6947574946087, 9163611272327
CROSSREFS
Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A093047.
Primes p whose order of primeness A078442(p) is at least 12.
+10
12
174440041, 3657500101, 88362852307, 414507281407, 2428095424619, 4952019383323, 12055296811267, 17461204521323, 28871271685163, 53982894593057, 75063692618249, 119543903707171, 156740126985437, 180252380737439, 222334565193649
COMMENTS
Primes p whose primeness is > 12: 3657500101, 88362852307, 2428095424619, 12055296811267, 75063692618249, 156740126985437, ..., . - Robert G. Wilson v, Mar 15 2000
CROSSREFS
Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A058332, A093047.
Primes p(n) such that n is a second-order nonprime number.
+10
12
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
EXAMPLE
Nonprime(4) = 8.
The 8th prime is 19, the second entry.
MAPLE
For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016
MATHEMATICA
nonPrime[n_Integer] := FixedPoint[n + PrimePi[ # ] &, n]; Prime /@ nonPrime /@ nonPrime /@ Range[54] (* Robert G. Wilson v, Feb 04 2005 *)
PROG
(PARI) \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) }
CROSSREFS
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.
The prime/nonprime compound sequence ABA.
+10
12
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
MAPLE
# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016
CROSSREFS
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.
The prime/nonprime compound sequence BAA.
+10
12
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
MAPLE
# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016
CROSSREFS
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.
The prime/nonprime compound sequence BBA.
+10
12
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
MAPLE
# For Maple code for the prime/nonprime compound sequences (listed in cross-references) see A003622. - N. J. A. Sloane, Mar 30 2016
CROSSREFS
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.
MM-numbers of labeled graphs with loops, without isolated vertices.
+10
12
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
COMMENTS
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.
EXAMPLE
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}}
MATHEMATICA
Select[Range[100], SquareFreeQ[#]&&FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; PrimeOmega[PrimePi[p]]!=2]&]
CROSSREFS
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. 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.
Composite numbers whose prime indices are also composite.
+10
11
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
COMMENTS
A prime index of n is a number m such that prime(m) divides n.
EXAMPLE
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)
MATHEMATICA
Select[Range[2, 1000], And[OddQ[#], !PrimeQ[#], And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]]&]
CROSSREFS
Cf. A000040, A006450, A007821, A018252, A050370, A056239, A076610, A112798, A302242, A302478, A320533, A320628, A320629.
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