A008364 -id:A008364 - cp's OEIS frontend

A166061 19-rough numbers: positive integers that have no prime factors less than 19.

1, 19, 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

Offset: 1

Michael B. Porter, Oct 05 2009

Or, positive integers relatively prime to 510510 = 2*3*5*7*11*13*17.

			437 = 19 * 23 is in the sequence since the two prime factors, 19 and 23, are not less than 19.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rough Number
Index entries for sequences related to smooth numbers [From _Michael B. Porter_, Oct 10 2009]

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

Join[{1},Select[Range[300],Min[Transpose[FactorInteger[#]][[1]]]>18&]] (* Harvey P. Dale, Feb 18 2015 *)

isA166061(n) = gcd(n,510510)==1 \\ Michael B. Porter, Oct 10 2009

a(n) = k*n + O(1) where k = 17017/3072 = 5.539388.... In particular, k*n - 31 < a(n) < k*n + 25. - Charles R Greathouse IV, Sep 24 2018

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

A038511 Composite numbers with smallest prime factor >= 11.

Original entry on oeis.org

121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 341, 361, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 529, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703, 713, 731, 737, 767, 779, 781, 793, 799, 803, 817, 841, 851

Offset: 1

Jeff Burch

nonn

Composite n such that n^6 is congruent to {1, 169} mod 210. All primes > 7 satisfy this condition. - Gary Detlefs, Dec 09 2012

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A008364.

Filtered([11..1000],n->(PowerMod(n,6,210)=1 or PowerMod(n,6,210)=169) and not IsPrime(n)); # Muniru A Asiru, Nov 24 2018

for n from 1 to 1000 do if (n^6 mod 210 = 1 or n^6 mod 210 = 169) and not isprime(n) then print(n) fi od; # Gary Detlefs, Dec 09 2012

Select[Range[1000], Not[PrimeQ[#]] && FactorInteger[#][[1, 1]] > 7 &] (* Alonso del Arte, Dec 09 2012 *)

is(n)=gcd(210,n)==1 && !isprime(n) \\ Charles R Greathouse IV, Dec 10 2012

a(n) ~ 4.375n. - Charles R Greathouse IV, Dec 10 2012

A270298 Numbers which are representable as a sum of eight but no fewer consecutive nonnegative integers.

Original entry on oeis.org

44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 484, 508, 524, 548, 556, 572, 596, 604, 628, 652, 668, 676, 692, 716, 724, 748, 764, 772, 788, 796, 836, 844, 884, 892, 908, 916, 932

Offset: 1

Martin Renner, Mar 14 2016

nonn

			36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 11 + 12 + 13 (not in sequence);
44 = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9;
52 = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10;
68 = 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A008364, A138591, A270296, A270297, A270299, A270300, A270301, A270302, A270303.

A163169(a(n)) = 8. - Ray Chandler, Mar 22 2016

a(n) = 4*A008364(n+1). - Hugo Pfoertner, Feb 04 2021

A078859 Least positive residues (mod 210) representing those residue classes which can be the lesser of twin prime pairs (A001359).

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 167, 179, 191, 197, 209

Offset: 1

Labos Elemer, Dec 13 2002

Cf. A001359, A008364.

With[{n = 4}, Function[P, Join[Select[Prime@ Range@ n, NextPrime@ # == # + 2 &], Select[Partition[Select[Range[P + 1], CoprimeQ[#, P] &], 2, 1], Differences@ # == {2} &][[All, 1]]]]@ Product[Prime@ i, {i, n}]] (* Michael De Vlieger, May 15 2017 *)

Intersection[RRS(210), 2+RRS{210)]-2 and {3, 5}. RRS(210)=reduced residue system of 210=first 48=phi(210) terms of A008364; two additional term 3 and 5 are singular cases; 210k+r generates complete A001359 with suitable k and r taken from these 15+2 numbers.

A092695 Number of positive integers less than or equal to n which are not divisible by the primes 2,3,5,7.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 0

Michael Somos, Mar 04 2004

nonn

This sequence is a special case of the following: Take different primes p_1, p_2,...,p_k. For a nonempty subset I of {1,2,...,k} denote by |I| the number of its elements. For a positive integer n denote A(n,I) = floor(n/Product_{i in I} p_i). Then the number of positive integers m <= n such that m is divisible by none of p_1,p_2,...,p_k is equal to n + Sum_{} (-1)^(|I|)*A(n,I), where I runs over all nonempty subsets of {1,2,...,k}. - Milan Janjic, Apr 23 2007

			x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + ...

John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 62.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for two-way infinite sequences

Cf. A020639, A008364.

a092695 n = a092695_list !! n
a092695_list = scanl (+) 0 $
               map (fromEnum . (> 7)) (8 : tail a020639_list)
-- Reinhard Zumkeller, Mar 26 2012

Accumulate @ Table[Boole @ CoprimeQ[n, 210], {n, 0, 100}] (* Amiram Eldar, Dec 06 2020 *)

{a(n) = n - n\2 - n\3 - n\5 - n\7 + n\6 + n\10 + n\14 + n\15 + n\21 - n\30 + n\35 - n\42 - n\70 - n\105 + n\210}

{a(n) = if( n<0, -a(-1 - n), sum( k=0, n, 1==gcd( k, 210)))}

G.f.: (x * P172 * P36) / (e(1) * e(210)) where e(n) = 1 - x^n, P36 = e(16) * e(20) * e(24) / (e(6) * e(8) * e(10)) is a polynomial of degree 36 and P172 is a polynomial of degree 172.
a(n + 210) = a(n) + 48.
a(n) = -a(-1 - n) for n < 0.
a(n) ~ (8/35)*n. - Amiram Eldar, Dec 06 2020

A322273 Smallest multiplication factors f, prime or 1, for all b (mod 840), coprime to 840 (= 47#), so that bf is a nonzero square mod 8, mod 3, mod 5, and mod 7.

Original entry on oeis.org

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 43, 71, 73, 79, 83, 41, 73, 101, 103, 107, 109, 113, 1, 127, 59, 113, 19, 47, 29, 79, 13, 43, 47, 1, 173, 11, 61, 283, 71, 193, 53, 31, 41, 211, 29, 103, 83, 61, 113, 71, 241, 127, 59, 37, 17, 23

Offset: 1

Hans Ruegg, Dec 01 2018

See sequence A322269 for further explanations. This sequence is related to A322269(4).
The sequence is periodic, repeating itself after phi(840) = 192 terms. Its largest term is 311, which is A322269(4). In order to satisfy the conditions, both f and b must be coprime to 840. Otherwise, the product would be zero mod a prime <= 7.
The b(n) corresponding to each a(n) is A008364(n).
The first 15 terms are trivial: f=b, and then the product b*f naturally is a square modulo everything.

			The 16th number coprime to 840 is 67. a(16) is 43, because 43 is the smallest prime by which we can multiply 67, so that the product (67*43 = 2881) is a square mod 8, mod 2, mod 3, mod 5, and mod 7.

Hans Ruegg, Table of n, a(n) for n = 1..192

Cf. A322269, A322271, A322272, A322274, A322275, A008364.

QresCode(n, nPrimes) = {
  code = bitand(n,7)>>1;
  for (j=2, nPrimes,
    x = Mod(n,prime(j));
    if (issquare(x), code += (1<A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.

A080672 Numbers having divisors 2 or 3 or 5 or 7.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Cino Hilliard, Mar 02 2003

A020639(a(n)) <= 7; A210679(a(n)) > 0. - Reinhard Zumkeller, Apr 02 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 162.

Cf. A020639, A008364 (complement).

Subsequences: A002473, A343597.

a080672 n = a080672_list !! (n-1)
a080672_list = filter ((<= 7) . a020639) [2..]
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],Length[Intersection[Divisors[#],{2,3,5,7}]]>0&] (* Harvey P. Dale, Apr 03 2024 *)

div2357(n)= for(x=1,n, if(gcd(x,210)<>1,print1(x" ")) )

is(n)=gcd(n,210)>1 \\ Charles R Greathouse IV, Sep 14 2015

From Charles R Greathouse IV, Sep 14 2015: (Start)
a(n) = 35n/27 + O(1).
For n > 162, a(n) = a(n-162) + 210. [Corrected by Peter Munn, Apr 22 2021]
(End)
For n < 162, a(n) = 210 - a(162-n). - Peter Munn, Apr 22 2021

Offset fixed by Reinhard Zumkeller, Apr 02 2012

A210679 Number of distinct prime factors <= 7 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 2, 2, 0, 1, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 2, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1

Offset: 1

Reinhard Zumkeller, Apr 01 2012

Periodic with period length 210. - Amiram Eldar, Sep 16 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-5,-6,-6,-5,-3,0,3,5,6,6,5,3,1).

Cf. A001221, A002473, A008364, A027748, A080672, A165743.

Number of distinct prime factors <= p: A171182 (p=3), A178146 (p=5), this sequence (p=7).

a210679 = length . takeWhile (<= 7) . a027748_row

Join[{0},Table[Count[FactorInteger[n][[All,1]],?(#<8&)],{n,2,100}]] (* _Harvey P. Dale, Aug 18 2021 *)
a[n_] := PrimeNu[GCD[n, 210]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

a(n) = omega(gcd(n, 210)); \\ Amiram Eldar, Sep 16 2023

a(n) <= 4.
a(A008364(n)) = 0; a(A080672(n)) > 0.
a(n) = A001221(n) iff n is 7-smooth: a(A002473(n)) = A001221(A002473(n)). [corrected by Amiram Eldar, Sep 16 2023]
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 7, and 0 otherwise.
a(n) = A001221(A165743(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 247/210. (End)

A236206 Numbers not divisible by 3, 5 or 7.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 17, 19, 22, 23, 26, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 88, 89, 92, 94, 97, 101, 103, 104, 106, 107, 109, 113, 116, 118, 121, 122, 124, 127, 128, 131, 134, 136

Offset: 1

Oleg P. Kirillov, Jan 20 2014

Numbers whose odd part is 11-rough: products of terms of A008364 and powers of 2 (terms of A000079). - Peter Munn, Aug 03 2020

Numbers coprime to 105. The asymptotic density of this sequence is 16/35. - Amiram Eldar, Oct 23 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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).

Subsequences: A000079, A008364.
Intersection of any 2 of A160545, A229829, A235933.
Other sequences with similar definitions: A007775, A236217.

Select[Range[300], Mod[#, 3] > 0 && Mod[#, 5] > 0 && Mod[#, 7] > 0 &] (* T. D. Noe, Feb 05 2014 *)
Select[Range[300],Or@@Divisible[#,{3,5,7}]==False&] (* Harvey P. Dale, Mar 13 2014 *)
Select[Range[150], CoprimeQ[105, #] &] (* Amiram Eldar, Oct 23 2020 *)

a(n) = a(n-1) + a(n-48) - a(n-49). - Amiram Eldar, Oct 23 2020

cp's OEIS Frontend

A166061 19-rough numbers: positive integers that have no prime factors less than 19.

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

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A038511 Composite numbers with smallest prime factor >= 11.

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A270298 Numbers which are representable as a sum of eight but no fewer consecutive nonnegative integers.

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A078859 Least positive residues (mod 210) representing those residue classes which can be the lesser of twin prime pairs (A001359).

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A092695 Number of positive integers less than or equal to n which are not divisible by the primes 2,3,5,7.

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A322273 Smallest multiplication factors f, prime or 1, for all b (mod 840), coprime to 840 (= 4*7#), so that b*f is a nonzero square mod 8, mod 3, mod 5, and mod 7.

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A322273 Smallest multiplication factors f, prime or 1, for all b (mod 840), coprime to 840 (= 47#), so that bf is a nonzero square mod 8, mod 3, mod 5, and mod 7.