A007775 -id:A007775 - cp's OEIS frontend

A076205 Numbers n such that 30*n+{1,7,11,13,17,19,23,29} are all composite.

360, 523, 654, 941, 1020, 1047, 1064, 1136, 1188, 1213, 1264, 1280, 1343, 1355, 1445, 1477, 1515, 1526, 1530, 1533, 1582, 1623, 1652, 1693, 1842, 1900, 1960, 2018, 2039, 2129, 2208, 2280, 2309, 2332, 2406, 2413, 2440, 2499, 2539, 2622, 2633, 2650, 2657

Offset: 1

Donald S. McDonald, Nov 02 2002

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 141.

Klaus Brockhaus, Table of n, a(n) for n = 1..6829 (terms < 100000)

Cf. A005776, A007775, A100418-A100423.

[ n: n in [0..3000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 23, 29] | not IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[3000],AllTrue[30#+{1,7,11,13,17,19,23,29},CompositeQ]&] (* Harvey P. Dale, Jan 06 2022 *)

{cav(mx)= local(wp=[1,7,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<9),m=!isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}

More terms from Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004
Edited by Don Reble, Nov 17 2005
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A264449 a(n) = n(n + 7)(n + 14)(n + 21)(n + 28)/120.

Original entry on oeis.org

0, 638, 1656, 3162, 5280, 8151, 11934, 16807, 22968, 30636, 40052, 51480, 65208, 81549, 100842, 123453, 149776, 180234, 215280, 255398, 301104, 352947, 411510, 477411, 551304, 633880, 725868, 828036, 941192, 1066185, 1203906, 1355289, 1521312, 1702998, 1901416, 2117682

Offset: 0

Peter Bala, Nov 13 2015

It is well-known, and easy to prove, that the product of 5 consecutive integers n*(n + 1)*(n + 2)*(n + 3)*(n + 4) is divisible by 5!. It can be shown that the product of 5 integers in arithmetic progression n*(n + r)*(n + 2*r)*(n + 3*r)*(n + 4*r) is divisible by 5! if and only if r is not divisible by 2, 3 or 5 (see A007775 for these numbers). This is the case r = 7.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A007775, A264443, A264444, A264445, A264446, A264447, A264448, A264450.

[n*(n+7)*(n+14)*(n+21)*(n+28)/120: n in [0..40]]; // Vincenzo Librandi, Nov 16 2015

seq( n*(n + 7)*(n + 14)*(n + 21)*(n + 28)/120, n = 0..35 );

Table[n (n + 7) (n + 14) (n + 21) (n + 28)/120, {n, 0, 40}] (* Vincenzo Librandi, Nov 16 2015 *)

vector(100, n, n--; n*(n+7)*(n+14)*(n+21)*(n+28)/120) \\ Altug Alkan, Nov 15 2015

O.g.f.: x*(351*x^4 - 1612*x^3 + 2796*x^2 - 2172*x + 638)/(1 - x)^6.

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), for n>5. - Vincenzo Librandi, Nov 16 2015

A264450 a(n) = n(n + 11)(n + 22)(n + 33)(n + 44)/120.

Original entry on oeis.org

0, 3519, 8372, 14805, 23088, 33516, 46410, 62118, 81016, 103509, 130032, 161051, 197064, 238602, 286230, 340548, 402192, 471835, 550188, 638001, 736064, 845208, 966306, 1100274, 1248072, 1410705, 1589224, 1784727, 1998360, 2231318, 2484846, 2760240, 3058848, 3382071

Offset: 0

Peter Bala, Nov 13 2015

It is well-known, and easy to prove, that the product of 5 consecutive integers n*(n + 1)*(n + 2)*(n + 3)*(n + 4) is divisible by 5!. It can be shown that the product of 5 integers in arithmetic progression n*(n + r)*(n + 2*r)*(n + 3*r)*(n + 4*r) is divisible by 5! if and only if r is not divisible by 2, 3 or 5 (see A007775 for these numbers). This is the case r = 11.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A007775, A264443, A264444, A264445, A264446, A264447, A264448, A264449.

[n*(n+11)*(n+22)*(n+33)*(n+44)/120: n in [0..40]]; // Vincenzo Librandi, Nov 16 2015

seq( n*(n + 11)*(n + 22)*(n + 33)*(n + 44)/120, n = 0..33 );

Table[n (n + 11) (n + 22) (n + 33) (n + 44)/120, {n, 0, 40}] (* Vincenzo Librandi, Nov 16 2015 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,3519,8372,14805,23088,33516},40] (* Harvey P. Dale, Nov 27 2015 *)

vector(100, n, n--; n*(n+11)*(n+22)*(n+33)*(n+44)/120) \\ Altug Alkan, Nov 15 2015

O.g.f.: x*(2408*x^4 - 10542*x^3 + 17358*x^2 - 12742*x + 3519)/(1 - x)^6.

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) for n>5. - Vincenzo Librandi, Nov 16 2015

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

A322272 Smallest multiplication factors f, prime or 1, for all a (mod 120), coprime to 120, so that b*f is a nonzero square mod 8, mod 3, and mod 5.

Original entry on oeis.org

1, 7, 11, 13, 17, 19, 23, 29, 31, 13, 41, 43, 23, 1, 53, 11, 61, 43, 71, 73, 53, 31, 83, 41, 19, 73, 29, 7, 83, 61, 17, 71

Offset: 1

Hans Ruegg, Dec 01 2018

See sequence A322269 for further explanations. This sequence is related to A322269(3).
The sequence is periodic, repeating itself after phi(120) terms. Its largest term is 83, which is A322269(3). In order to satisfy the conditions, both f and b must be coprime to 120. Otherwise, the product would be zero mod a prime <= 5.
The b(n) corresponding to each a(n) is A007775(n).

			The 10th number coprime to 120 is 37. a(10) is 13, because 13 is the smallest prime by which we can multiply 37, so that the product (37*13 = 481) is a square mod 8, mod 3 and mod 5.

Cf. A322269, A322271, A322273, A322274, A322275, A007775.

QresCode(n, nPrimes) = {
  code = bitand(n,7)>>1;
  for (j=2, nPrimes,
    x = Mod(n,prime(j));
    if (issquare(x), code += (1<

A080671 Numbers having divisors 2 or 3 or 5.

Original entry on oeis.org

2, 3, 4, 5, 6, 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, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 96, 98

Offset: 1

Cino Hilliard, Mar 02 2003

Complement of A007775. - Gary Detlefs, Oct 06 2013

The asymptotic density of this sequence is 11/15. - Amiram Eldar, Dec 07 2020

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).

Cf. A005843, A007775, A281746 (divisors 3 or 5).

A080671 := proc(n) local s; option remember;
s:=[2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30];
if n <= 22 then s[n] else 30 + A080671(n-22); fi; end proc; # N. J. A. Sloane, Sep 01 2022

Select[Range[98], Mod[#, 2]*Mod[#, 3]*Mod[#, 5] == 0 &] (* T. D. Noe, Oct 07 2013 *)
d235Q[n_]:=AnyTrue[Divisors[n],MemberQ[{2,3,5},#]&]; Select[Range[100],d235Q] (* Harvey P. Dale, Sep 22 2024 *)

div235(n) = { for(x=1,n, if(gcd(x,30)<>1,print1(x", ")) ) }

a(n+22) = a(n) + 30. - Gary Detlefs, Oct 06 2013
G.f.: x *( 2 -x +2*x^2 -x^3 +2*x^4 +x^6 +2*x^8 +x^10 +2*x^12 +x^14 +2*x^16 -x^17 +2*x^18 -x^19 +2*x^20 ) / ( (x^10 -x^9 +x^8 -x^7 +x^6 -x^5 +x^4 -x^3 +x^2 -x+1)*(1 +x +x^5 +x^6 +x^7 +x^8 +x^9 +x^2 +x^4 +x^3 +x^10)*(x-1)^2 ). - R. J. Mathar, Jul 11 2024
Union of A281746 and A005843, without {0}. - R. J. Mathar, Jul 11 2024

A229968 Numbers not divisible by 3 or 11.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106

Offset: 1

Gary Detlefs, Oct 04 2013

Numbers coprime to 33.
For n from 1 to 20, a(n) mod 33 - n - floor(8*n/19) - 2*floor(n/7) has a period of 20 consisting of all zeros except for a -1 at index 7.
The first index where this differs from A192817 is n = 68; A192817(68) = 110 whereas a(68) = 112. - Tom Edgar, Feb 05 2015
The asymptotic density of this sequence is 20/33. - 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 (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A007775, A192817.

for n from 1 to 500 do if n mod 3<>0 and n mod 11 <>0 then print(n) fi od

Select[Range[132], GCD[#, 33] == 1 &] (* Alonso del Arte, Oct 05 2013 *)
Select[Range[200], Mod[#, 3]>0 && Mod[#, 11]>0 &] (* Vincenzo Librandi, Feb 08 2014 *)

a(n+20) = a(n) + 33.
a(n) = 33*floor((n-1)/20) + f(n) + floor(8*f(n)/19) + 2*floor(f(n)/7) - floor(f(n+12)/19) + 32*floor(f(n-1)/19), where f(n) = n mod 20.
a(n) = a(n-1)+a(n-20)-a(n-21). G.f.: x*(x^20 +x^19 +2*x^18 +x^17 +2*x^16 +x^15 +2*x^14 +3*x^13 +x^12 +2*x^11 +x^10 +2*x^9 +x^8 +3*x^7 +2*x^6 +x^5 +2*x^4 +x^3 +2*x^2 +x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)*(x^8 -x^6 +x^4 -x^2 +1)). - Colin Barker, Oct 08 2013

A038510 Composite numbers with smallest prime factor >= 7.

Original entry on oeis.org

49, 77, 91, 119, 121, 133, 143, 161, 169, 187, 203, 209, 217, 221, 247, 253, 259, 287, 289, 299, 301, 319, 323, 329, 341, 343, 361, 371, 377, 391, 403, 407, 413, 427, 437, 451, 469, 473, 481, 493, 497, 511, 517, 527, 529, 533, 539, 551, 553, 559, 581, 583

Offset: 1

Jeff Burch

nonn

Let A = set of numbers of form 6n + 1, B = numbers of form 6n - 1. Eliminating numbers of form 25 + 30s from A and those of form 35 + 30s from B we obtain sets A* and B*. Removing all terms of the sequence from the union of A* and B*, only prime numbers remain. - Hisanobu Shinya (ilikemathematics(AT)hotmail.com), Jul 14 2002
Divide n by a*b*c where a = 2^(A001511(n)-1), b = 3^(A051064(n)-1) and c = 5^(A055457(n) -1). Then the resulting sequence includes only primes and a(n). - Alford Arnold, Sep 08 2003
Composite numbers not divisible by 2, 3 or 5. - Lekraj Beedassy, Jun 30 2004
Composite numbers k such that k^4 mod 30 = 1. - Gary Detlefs, Dec 09 2012
Composite numbers congruent to 1, 7, 11, 13, -13, -11, -7, -1 (mod 30). Since asymptotically, 100% of integers are composite, we have a(n)/n ~ 30/phi(30) = 30/8 = 3.75. - Daniel Forgues, Mar 16 2013
"John [Conway] recommends the more refined partition [of the positive numbers]: 1, prime, trivially composite, or nontrivially composite. Here, a composite integer is trivially composite if it is divisible by 2, 3, or 5." See link to (van der Poorten, Thomsen, and Wiebe; 2006) pp. 73-74. - Daniel Forgues, Jan 30 2015, Feb 04 2015
For the eight congruences coprime to 30, we can use one byte to encode the "primality/non-primality (unit or composite)" for each [30*n, 30*(n+1)[, n >= 0, closed-open interval, either as little endian binary sequence {01111111, 11111011, 11110111, 01111110, ...}, or as big endian binary sequence {11111110, 11011111, 11101111, 01111110, ...}, which we may then express in base 10. - Daniel Forgues, Feb 05 2015

J. H. Silverman, A Friendly Introduction to Number Theory, 2nd Edn. "Appendix A: Factorization of Small Composite Integers", Prentice Hall NY 2001.

T. D. Noe, Table of n, a(n) for n = 1..10000
Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, A Curious Cubic Identity and Self-similar Sums of Squares, 2006, pp. 73-74.

Cf. A001511, A051064, A055457, A038509, A071904.
Cf. A070884, A038511.
Intersection of A002808 and A007775.

for n from 1 to 583 do if n^4 mod 30 = 1 and not isprime(n) then print(n)fi od; # Gary Detlefs, Dec 09 2012

Select[Range[1000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 7 &] (* T. D. Noe, Mar 16 2013 *)

is(n)=gcd(n,30)==1 && !isprime(n) \\ Charles R Greathouse IV, Dec 09 2012

a(n) ~ 3.75n. - Charles R Greathouse IV, Dec 09 2012

Corrected by Ralf Stephan, Apr 04 2003

A100419 Numbers k such that 30*k+{1,7,13,17,19,23,29} are all prime.

Original entry on oeis.org

89, 6627, 18674, 223949, 229269, 240007, 267356, 606681, 638454, 771496, 951060, 1068030, 1150693, 1254839, 1688923, 1920084, 2413577, 2433289, 2649414, 3053398, 3080572, 3337444, 3586658, 3604256, 3830335, 4137166

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

nonn

Values are 5 mod 7.

In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2020 from Robert Israel)

Cf. A005776, A007775, A076205, A100418, A100420-A100423.

[ n: n in [5..70000000 by 7] | forall{ q: q in [1, 7, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

filter:= proc(n) local j; andmap(isprime, [seq(30*n+j,j=[1,7,13,17,19,23,29])]) end proc:
select(filter, [seq(i,i=5..5*10^6,7)]); # Robert Israel, Nov 04 2024

Select[Range[42*10^5],AllTrue[30#+{1,7,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 10 2018 *)

Edited by Don Reble, Nov 17 2005

A206547 Positive odd numbers relatively prime to 21.

Original entry on oeis.org

1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 131, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 209, 211

Offset: 1

Wolfdieter Lang, Feb 10 2012

These are the positive integers not divisible by 2, 3, or 7.

Numbers coprime to 42. The asymptotic density of this sequence is 2/7. - Amiram Eldar, Oct 23 2020

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A007775, A204458.

Select[Range[1,211,2],CoprimeQ[#,21]&] (* Harvey P. Dale, Jul 28 2020 *)

a(n) = a(n-12) + 42, n>=13.
a(n) = a(n-1) + a(n-12) - a(n-13), n>=13, with a(0)=-1.
a(n) = 2*n-1 + 2*sum(F21[j]*floor((n+(j-1))/12),j=1..12), with F21=[1,2,0,1,0,1,0,1,0,2,1,0], n>=1. For n=0 this becomes -1, but the following o.g.f. has a(0)=0 if it starts with x^0.
O.g.f.: x*(1+x^12+4*x*(1+x^10)+6*x^2*(1+x^8)+2*x^3*(1+x^6)+4*x^4*(1+x^4)+2*x^5*(1+x^2)+4*x^6)/((1-x^12)*(1-x)). The denominator could be factored into cyclotomic polynomials. Compare with the formula contribution from R. J. Mathar in A007775.

cp's OEIS Frontend

A076205 Numbers n such that 30*n+{1,7,11,13,17,19,23,29} are all composite.

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A264449 a(n) = n*(n + 7)*(n + 14)*(n + 21)*(n + 28)/120.

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A264450 a(n) = n*(n + 11)*(n + 22)*(n + 33)*(n + 44)/120.

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A236206 Numbers not divisible by 3, 5 or 7.

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A322272 Smallest multiplication factors f, prime or 1, for all a (mod 120), coprime to 120, so that b*f is a nonzero square mod 8, mod 3, and mod 5.

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A080671 Numbers having divisors 2 or 3 or 5.

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A229968 Numbers not divisible by 3 or 11.

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A264449 a(n) = n(n + 7)(n + 14)(n + 21)(n + 28)/120.

A264450 a(n) = n(n + 11)(n + 22)(n + 33)(n + 44)/120.