A332798 -id:A332798 - cp's OEIS frontend

A084968 Multiples of 7 coprime to 30.

7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511, 539, 553, 581, 623, 637, 679, 707, 721, 749, 763, 791, 833, 847, 889, 917, 931, 959, 973, 1001, 1043, 1057, 1099, 1127, 1141, 1169, 1183, 1211, 1253, 1267, 1309

Offset: 1

Robert G. Wilson v, Jun 15 2003

Numbers 7*k such that gcd(k,30) = 1.

Numbers congruent to 7, 49, 77, 91, 119, 133, 161, 203 modulo 210. - Jianing Song, Nov 18 2022

			77 is in the sequence because gcd(77, 30) = 1.
84 is not in the sequence because gcd(84, 3) = 6.
91 is in the sequence because gcd(91, 30) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Subsequence of A008589.
Fourth row of A083140.
Cf. A084967 (5), A084969 (11), A084970 (13), A332799 (17), A332798 (19), A332797 (23), A007775 (7-rough numbers).

q:= k-> igcd(k, 30)=1:
select(q, [7*i$i=1..300])[];  # Alois P. Heinz, Feb 25 2020

7Select[ Range[190], GCD[ #, 2*3*5] == 1 & ]

is(n)=gcd(210,n)==7 \\ Charles R Greathouse IV, Aug 05 2013

G.f.: 7*x*(x^8 + 6*x^7 + 4*x^6 + 2*x^5 + 4*x^4 + 2*x^3 + 4*x^2 + 6*x + 1) / ((x-1)^2*(x+1)*(x^2+1)*(x^4+1)). - Colin Barker, Feb 24 2013
Lim_{n->oo} a(n)/n = A038111(4)/A038110(4) = 105/4. - Vladimir Shevelev, Jan 20 2015
a(n) = 7*A007775(n).
a(n+8) = a(n) + 210. - Jianing Song, Nov 18 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(23 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/105. - Amiram Eldar, Jul 15 2023

A084970 Numbers whose smallest prime factor is 13.

Original entry on oeis.org

13, 169, 221, 247, 299, 377, 403, 481, 533, 559, 611, 689, 767, 793, 871, 923, 949, 1027, 1079, 1157, 1261, 1313, 1339, 1391, 1417, 1469, 1651, 1703, 1781, 1807, 1937, 1963, 2041, 2119, 2171, 2197, 2249, 2327, 2353, 2483, 2509, 2561, 2587, 2743, 2873

Offset: 1

Robert G. Wilson v, Jun 15 2003

			a(2) = 13*13, a(3) = 13*17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 481.

Sixth row of A083140.

Cf. A084967 (5), A084968 (7), A084969 (11), A332799 (17), A332798 (19), A332797 (23), A008365 (13-rough numbers).

Select[ 13Range@ 225, GCD[#, 2310] == 1 &] (* Robert G. Wilson v, Dec 18 2014 *)

is(n)=n%13==0 && gcd(n,2310)==1 \\ Charles R Greathouse IV, Nov 19 2014

a(n) = a(n-480) + 30030 = a(n-1) + a(n-480) - a(n-481). - Charles R Greathouse IV, Nov 19 2014
Lim_{n->infinity} a(n)/n = A038111(6)/A038110(6) = 1001/16 = 62.5625. - Vladimir Shevelev, Jan 20 2015
a(n) = 13*A008365(n).

More terms from David Wasserman, Oct 19 2004

A084969 Numbers whose smallest prime factor is 11.

Original entry on oeis.org

11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1573, 1639, 1661, 1727, 1793, 1837, 1859, 1903, 1969, 1991, 2057, 2101, 2123, 2167, 2189, 2299, 2321

Offset: 1

Robert G. Wilson v, Jun 15 2003

Fifth row of A083140.

Integers k such that gcd(11*k, 210) = 1.

			a(2) = 11*11, a(3) = 11*13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A084967 (5), A084968 (7), A084970 (13), A332799 (17), A332798 (19), A332797 (23), A008364 (11-rough numbers).

Cf. A008364, A038110, A038111, A083140.

11Select[ Range[210], GCD[ #, 2*3*5*7] == 1 & ]
Select[11*Range[0,200],GCD[#,210]==1&] (* Harvey P. Dale, Dec 23 2013 *)

is(n)=gcd(n,2310)==11 \\ Charles R Greathouse IV, Nov 19 2014

G.f.: 11*x*(x^48 +10*x^47 +2*x^46 +4*x^45 +2*x^44 +4*x^43 +6*x^42 +2*x^41 +6*x^40 +4*x^39 +2*x^38 +4*x^37 +6*x^36 +6*x^35 +2*x^34 +6*x^33 +4*x^32 +2*x^31 +6*x^30 +4*x^29 +6*x^28 +8*x^27 +4*x^26 +2*x^25 +4*x^24 +2*x^23 +4*x^22 +8*x^21 +6*x^20 +4*x^19 +6*x^18 +2*x^17 +4*x^16 +6*x^15 +2*x^14 +6*x^13 +6*x^12 +4*x^11 +2*x^10 +4*x^9 +6*x^8 +2*x^7 +6*x^6 +4*x^5 +2*x^4 +4*x^3 +2*x^2 +10*x +1) / (x^49 -x^48 -x +1). - Colin Barker, Feb 22 2013
a(n) = a(n-48) + 2310 = a(n-1) + a(n-48) - a(n-49). - Charles R Greathouse IV, Nov 19 2014
Lim_{n->infinity} a(n)/n = A038111(5)/A038110(5) = 385/8 = 48.125. - Vladimir Shevelev, Jan 20 2015
a(n) = 11*A008364(n).

a(47)-a(49) from Georg Fischer, Nov 07 2019

New name from Frank Ellermann, Feb 25 2020

A166063 23-rough numbers: positive integers that have no prime factors less than 23.

Original entry on oeis.org

1, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499

Offset: 1

Michael B. Porter, Oct 05 2009

Or, positive integers relatively prime to 9699690 = 2*3*5*7*11*13*17*19.

First composite term is 529 = 23^2.

			667 = 23 * 29 is in the sequence since the two prime factors, 23 and 29, are not less than 23.

G. C. Greubel, Table of n, a(n) for n = 1..1200
Eric Weisstein's World of Mathematics, Rough Number
Index entries for sequences related to smooth numbers

k-rough numbers: A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063.

Cf. A332797 (subsequence).

A166063 := proc(n)
    option remember;
    local a;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            numtheory[factorset](a) ;
            if min(op(%)) >= 23 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A166063(n),n=1..80) ; # R. J. Mathar, Nov 05 2024

Select[Range[500],FactorInteger[#][[1,1]]>22&] (* Harvey P. Dale, Nov 22 2010 *)

isA166063(n) = gcd(n,9699690)==1 \\ Michael B. Porter, Oct 10 2009

a(n) = k*n + O(1) where k = 323323/55296 = 5.8471.... In particular, k*n - 51 < a(n) < k*n + 45. - Charles R Greathouse IV, Sep 21 2018

A166061 SETMINUS A332798 - R. J. Mathar, Nov 05 2024

Additional terms provided provided by Harvey P. Dale, Nov 22 2010

A332797 Numbers whose smallest prime factor is 23.

Original entry on oeis.org

23, 529, 667, 713, 851, 943, 989, 1081, 1219, 1357, 1403, 1541, 1633, 1679, 1817, 1909, 2047, 2231, 2323, 2369, 2461, 2507, 2599, 2921, 3013, 3151, 3197, 3427, 3473, 3611, 3749, 3841, 3979, 4117, 4163, 4393, 4439, 4531, 4577, 4853, 5129, 5221, 5267, 5359, 5497

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 55296/7436429. - Amiram Eldar, Dec 06 2020

			a(2) = 23*23, a(3) = 23*29.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332799 (17), A332798 (19), A166063 (23-rough numbers).

23 * Select[Range[240], CoprimeQ[#, 9699690] &] (* Amiram Eldar, Feb 24 2020 *)
Select[Range[6000],FactorInteger[#][[1,1]]==23&] (* Harvey P. Dale, Aug 29 2025 *)

P = 23         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 23*A166063(n).

A332799 Numbers whose smallest prime factor is 17.

Original entry on oeis.org

17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037, 1139, 1207, 1241, 1343, 1411, 1513, 1649, 1717, 1751, 1819, 1853, 1921, 2159, 2227, 2329, 2363, 2533, 2567, 2669, 2771, 2839, 2941, 3043, 3077, 3247, 3281, 3349, 3383, 3587, 3791, 3859, 3893

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 192/17017. - Amiram Eldar, Dec 06 2020

			a(2) = 17*17, a(3) = 17*19.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332798 (19), A332797 (23), A008366 (17-rough numbers).

17 * Select[Range[230], CoprimeQ[#, 30030] &] (* Amiram Eldar, Feb 24 2020 *)

P = 17         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 17*A008366(n).

cp's OEIS Frontend

A084968 Multiples of 7 coprime to 30.

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A084970 Numbers whose smallest prime factor is 13.

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A084969 Numbers whose smallest prime factor is 11.

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A166063 23-rough numbers: positive integers that have no prime factors less than 23.

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A332797 Numbers whose smallest prime factor is 23.

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A332799 Numbers whose smallest prime factor is 17.

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