A007775 -id:A007775 - cp's OEIS frontend

A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

1, 2, 2, 3, 3, 3, 4, 6, 6, 4, 5, 8, 5, 8, 5, 6, 10, 12, 12, 10, 6, 7, 5, 15, 9, 15, 5, 7, 8, 14, 10, 20, 20, 10, 14, 8, 9, 12, 21, 24, 7, 24, 21, 12, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 15, 27, 18, 35, 15, 35, 18, 27, 15, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24, 33, 40, 45, 20, 11, 20, 45, 40, 33, 24, 13

Offset: 1

Peter Munn, Jan 21 2020

As a binary operation, this sequence defines a commutative monoid over the positive integers that is isomorphic to multiplication. The self-inverse permutation A225546(.) provides an isomorphism. This monoid therefore has unique factorization. Its primes are the even terms of A050376: 2, 4, 16, 256, ..., which in standard integer multiplication are the powers of 2 with powers of 2 as exponents.
In this monoid, in contrast, the powers of 2 run through the squarefree numbers, the k-th power of 2 being A019565(k). 4 is irreducible and its powers are the squares of the squarefree numbers, the k-th power of 4 being A019565(k)^2 (where "^2" denotes standard integer squaring); and so on with powers of 16, 256, ...
In many cases the product of two numbers is the same here as in standard integer multiplication. See the formula section for details.

			From _Antti Karttunen_, Feb 02 2020: (Start)
The top left 16 X 16 corner of the array:
   1,  2,  3,  4,  5,  6,   7,   8,   9,  10,  11,  12,  13,  14,  15,  16, ...
   2,  3,  6,  8, 10,  5,  14,  12,  18,  15,  22,  24,  26,  21,  30,  32, ...
   3,  6,  5, 12, 15, 10,  21,  24,  27,  30,  33,  20,  39,  42,   7,  48, ...
   4,  8, 12,  9, 20, 24,  28,  18,  36,  40,  44,  27,  52,  56,  60,  64, ...
   5, 10, 15, 20,  7, 30,  35,  40,  45,  14,  55,  60,  65,  70,  21,  80, ...
   6,  5, 10, 24, 30, 15,  42,  20,  54,   7,  66,  40,  78,  35,  14,  96, ...
   7, 14, 21, 28, 35, 42,  11,  56,  63,  70,  77,  84,  91,  22, 105, 112, ...
   8, 12, 24, 18, 40, 20,  56,  27,  72,  60,  88,  54, 104,  84, 120, 128, ...
   9, 18, 27, 36, 45, 54,  63,  72,  25,  90,  99, 108, 117, 126, 135, 144, ...
  10, 15, 30, 40, 14,  7,  70,  60,  90,  21, 110, 120, 130, 105,  42, 160, ...
  11, 22, 33, 44, 55, 66,  77,  88,  99, 110,  13, 132, 143, 154, 165, 176, ...
  12, 24, 20, 27, 60, 40,  84,  54, 108, 120, 132,  45, 156, 168,  28, 192, ...
  13, 26, 39, 52, 65, 78,  91, 104, 117, 130, 143, 156,  17, 182, 195, 208, ...
  14, 21, 42, 56, 70, 35,  22,  84, 126, 105, 154, 168, 182,  33, 210, 224, ...
  15, 30,  7, 60, 21, 14, 105, 120, 135,  42, 165,  28, 195, 210,  35, 240, ...
  16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240,  81, ...
(End)

Antti Karttunen, Antidiagonals n = 1..144, flattened
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..80200; (antidiagonals n = 1..400)
Eric Weisstein's World of Mathematics, Monoid

Isomorphic to A003991 with A225546 as isomorphism.
Cf. A003961(main diagonal), A048675, A059895, A059896, A059897.
Rows/columns, sorted in ascending order: 2: A000037, 3: A028983, 4: A252849.
A019565 lists powers of 2 in order of increasing exponent.
Powers of k, sorted in ascending order: k=2: A005117, k=3: A056911, k=4: A062503, k=5: A276378, k=6: intersection of A325698 and A005117, k=7: intersection of A007775 and A005117, k=8: A062838.
Irreducibles are A001146 (even terms of A050376).

up_to = 1275;
A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331590sq(x,y) = if(1==x,y,if(1==y,x, my(fx=factor(x),fy=factor(y),u=max(#binary(vecmax(fx[, 2])),#binary(vecmax(fy[, 2]))),prodsx=vector(u,x,1),m=1); for(i=1,u,for(k=1,#fx~, if(bitand(fx[k,2],m),prodsx[i] *= fx[k,1])); for(k=1,#fy~, if(bitand(fy[k,2],m),prodsx[i] *= fy[k,1])); m<<=1); prod(i=1,u,A019565(A048675(prodsx[i]))^(1<<(i-1)))));
A331590list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A331590sq(col,(a-(col-1))))); (v); };
v331590 = A331590list(up_to);
A331590(n) = v331590[n]; \\ Antti Karttunen, Feb 02 2020

Alternative definition: A(n,1) = n; A(n,k) = A(A059897(n,k), A003961(A059895(n,k))).
Main derived identities: (Start)
A(n,k) = A(k,n).
A(1,n) = n.
A(n, A(m,k)) = A(A(n,m), k).
A(m,m) = A003961(m).
A(n^2, k^2) = A(n,k)^2.
A(A003961(n), A003961(k)) = A003961(A(n,k)).
A(A019565(n), A019565(k)) = A019565(n+k).
(End)
Characterization of conditions for A(n,k) = n * k: (Start)
The following 4 conditions are equivalent:
(1) A(n,k) = n * k;
(2) A(n,k) = A059897(n,k);
(3) A(n,k) = A059896(n,k);
(4) A059895(n,k) = 1.
If gcd(n,k) = 1, A(n,k) = n * k.
If gcd(n,k) = 1, A(A225546(n), A225546(k)) = A225546(n) * A225546(k).
The previous formula implies A(n,k) = n * k in the following cases:
(1) for n = A005117(m), k = j^2;
(2) more generally for n = A005117(m_1)^(2^i_1), k = A005117(m_2)^(2^i_2), with A004198(i_1, i_2) = 0.
(End)

A084968 Multiples of 7 coprime to 30.

Original entry on oeis.org

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

A008365 13-rough numbers: positive integers that have no prime factors less than 13.

Original entry on oeis.org

1, 13, 17, 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, 169, 173, 179, 181, 191, 193, 197, 199, 211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 257, 263, 269

Offset: 1

N. J. A. Sloane

For n > 1, the smallest prime factor of a(n) is >= 13.
Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - Gary Detlefs, Dec 30 2011
This sequence is exactly the set of positive values of r such that ( Product_{k = 0..10} n + k*r )/11! is an integer for all n. - Peter Bala, Nov 14 2015
The asymptotic density of this sequence is 16/77. - Amiram Eldar, Sep 30 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Peter Bala, A property of p-rough numbers.
Benedict W. J. Irwin, Generating Function.
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for linear recurrences with constant coefficients, order 481.
Index entries for sequences related to smooth numbers

For k-rough numbers with other values of k, see A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063.

a008365 n = a008365_list !! (n-1)
a008365_list = 1 : filter ((> 11) . a020639) [1..]
-- Reinhard Zumkeller, Jan 06 2013

for i from 1 to 500 do if gcd(i,2310) = 1 then print(i); fi; od;

Select[ Range[ 300 ], GCD[ #1, 2310 ]==1& ]

isA008365(n) = gcd(n,2310)==1 \\ Michael B. Porter, Oct 10 2009

G.f: x*P(x)/(1 - x - x^480 + x^481) where P(x) is a polynomial of degree 480. - Benedict W. J. Irwin, Mar 18 2016
77*n/16 - 13 < a(n) < 77*n/16 + 8. - Charles R Greathouse IV, Mar 21 2023
a(n) = a(n-1) + a(n-480) - a(n-481). - Charles R Greathouse IV, Mar 21 2023

New name following a comment of Michael B. Porter, Mar 21 2023

A008366 Smallest prime factor is >= 17.

Original entry on oeis.org

1, 17, 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, 277, 281, 283

Offset: 1

N. J. A. Sloane

Also the 17-rough numbers: positive integers that have no prime factors less than 17. - Michael B. Porter, Oct 10 2009
a(n) - (1001/192) n is periodic with period 5760. - Robert Israel, Mar 18 2016
From Peter Bala, May 12 2018: (Start)
The product of two 17-rough numbers is a 17-rough number and the prime factors of a 17-rough number are 17-rough numbers.
Let k equal either 13, 14, 15 or 16. Then the product of k numbers n*(n + a)*(n + 2*a)*...*(n + (k-1)*a) in arithmetical progression is divisible by k! for all integer n if and only if a is a 17-rough number.
The sequence terms satisfy the congruence x^60 = 1 (mod 30030), where 30030 = 2*3*5*7*11*13. (End)
The asymptotic density of this sequence is 192/1001. - Amiram Eldar, Sep 30 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Peter Bala, A property of p-rough numbers.
Benedict Irwin, Generating Function.
Eric Weisstein's World of Mathematics, Rough Number.
Index entries for sequences related to smooth numbers

For k-rough numbers with other values of k, see A000027 A005408 A007310 A007775 A008364 A008365 A008366 A166061 A166063.

Cf. A005867.

for i from 1 to 500 do if gcd(i,30030) = 1 then print(i); fi; od;

Select[ Range[ 300 ], GCD[ #1, 30030 ]==1& ]
Join[{1},Select[Range[300],FactorInteger[#][[1,1]]>=17&]] (* Harvey P. Dale, Mar 28 2020 *)

isA008366(n) = gcd(n,30030)==1 \\ Michael B. Porter, Oct 10 2009

Numbers n > 1 such that ((Sum_{k=1..n} k^10) mod n = 0) and ((Sum_{k=1..n} k^12) mod n = 0) (conjecture). - Gary Detlefs, Dec 27 2011
a(n) = a(n-1) + a(n-5760) - a(n-5761). - Vaclav Kotesovec, Mar 18 2016
G.f: x*P(x)/(1 - x - x^5760 + x^5761) where P(x) is a polynomial of degree 5760. - Benedict W. J. Irwin, Mar 23 2016
a(n) = (1001/192)*n + O(1), where the O(1) term is bounded by +/- 19. - Charles R Greathouse IV, Oct 13 2022
A008365 SETMINUS A084970 . - R. J. Mathar, Nov 05 2024

A236185 Differences between terms of compacting Eratosthenes sieve for prime(4) = 7.

Original entry on oeis.org

4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 4, 6, 2

Offset: 1

Christopher J. Hanson, Jan 21 2014

nonn

P(x) is a function which represents a prime number at a ordinal x.
This pattern, dp(x) is a sequence of the differences between consecutive prime numbers as described by p(x).
For P(4), dp(4)is the relative offsets of the next 7 primes: 7, +4 = 11, +2 = 13, +4 = 17, +2 = 19, +4 = 23, +6 = 29, +2 = 31
The Eratosthenes sieve can be expressed as follows. Start with S1 = [2, 3, 4, 5, ...] the list of numbers bigger than 1. Removing all multiples of the first element 2 yields the list S2 = [3, 5, 7, 9, ...]. Removing all multiples of the first element 3 yields S3 = [5, 7, 11, 13, 17, 19, ...], Removing all multiples of the first element 5 yields S4 = [7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, ...], and so on. The list of first differences of S4 is [4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, ...] which is this sequence. - Michael Somos, Mar 12 2014

Christopher J. Hanson, The structure of prime numbers and twin prime gaps

Cf. A236175-A236180, A236185-A236190.

Cf. A007775. Essentially the same as A145011.

{a(n) = my(A); if( n<1, 0, A = vector( n*28 + 48, k, k+1); for( i = 1, 3, A = select( k -> k%prime(i), A) ); polcoeff( (1 - x) * Ser( select( k -> k>7 && (k%7) == 0, A) / 7), n)) }; /* Michael Somos, Mar 10 2014 */
(C#)
// dp(4) = GeneratePrimeDifferencialPattern( 4 );
static void GeneratePrimeDifferencialPattern( int ordinal )
{
    // Contract
    if( ordinal < 1 )
        throw new ArgumentOutOfRangeException( "ordinal" );
    // Local data
    int size = 1 << 18;
    int[] numberLine = Enumerable.Range( 2, size ).ToArray();
    int pointer = 0;
    // Apply sieve: for each integer greater than 1
    while( pointer < numberLine.Length )
    {
        // Locals
        int x = numberLine[pointer];
        int index = pointer;
        List pattern = new List();
        int skips = 0;
        int count = 0;
        bool counting = true;
        // Find all products
        for( int n = x + x; n < size; n += x )
        {
            // Fast forward through number-line
            while( numberLine[++index] < n )
                skips++;
            // If the number was not already removed
            if( numberLine[index] == n )
            {
                // Add skip count to pattern
                pattern.Add( numberLine[index] );
                // Mark as not prime
                numberLine[index] = 0;
                // Reset skips
                if( counting )
                {
                    count++;
                    if( skips <= 2 )
                        counting = false;
                }
                skips = 0;
            }
            // Otherwise we've skipped again
            else skips++;
        }
        // Reduce number-line
        numberLine = numberLine.Where( n => n > 0 ).ToArray();
        // If we have a pattern we want
        if( pattern.Any() && pointer == ordinal - 1 )
        {
            // Report pattern
            int prime = numberLine[ordinal-1];
            var d = pattern.Take( count ).ToArray();
            List dp = new List();
            for( int y = 1; y < count; y++ )
                dp.Add( ( d[y] - d[y - 1] ) / prime );
            System.Console.WriteLine( "Pattern P({0}) = {1} :: dp({0}) = {2}", pointer + 1, numberLine[pointer], String.Join( ", ", dp ) );
            return;
        }
        // Move number-line pointer forward
        pointer++;
    }
}

a(n + 8) = a(n). - Michael Somos, Mar 10 2014

a(n) = 4*((n+2) mod 2) + 2*((n+1) mod 2) + 4*(f(8,n+2)+f(8,n)) - 2*f(8,n+1), where f(x,n)= floor(n/x)-floor((n-1)/x). - Gary Detlefs, Nov 16 2020

Edited by Michael Somos, Mar 12 2014. (Added code and comments, refined description.)

A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.

Original entry on oeis.org

49, 34083, 41545, 48713, 140609, 524027, 616812, 855281, 1314397, 1324750, 1636152, 2281293, 2927134, 3401412, 3605413, 4989341, 5212221, 5284979, 5406303, 5645269, 6141254, 6342728, 7231434, 7347697, 7637329, 8027068, 8161657, 8372756, 8392776, 8567216, 8986096, 9145563

Offset: 1

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

Values are 0 mod 7.
From Peter Munn, Sep 06 2023: (Start)
In each case, the 7 primes are necessarily consecutive.
As A065706 demonstrates, many intervals of 27 integers contain 8 primes, but only A364678(30) = 7 primes can occur between adjacent positive multiples of 30. This is because there are 8 values {1,7,11,13,17,19,23,29} coprime to 30, but they cover every residue class modulo 7, which means at least one of 30*k + {1,7,11,13,17,19,23,29} is divisible by 7.
1 and 29 are in the same residue class, but if we remove any of the other coprime integers there is a class that is not represented in the set. For this sequence, we remove 7, so when k is a multiple of 7, none of 30*k + {1,11,13,17,19,23,29} is a multiple of 2, 3, 5 or 7 and the set can potentially be 7 consecutive primes.
The sequences for the other appropriate subsets of 7 coprime values are A100419-A100423.
(End)

David A. Corneth, Table of n, a(n) for n = 1..10309

Cf. A005776, A007775, A056956, A065706, A076205, A100419-A100423, A364678.

[ n: n in [0..70000000 by 7] | forall{ q: q in [1, 11, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[803*10^4],AllTrue[30#+{1,11,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 11 2019 *)

{pav7(mx)= local(wp=[1,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<8),m=isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}

Edited by Don Reble, Nov 17 2005

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

Original entry on oeis.org

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

A229829 Numbers coprime to 15.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 86, 88, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 119

Offset: 1

Gary Detlefs, Oct 01 2013

A001651 INTERSECT A047201.
a(n) - 15*floor((n-1)/8) - 2*((n-1) mod 8) has period 8, repeating [1,0,0,1,0,1,1,0].
Numbers whose odd part is 7-rough: products of terms of A007775 and powers of 2 (terms of A000079). - Peter Munn, Aug 04 2020
The asymptotic density of this sequence is 8/15. - Amiram Eldar, Oct 18 2020

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).

Lists of numbers coprime to other semiprimes: A007310 (6), A045572 (10), A162699 (14), A160545 (21), A235933 (35).
Cf. A008676, A078588, A113763, A160542, A160543, A160544.
Subsequence of: A001651, A047201.
Subsequences: A000079, A007775.

[n: n in [1..120] | IsOne(GCD(n,15))]; // Bruno Berselli, Oct 01 2013

for n from 1 to 500 do if n mod 3<>0 and n mod 5<>0 then print(n) fi od

Select[Range[120], GCD[#, 15] == 1 &] (* or *) t = 70; CoefficientList[Series[(1 + x + 2 x^2 + 3 x^3 + x^4 + 3 x^5 + 2 x^6 + x^7 + x^8)/((1 - x)^2 (1 + x) (1 + x^2) (1 + x^4)) , {x, 0, t}], x] (* Bruno Berselli, Oct 01 2013 *)
Select[Range[120],CoprimeQ[#,15]&] (* Harvey P. Dale, Oct 31 2013 *)

[i for i in range(120) if gcd(i, 15) == 1] # Bruno Berselli, Oct 01 2013

a(n+8) = a(n) + 15.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor(2*phi*(f(n+1)+2)) -2*floor(phi*(f(n+1)+2)), where f(n) = (n-1) mod 8 and phi=(1+sqrt(5))/2.
a(n) = 15*floor((n-1)/8) +2*f(n) +floor((2*f(n)+5)/5) -floor((f(n)+2)/3), where f(n) = (n-1) mod 8.
From Bruno Berselli, Oct 01 2013: (Start)
G.f.: x*(1 +x +2*x^2 +3*x^3 +x^4 +3*x^5 +2*x^6 +x^7 +x^8) / ((1-x)^2*(1+x)*(1+x^2)*(1+x^4)). -
a(n) = a(n-1) +a(n-8) -a(n-9) for n>9. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(7 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/15. - Amiram Eldar, Dec 13 2021

A100423 Numbers n such that 30*n+{1,7,11,13,17,19,29} are all prime.

Original entry on oeis.org

62, 188, 9491, 31982, 38226, 38520, 89459, 168237, 175125, 368248, 471078, 634892, 704416, 803102, 994748, 1436315, 1488857, 1605484, 1842553, 1945824, 2282958, 2465266, 2620715, 2627029, 2705037, 4282305, 5569899, 5914824

Offset: 1

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

Values are 6 mod 7.

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

Cf. A005776, A007775, A076205, A100418-A100422.

[ n: n in [0..6000000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[6*10^6],AllTrue[30#+{1,7,11,13,17,19,29},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 21 2021 *)

Edited by Don Reble, Nov 17 2005

cp's OEIS Frontend

A331590 Square array A(n,k) = A225546(A225546(n) * A225546(k)), n >= 1, k >= 1, read by descending antidiagonals.

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A084968 Multiples of 7 coprime to 30.

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A008365 13-rough numbers: positive integers that have no prime factors less than 13.

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Haskell

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A008366 Smallest prime factor is >= 17.

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A236185 Differences between terms of compacting Eratosthenes sieve for prime(4) = 7.

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A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.

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

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