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 A007775 Numbers not divisible by 2, 3 or 5. 59
 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also numbers n such that the sum of the 4th powers of the first n positive integers is divisible by n, or A000538(n) = n*(n+1)(2*n+1)(3*n^2+3*n-1)/30 is divisible by n. - Alexander Adamchuk, Jan 04 2007 Also the 7-rough numbers: positive integers that have no prime factors less than 7. - Michael B. Porter, Oct 09 2009 a(n) mod 3 has period 8, repeating [1,1,2,1,2,1,2,2] = (n mod 2) + floor(((n-1) mod 8)/7) - floor(((n-2) mod 8)/7) + 1. floor(a(n)/3) is the set of numbers k such that k is congruent to {0,2,3,4,5,6,7,9} mod 10 = floor((5*n-2)/4)-floor((n mod 8)/6). - Gary Detlefs, Jan 08 2012 Numbers k such that C(k+3,3)==1 (mod k) and C(k+5,5)==1 (mod k). - Gary Detlefs, Sep 15 2013 a(n) mod 30 has period 8 repeating [1, 7, 11, 13, 17, 19, 23, 29]. The mean of these 8 numbers is 120/8 = 15. (a(n)-15) mod 30 has period 8 repeating [-14, -8, -4, -2, 2, 4, 8, 14]. One half of the absolute value produces the symmetric sequence [7, 4, 2, 1, 1, 2, 4, 7] = A061501(((n-1) + 16) mod 8). - Gary Detlefs, Sep 24 2013 a(n) are exactly those positive integers m such that the sequence b(n) = n*(n + m)*(n + 2*m)*(n + 3*m)(n + 4*m)/120 is integral. Cf. A007310. - Peter Bala, Nov 13 2015 The asymptotic density of this sequence is 4/15. - Amiram Eldar, Sep 30 2020 LINKS N. J. A. Sloane, Table of n, a(n) for n = 1..8000 Peter Bala, A note on A007775. Abel Jansma, E_8 Symmetry Structures in the Ising model, Master's thesis, University of Amsterdam, 2018. Eric Weisstein's World of Mathematics, Rough Number. Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1). FORMULA A141256(a(n)) = n+1. - Reinhard Zumkeller, Jun 17 2008 From R. J. Mathar, Feb 27 2009: (Start) a(n+8) = a(n) + 30. a(n) = a(n-1) + a(n-8) - a(n-9). G.f.: x*(1 + 6*x + 4*x^2 + 2*x^3 + 4*x^4 + 2*x^5 + 4*x^6 + 6*x^7 + x^8)/((1 + x)*(x^2 + 1)*(x^4 + 1)*(x-1)^2). (End) a(n) = 4*n - 3 - 2*floor((n-1)/8) + (1 + (-1)^floor((n-2)/2))*(-1)^floor((n-2)/4), n >= 1. - Timothy Hopper, Mar 14 2010 a(1 - n) = -a(n). - Michael Somos, Feb 05 2011 Numbers k such that ((k^2 mod 48=1) or (k^2 mod 48=25)) and ((k^2 mod 120=1) or (k^2 mod 120=49)). - Gary Detlefs, Dec 30 2011 Numbers k such that k^2 mod 30 is 1 or 19. - Gary Detlefs, Dec 31 2011 a(n) = 3*(floor((5*n-2)/4) - floor((n mod 8)/6)) + (n mod 2) + floor(((n-1) mod 8)/7) - floor(((n-2) mod 8)/7) + 1. - Gary Detlefs, Jan 08 2012 a(n) = 4*n - 3 + 2*(floor((n+6)/8) - floor((n+4)/8) - floor((n+2)/8) + floor(n/8) - floor((n-1)/8)), n >= 1. From the o.g.f. given above by R. J. Mathar (with the denominator written as (1-x^8)*(1-x)), and a two-step reduction of the floor functions. Compare with Hopper's and Detlefs's formulas above. - Wolfdieter Lang, Jan 26 2012 a(n) = (6*f(n) - 3 + (-1)^f(n))/2, where f(n)= n + floor(n/4)+ floor(((n+4) mod 8)/6). - Gary Detlefs, Sep 15 2013 a(n) = 30*floor((n-1)/8) + 15 + 2*f((n-1) mod 8 + 16)*(-1)^floor(((n+3) mod 8)/4), where f(n) = (n*(n+1)/2+1) mod 10. - Gary Detlefs, Sep 24 2013 a(n) = 3*n + 6*floor(n/8) + (n mod 2) - 2*floor(((n-2) mod 8)/6) - 2*floor(((n-2) mod 8)/7) + 1. - Gary Detlefs, Jun 01 2014 a(n+1) = ((n << 2 - n >> 2) || 1) + ((n << 1 - n >> 1) && 2), where << and >> are bitwise left and right shifts, || and && are bitwise "or" and "and". - Andrew Lelechenko, Jul 08 2017 From Mikk Heidemaa, Dec 06 2017: (Start) a(n) = 2*n + 2*floor(1/2 + (7*n)/8) + 2*(91 mod (2 - ((3*n)/4 + n^2/4) mod 2)) - 3 (n > 0). a(n+1) = (1/8)*i^(-n)*((-1 - i) - (1 - i)*(-1)^n - (-i)^n + 15*i^n + 30*i^n*n - 2*cos((n*Pi)/4) + 2*i*sqrt(2)*cos((n*Pi)/4) - 2*cos((3*n*Pi)/4) - 2*i*sqrt(2)*cos((3*n*Pi)/4) - 6*sin((n*Pi)/4) + 4*i*sqrt(2)*sin((n*Pi)/4) + 6*sin((3*n*Pi)/4) + 4*i*sqrt(2)*sin((3*n*Pi)/4)) (for n >= 0), where i is the imaginary unit. (End) MAPLE for i from 1 to 500 do if gcd(i, 30) = 1 then print(i); fi; od; for k from 1 to 300 do if ((k^2 mod 48=1) or (k^2 mod 48=25)) and ((k^2 mod 120=1) or (k^2 mod 120=49)) then print(k) fi od. # Gary Detlefs, Dec 30 2011 MATHEMATICA Select[ Range[ 300 ], GCD[ #1, 30 ]==1& ] Select[Range[250], Mod[#, 2]>0&&Mod[#, 3]>0&&Mod[#, 5]>0&] (* Vincenzo Librandi, Feb 08 2014 *) a[ n_] := Quotient[ n, 8, 1] 30 + {1, 7, 11, 13, 17, 19, 23, 29}[[Mod[n, 8, 1]]]; (* Michael Somos, Jun 02 2014 *) LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 7, 11, 13, 17, 19, 23, 29, 31}, 100] (* Mikk Heidemaa, Dec 07 2017 *) Cases[Range@1000, x_ /; NoneTrue[Array[Prime, 3], Divisible[x, #] &]] (* Mikk Heidemaa, Dec 07 2017 *) CoefficientList[ Series[(x^8 + 6x^7 + 4x^6 + 2x^5 + 4x^4 + 2x^3 + 4x^2 + 6x + 1)/((x - 1)^2 (x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 55}], x] (* Robert G. Wilson v, Dec 07 2017 *) PROG (PARI) isA007775(n) = gcd(n, 30)==1 \\ Michael B. Porter, Oct 09 2009 (PARI) {a(n) = n\8 * 30 + [ -1, 1, 7, 11, 13, 17, 19, 23][n%8 + 1]} /* Michael Somos, Feb 05 2011 */ (PARI) {a(n) = n\8 * 6 + 9 + 3 * (n+1)\2 * 2 - max(5, (n-2)%8) * 2} /* Michael Somos, Jun 02 2014 */ (PARI) Vec(x*(1+6*x+4*x^2+2*x^3+4*x^4+2*x^5+4*x^6+6*x^7+x^8)/((1+x)*(x^2+1)*(x^4+1)*( x-1)^2) + O(x^100)) \\ Altug Alkan, Nov 16 2015 (Haskell) a007775 n = a007775_list !! (n-1) a007775_list = 1 : filter ((> 5) . a020639) [1..] -- Reinhard Zumkeller, Jan 06 2013 (Sage) a = lambda n: ((((n-1)<< 2)-((n-1)>>2))|1) + ((((n-1)<<1)-((n-1)>> 1)) & 2) print([a(n) for n in (1..56)]) # after Andrew Lelechenko, Peter Luschny, Jul 08 2017 (MAGMA) I:=[1, 7, 11, 13, 17, 19, 23, 29, 31]; [n le 9 select I[n] else Self(n-1) +Self(n-8) - Self(n-9): n in [1..80]]; // G. C. Greubel, Oct 22 2018 CROSSREFS Cf. A000538, A054403, A145011 (first differences). For k-rough numbers with other values of k, see A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063. Complement is A080671. For digital root of Fibonacci numbers indexed by this sequence, see A227896. Sequence in context: A005776 A322272 A161850 * A070884 A135777 A090459 Adjacent sequences:  A007772 A007773 A007774 * A007776 A007777 A007778 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified December 1 03:31 EST 2020. Contains 338833 sequences. (Running on oeis4.)