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Search: A001222
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| A001222 |
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Number of prime divisors of n (counted with multiplicity).
(Formerly M0094 N0031)
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+30
601
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| 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3, 2
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Also called bigomega(n) or Omega(n).
Maximal number of terms in any factorization of n.
Number of prime powers (not including 1) that divide n.
Sum of exponents in prime-power factorization of n. [From Daniel Forgues (squid(AT)zensearch.com), Mar 29 2009]
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 119, #12, omega(n)..
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
Amarnath Murthy, Generalization of Parition Function and Introducing Smarandache Factor Partitions, Smarandache Notions Journal Vol. 11, 1-2-3 Spring 2000.
Amarnath Murthy, Length and Extent of Smarandache Factor Partitions, Smarandache Notions Journal Vol. 11, 1-2-3 Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.10.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..100000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. L. Perez et al., eds., Smarandache Notions Journal
S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wolfram Research, First 50 numbers factored
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FORMULA
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n = Product (p_j^k_j) -> a(n) = Sum (k_j).
Dirichlet generating function: ppzeta(s)*zeta(s). Here ppzeta(s) = sum_{p prime} sum_{k=1}^{infinity} 1/(p^)k^s. Note that ppzeta(s) = sum_{p prime} 1/(p^s-1) and ppzeta(s) = sum_{k=1}^{infinity} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005.
Totally additive with a(p) = 1.
a(n) = if n=1 then 0 else a(n/A020639(n)) + 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 25 2008
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EXAMPLE
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16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3.
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MAPLE
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with(numtheory): seq(bigomega(n), n=1..111);
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MATHEMATICA
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Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105]
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PROGRAM
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(PARI) v=[ ]; for (n=1, 100, v=concat(v, bigomega(n))); v
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CROSSREFS
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Cf. A001221 (primes counted without multiplicity), A046660, A144494. Bisections give A091304 and A073093. A086436 is essentially the same sequence.
a(n) = A091222(A091202(n)).
Sequence in context: A116479 A122810 A086436 this_sequence A098893 A069248 A008481
Adjacent sequences: A001219 A001220 A001221 this_sequence A001223 A001224 A001225
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KEYWORD
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nonn,easy,nice,core
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net).
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| A079148 |
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Primes p such that p-1 has at most 2 prime factors, counted with multiplicity; i.e. primes p such that bigomega(p-1) = A001222(p-1) <= 2. |
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+20
12
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| 2, 3, 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Sum of reciprocals ~ 1.477.
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EXAMPLE
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83 is in the sequence because 83-1 = 2*41 has 2 prime factors.
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PROGRAM
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(PARI) s(n) = {sr=0; forprime(x=2, n, if(bigomega(x-1) < 3, print1(x" "); sr+=1.0/x; ); ); print(); print(sr); } \\ Lists primes p<=n such that p-1 has at most 2 prime factors.
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CROSSREFS
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Except for 2 and 3, this is identical to A005385. Cf. A079147, A079149, A079151.
Except for 2, this is identical to A005385.
Sequence in context: A165802 A107798 A119660 this_sequence A107367 A036342 A114421
Adjacent sequences: A079145 A079146 A079147 this_sequence A079149 A079150 A079151
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KEYWORD
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easy,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Dec 27 2002
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| A124508 |
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2^BigO(n) * 3^omega(n), where BigO=A001222 and omega=A001221, the numbers of prime factors of n with and without repetitions. |
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+20
8
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| 1, 6, 6, 12, 6, 36, 6, 24, 12, 36, 6, 72, 6, 36, 36, 48, 6, 72, 6, 72, 36, 36, 6, 144, 12, 36, 24, 72, 6, 216, 6, 96, 36, 36, 36, 144, 6, 36, 36, 144, 6, 216, 6, 72, 72, 36, 6, 288, 12, 72, 36, 72, 6, 144, 36, 144, 36, 36, 6, 432, 6, 36, 72, 192, 36, 216, 6, 72, 36, 216, 6, 288, 6
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n) = A061142(n)*A074816(n) = A000079(A001222(n))*A000244(A001221(n));
A124509 gives the range: A124509(n) = a(A124510(n)) and a(m) <> a(A124510(n)) for m < A124510(n);
for primes p, q with p<>q: a(p) = 6; a(p*q) = 36; a(p^k) = 3*2^k, k>0;
for squarefree numbers m: a(m) = 6^omega(m);
A001222(a(n)) = A001222(n)+1; A001221(a(n)) = 2 for n > 1;
A124511(n) = a(a(n)); A124512(n) = a(a(a(n)));
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LINKS
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R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization
Eric Weisstein's World of Mathematics, Smooth number
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FORMULA
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Multiplicative with p^e -> 3*2^e, p prime and e>0.
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CROSSREFS
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Cf. A007283, A000400, A003586, A005117.
Sequence in context: A163757 A109538 A040031 this_sequence A028317 A055665 A127402
Adjacent sequences: A124505 A124506 A124507 this_sequence A124509 A124510 A124511
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KEYWORD
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nonn,mult
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2006
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| A068936 |
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Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(n)<=A001222(n). |
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+20
7
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| 1, 4, 8, 16, 27, 32, 48, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 320, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 800, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1792, 1944, 2000, 2048, 2187, 2304, 2560, 2592, 2916
(list; graph; listen)
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| A069346 |
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Primes of the form n - BigOmega(n), where BigOmega(n) is the number of prime-factors of n, A001222(n). |
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+20
6
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| 2, 5, 7, 13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 67, 73, 83, 89, 103, 107, 109, 113, 127, 131, 139, 151, 157, 163, 167, 179, 181, 199, 211, 227, 233, 239, 241, 251, 257, 263, 281, 283, 293, 307, 311, 313, 317, 337, 347, 353, 359
(list; graph; listen)
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| 0, 0, 1, 0, 0, -2, -2, -2, -2, -2, -6, -2, -4, -2, -7, -1, -5, 0, -7, -3, -9, 1, -11, 2, -7, 1, -12, 1, -11, 7, -8, -5, -8, -1, -18, 3, -10, 1, -13, 1, -7, 13, -12, -2, -13, 6, -16, 3, -11, 3, -15, -4, -16, 13, -15, -4, -15, 4, -17, 11, -14, 4, -13, 7, -12, 15, -17, -5, -15, 16, -13, 3, -12, 3, -20, 3, -27, 19, -20, -3, -11, 3
(list; graph; listen)
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| A064612 |
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Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n). |
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+20
5
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OFFSET
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1,2
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COMMENT
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Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
Partial sums of A001222, similarly to summatory A001221 increases like loglog(n), explaining small quotients.
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FORMULA
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Mod[A022559(n), n]=0
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EXAMPLE
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Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.
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CROSSREFS
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A001222, A022559, A050226, A056650, A064602-A064611, A048290, A062982, A045345.
Sequence in context: A042717 A134463 A058916 this_sequence A005927 A079207 A056945
Adjacent sequences: A064609 A064610 A064611 this_sequence A064613 A064614 A064615
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KEYWORD
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nonn
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Sep 24 2001
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| A064800 |
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n plus the number of its prime-factors: a(n) = n + A001222(n). |
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+20
5
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| 1, 3, 4, 6, 6, 8, 8, 11, 11, 12, 12, 15, 14, 16, 17, 20, 18, 21, 20, 23, 23, 24, 24, 28, 27, 28, 30, 31, 30, 33, 32, 37, 35, 36, 37, 40, 38, 40, 41, 44, 42, 45, 44, 47, 48, 48, 48, 53, 51, 53, 53, 55, 54, 58, 57, 60, 59, 60, 60, 64, 62, 64, 66, 70, 67, 69, 68, 71, 71, 73, 72
(list; graph; listen)
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OFFSET
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1,2
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,1000
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EXAMPLE
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a(42) = 45 = 42 + 3 (as 42 = 2 * 3 * 7)
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PROGRAM
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(PARI) { for (n=1, 1000, write("b064800.txt", n, " ", n + bigomega(n)) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Sep 26 2009]
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CROSSREFS
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A001222.
Sequence in context: A004219 A077542 A023836 this_sequence A078574 A162625 A033095
Adjacent sequences: A064797 A064798 A064799 this_sequence A064801 A064802 A064803
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 21 2001
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| A068935 |
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Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n)<A001222(n). |
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+20
5
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| 8, 16, 32, 64, 81, 96, 128, 144, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1944, 2048, 2187
(list; graph; listen)
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| A068937 |
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Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n). |
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+20
5
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| 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76
(list; graph; listen)
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