A007775 -id:A007775 - cp's OEIS frontend

A235583 Numbers not divisible by 2, 5 or 7.

1, 3, 9, 11, 13, 17, 19, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 81, 83, 87, 89, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 121, 123, 127, 129, 131, 137, 139, 141, 143, 149, 151, 153, 157, 159, 163, 167, 169, 171, 173, 177, 179, 181, 183

Offset: 1

Oleg P. Kirillov, Jan 12 2014

All primes, except 2, 5 and 7, are in this sequence. Any product of terms is also a term in the sequence. For example, a(2)a(4) = 3 * 11 = 33 = a(12). - Alonso del Arte, Jan 12 2014
In other words, numbers equivalent 1,3,9,...,69 modulo 70. This means the first differences of the sequence are 24-periodic. - Ralf Stephan, Jan 14 2014
Numbers coprime to 70. The asymptotic density of this sequence is 12/35. - Amiram Eldar, Oct 23 2020

			51 = 3 * 17, and gcd(51, 70) = 1, so it is in the sequence.
53 is prime, so it is in the sequence.
55 = 5 * 11, and gcd(55, 70) = 5, so it is not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1).

Cf. A007775, A008364 (subsequence).

Select[Range[210], GCD[#, 70] == 1 &] (* Alonso del Arte, Jan 12 2014 *)
Select[Range[300], Mod[#, 2]>0 &&Mod[#, 5]>0 &&Mod[#, 7]>0&] (* Vincenzo Librandi, Feb 08 2014 *)

G.f.: x*(x^22 +3*x^21 +8*x^20 +7*x^19 +x^18-2*x^17 -x^16 +5*x^15 +10*x^14 +7*x^13 -x^12 -6*x^11 -x^10 +7*x^9 +10*x^8 +5*x^7 -x^6 -2*x^5 +x^4 +7*x^3 +8*x^2 +3*x +1) / ((x+1) *(x^2+1) *(x^2+x+1) *(x^4-x^2+1) *(x^4+1) *(x^8-x^4+1) *(x-1)^2). - Alois P. Heinz, Jan 12 2014

A354178 Numbers whose number of divisors is coprime to 30.

Original entry on oeis.org

1, 64, 729, 1024, 4096, 15625, 46656, 59049, 65536, 117649, 262144, 531441, 746496, 1000000, 1771561, 2985984, 3779136, 4194304, 4826809, 7529536, 9765625, 11390625, 16000000, 24137569, 34012224, 43046721, 47045881, 47775744, 60466176, 64000000, 85766121, 113379904

Offset: 1

Amiram Eldar, May 18 2022

nonn

Numbers k such that gcd(d(k), 30) = 1, where d(k) is the number of divisors of k (A000005).

All the terms are squares since their number of divisors is odd.

			64 is a term since A000005(64) = 7 and gcd(7, 30) = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1709 terms from Amiram Eldar)
Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.
Index entries for sequences computed from exponents in factorization of n.

Subsequence of other sequences of numbers k such that gcd(d(k), m) = 1: A000290 (m=2), A336590 (m=3), A352475 (m=6).

Cf. A000005, A007775, A354179.

Select[Range[10^4]^2, CoprimeQ[DivisorSigma[0, #], 30] &]

isok(k) = gcd(numdiv(k), 30) == 1;
for(k=1, 10650, if(isok(k^2), print1(k^2,", ")))

a(n) = A354179(n)^2.
The number of terms <= x is (zeta(5)*zeta(5/3))/(zeta(4)*zeta(10/3))*x^(1/6) + (zeta(3)*zeta(3/5))/(zeta(2)*zeta(12/5))*x^(1/10) + O(x^(1/20 + eps)) for all eps > 0 (Hilberdink, 2022).
Sum_{n>=1} 1/a(n) = Product_{p prime} (p^2 + p^8 + p^12 + p^14 + p^18 + p^20 + p^24 + p^30)/(p^30 - 1) = 1.0183538548...

A063118 Dimension of the space of weight 2n cusp forms for Gamma_0(50).

Original entry on oeis.org

2, 17, 31, 47, 61, 77, 91, 107, 121, 137, 151, 167, 181, 197, 211, 227, 241, 257, 271, 287, 301, 317, 331, 347, 361, 377, 391, 407, 421, 437, 451, 467, 481, 497, 511, 527, 541, 557, 571, 587, 601, 617, 631, 647, 661, 677, 691, 707, 721, 737, 751, 767, 781

Offset: 1

N. J. A. Sloane, Jul 08 2001

nonn

Appears to agree with the first 11-section of A186042 except for the first term of both sequences (verified up to a(10000)). - Klaus Brockhaus, Mar 10 2011

			G.f. = 2*x + 17*x^2 + 31*x^3 + 47*x^4 + 61*x^5 + 77*x^6 + 91*x^7 + 107*x^8 + 121*x^9 + ...

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database

Cf. A186042, A007775.

[ Dimension(CuspForms(Gamma0(50), 2*n)): n in [1..55] ]; // Klaus Brockhaus, Mar 10 2011

def a(n) : return( len( CuspForms( Gamma0( 50), 2*n, prec=1) . basis())); # Michael Somos, May 29 2013

From Klaus Brockhaus, Mar 10 2011: (Start)
G.f. (conjectured): x*(x^3 + 12*x^2 + 15*x + 2) / ((x - 1)^2*(x + 1)).
Recurrences (conjectured):
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4;
a(n) = a(n-2) + 30 for n > 3. (End)
Closed formula (conjectured): a(n) = (30*n+(-1)^n-27)/2 for n > 1. - Bruno Berselli, Mar 10 2011
Recurrence (conjectured): a(n) = 2*a(n-1) -a(n-2) +2*(-1)^n, n > 3. - Vincenzo Librandi, Mar 24 2011
Conjecture: a(n) = A007775(4*n - 3), n > 1. - Bill McEachen, May 15 2022

A098591 a(k) contains primality information for the numbers in the interval (k30,...,(k+1)30) packed into one byte using the fact that only numbers == 1, 7, 11, 13, 17, 19, 23, 29 mod 30 can be prime.

Original entry on oeis.org

223, 239, 126, 182, 219, 61, 249, 213, 79, 30, 243, 234, 166, 237, 158, 230, 12, 211, 211, 59, 221, 89, 165, 106, 103, 146, 189, 120, 30, 166, 86, 86, 227, 173, 45, 222, 42, 76, 85, 217, 163, 240, 159, 3, 84, 161, 248, 46, 253, 68, 233, 102, 246, 19, 58, 184, 76

Offset: 1

Hugo Pfoertner, Sep 16 2004

nonn

This sequence illustrates an efficient method for storing all prime numbers up to some moderate limit. With this method all prime numbers < 2^31 can be stored in a 70-MByte file.
Because of divisibility by 7, 254 appears only as the zeroth term, and 127 and 255 do not appear at all. All other single-byte numbers (0..255) appear. 247 is the last to appear, first appearing as the 22621st term.
0 and at least one nonzero term must both appear infinitely often. (Probably every number 0..126 and 128..253 appears infinitely often, but this may be hard to prove.) - Keith F. Lynch, Sep 09 2018

			a(1)=223: From the list of prime candidates between 30 and 60 only the number 49 is composite. Therefore
a(1) =   2^0 (representing 30 +  1)
       + 2^1 (representing 30 +  7)
       + 2^2 (representing 30 + 11)
       + 2^3 (representing 30 + 13)
       + 2^4 (representing 30 + 17)
       + 2^6 (representing 30 + 23)
       + 2^7 (representing 30 + 29)
     = 1 + 2 + 4 + 8 + 16 + 64 + 128 = 223.
a(17): There are 2 primes in the interval (17*30, 17*30 + 30) = (510,540): 521 == 11 (mod 30) and 523 == 13 (mod 30). Therefore a(17) = 2^2 (representing 510 + 11) + 2^3 (representing 510 + 13) = 4 + 8 = 12.
a(360) = 0 (1st occurrence), no primes between 360*30 = 10800 and 10830. - _Frank Ellermann_, Apr 03 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Generation of a file with packed primes. FORTRAN program.
Hugo Pfoertner, Primality testing using a packed lookup table. FORTRAN program.
Hugo Pfoertner, Programs to generate and access a packed prime lookup table. Executable programs and source code.

Cf. A000040 (prime numbers), A006880 (number of primes < 10^n), A098592 (number of primes in intervals (30*k, 30*(k+1))), A005867 (primorial sieving candidates), A007775 (7-rough numbers, corresponding to the bits).

With[{s = Select[Range@ 30, CoprimeQ[#, 30] &]}, Array[Total[2^(Position[30 # + s, ?PrimeQ][[All, 1]] - 1) ] &, 57]] (* _Michael De Vlieger, Sep 10 2018 *)

a(k) = {vec = [1, 7, 11, 13, 17, 19, 23, 29]; return (sum(i=1, length(vec), isprime(30*k+vec[i])*(1 << (i-1))));} \\ Michel Marcus, Jan 31 2013

from sympy import isprime
v = [1, 7, 11, 13, 17, 19, 23, 29]
def a(n): return sum(2**k for k, vk in enumerate(v) if isprime(n*30+vk))
print([a(n) for n in range(1, 58)]) # Michael S. Branicky, Oct 10 2021

a(n) = Sum_{k=0..7} (2^k)*isprime(30*n + offset(k)), where isprime(x)=1 for x prime, otherwise 0, and offset(k) = {1, 7, 11, 13, 17, 19, 23, 29} for k=0..7.

A100420 Numbers n such that 30*n+{1,7,11,17,19,23,29} are all prime.

Original entry on oeis.org

22621, 103205, 149125, 237794, 288467, 321451, 364921, 373370, 404002, 851099, 985933, 1106235, 1594044, 1696874, 1780265, 1824421, 1851756, 2249881, 3112939, 3257538, 3397608, 3601651, 3747356, 4347340, 4710990, 4886284

Offset: 1

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

Values are 4 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, A100419, A100421-A100423.

[ n: n in [4..70000000 by 7] | forall{ q: q in [1, 7, 11, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[5000000],And@@PrimeQ[30 #+{1,7,11,17,19,23,29}]&]  (* Harvey P. Dale, Mar 06 2011 *)

Edited by Don Reble, Nov 17 2005

A100422 Numbers n such that 30*n+{1,7,11,13,17,23,29} are all prime.

Original entry on oeis.org

1, 53887, 114731, 123306, 139742, 210554, 471745, 480859, 619039, 630862, 858929, 1075873, 1306614, 1714945, 1913514, 2767458, 3014285, 3454137, 3518243, 3699151, 3864512, 3874291, 4274376, 4862362, 4878329, 4937822

Offset: 1

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

Values are 1 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-A100423.

[ n: n in [0..5000000] | forall{ q: q in [1, 7, 11, 13, 17, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 23 2011

a:= proc(n) option remember;
      local m;
      if n=1 then 1
      else for m from 30*(a(n-1)+7) by 210
           while not (isprime (m+1) and isprime (m+7) and
                 isprime (m+11) and isprime (m+13) and
                 isprime (m+17) and isprime (m+23) and
                 isprime (m+29))
           do od; m/30
        fi
    end:
seq (a(n), n=1..10);

Select[Range[5000000],And@@PrimeQ/@(30(#)+{1,7,11,13,17,23,29})&]  (* Harvey P. Dale, Feb 23 2011 *)

Edited by Don Reble, Nov 17 2005

A158725 Non-repdigit composite numbers not divisible by 2, 3, 5 or 11.

Original entry on oeis.org

49, 91, 119, 133, 161, 169, 203, 217, 221, 247, 259, 287, 289, 299, 301, 323, 329, 343, 361, 371, 377, 391, 403, 413, 427, 437, 469, 481, 493, 497, 511, 527, 529, 533, 551, 553, 559, 581, 589, 611, 623, 629, 637, 667, 679, 689, 697, 703, 707, 713, 721, 731

Offset: 1

Lekraj Beedassy, Mar 24 2009

nonn

Non-repdigit composite numbers ending in 1, 3, 7 or 9, with digital root not a multiple of 3 and whose alternate digit sums do not differ by a multiple of 11.
The "compositeness" of larger entries of the sequence is not obvious right away or deducible by mere inspection, and hence these terms readily lend themselves to be (erroneously) suspected as primes to the casual glance.
This differs from the corresponding sequence without the repunit condition starting at a(1351) = 11123 rather than 11111. - Charles R Greathouse IV, Sep 08 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007775, A053795, A008364.

is(n)=gcd(n,330)==1&&!isprime(n)&&n%(10^#Str(n)\9) \\ Charles R Greathouse IV, Sep 08 2012

a(n) ~ kn with k = 33/8. - Charles R Greathouse IV, Sep 08 2012

Corrected and extended by Ray Chandler, Mar 27 2009

A375536 The maximum exponent in the prime factorization of the largest 5-smooth divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 2, 1, 3, 2, 0, 1, 0, 5, 1, 1, 1, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 6, 1, 1, 0, 2, 1, 1, 0, 3, 0, 1, 2, 2, 0, 1, 0, 4, 4, 1, 0, 2, 1, 1, 1, 3, 0, 2, 0, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 1, 0, 3, 1

Offset: 1

Amiram Eldar, Aug 19 2024

Cf. A007775, A051903, A244417 (3-smooth analog), A355582, A375537, A375538, A375539.

Cf. A007814, A007949, A112765.

a[n_] := Max[IntegerExponent[n, {2, 3, 5}]]; Array[a, 100]

a(n) = max(max(valuation(n, 2), valuation(n, 3)), valuation(n, 5));

a(n) = A051903(A355582(n)).
a(n) = max(A007814(n), A007949(n), A112765(n)).
a(n) = 0 if and only if n is a 7-rough number (A007775).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A375538(3)/A375539(3) = 51227/36540 = 1.401943076...

A086748 Odd numbers m such that when C(2k, k) == 1 (mod m) then k is necessarily even.

Original entry on oeis.org

3, 5, 9, 15, 21, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 81, 85, 87, 93, 95, 99, 105, 111, 115, 117, 123, 125, 129, 135, 141, 145, 147, 153, 155, 159, 165, 171, 175, 177, 183, 185, 189, 195, 201, 205, 207, 213, 215, 219, 225, 231, 235, 237, 243, 245

Offset: 1

Benoit Cloitre, Jul 30 2003

nonn

From Jinyuan Wang, Apr 05 2020: (Start)
All terms are odd, because C(2k, k) is always divisible by 2.
If m is a term, then m*t is also a term for odd numbers t.
Theorem 1: if C(2k, k) == 1 (mod 3) then k is necessarily even. If C(2k, k) == 2 (mod 3) then k is necessarily odd.
Proof: for k < 6 it is correct. We have C(6r, 3r) == C(2r, r) (mod 3) and C(6r+4, 3r+2) == C(2r, r)*C(4, 2) == 0 (mod 3). Suppose k is the least value such that theorem 1 is incorrect, then k must be of the form 3r+1. But C(6r+2, 3r+1) == C(2r, r)*C(2, 1) (mod 3), which means that r is a smaller counterexample, a contradiction!
Theorem 2: if C(2k, k) == 1 or 4 (mod 5) then k is necessarily even. If C(2k, k) == 2 or 3 (mod 5) then k is necessarily odd.
Note that C(10r, 5r) == C(2r, r) (mod 5), C(10r+2, 5r+1) == C(2r, r)*C(2, 1) (mod 5), C(10r+4, 5r+2) == C(2r, r)*C(4, 2) (mod 5), C(10r+6, 5r+3) == C(2r, r)*C(6, 3) (mod 5) and C(10r+8, 5r+4) == C(2r, r)*C(8, 4) (mod 5). The proof is similar to that of theorem 1. (End)
Up to m < 1000, all odd m are either of the form 3*(2t-1) or 5*(2t-1) (as proved by Jinyuan Wang) and in the sequence, or not in the sequence because an odd k <= 7412629 exists such that C(2k, k) == 1 (mod m). - Giovanni Resta, Apr 05 2020
Numbers m such that A099976(k) = 1 (mod m) has no solutions k. - R. J. Mathar, Jul 11 2024
Is this  A005408 \ A007775 ? - Antti Karttunen, Jul 11 2024
It is likely that 1001 = 7*11*13 or other products of at least 3 primes > 5 provide counterexamples, but it is difficult to prove. - M. F. Hasler, Jul 13 2024

			m = 7 is not a term because C(2k,k) = 1 (mod 7) is solvable by the odd k=17.
m = 11 is not a term because C(2k,k) = 1 (mod 11) is solvable by the odd k=13.
m = 13 is not a term because C(2k,k) = 1 (mod 13) is solvable by the odd k=2383.
m = 23 is not a term because C(2k,k) = 1 (mod 23) is solvable by the odd k=3391. - _R. J. Mathar_, Jul 11 2024
m=2261 = 7*17*19 is not a term because C(2k,k) = 1 (mod 2261) is solvable by k=57. - _R. J. Mathar_, Aug 09 2024

Cf. A000984.

Cf. A373469 (least k such that C(2k,k)=1 mod A007775(n)).

13 removed and offset changed by Jinyuan Wang, Apr 04 2020
23 removed and more terms added by Giovanni Resta, Apr 05 2020
Definition corrected by Max Alekseyev, Jul 12 2024

A100421 Numbers n such that 30*n+{1,7,11,13,19,23,29} are all prime.

Original entry on oeis.org

2, 79, 391701, 505017, 740413, 787187, 933025, 1169863, 1333719, 1406792, 2212261, 2719950, 2962738, 3125992, 3284955, 3384586, 3727271, 3821295, 3861881, 4320864, 4439878, 4764356, 5014865, 5480190, 5879274, 6124442

Offset: 1

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

Values are 2 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-A100420, A100422, A100423.

[ n: n in [2..70000000 by 7] | forall{ q: q in [1, 7, 11, 13, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[7*10^6],AllTrue[30#+{1,7,11,13,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 16 2016 *)

Edited by Don Reble, Nov 17 2005

cp's OEIS Frontend

A235583 Numbers not divisible by 2, 5 or 7.

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A354178 Numbers whose number of divisors is coprime to 30.

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A063118 Dimension of the space of weight 2n cusp forms for Gamma_0(50).

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A098591 a(k) contains primality information for the numbers in the interval (k*30,...,(k+1)*30) packed into one byte using the fact that only numbers == 1, 7, 11, 13, 17, 19, 23, 29 mod 30 can be prime.

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A100420 Numbers n such that 30*n+{1,7,11,17,19,23,29} are all prime.

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A100422 Numbers n such that 30*n+{1,7,11,13,17,23,29} are all prime.

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A158725 Non-repdigit composite numbers not divisible by 2, 3, 5 or 11.

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A098591 a(k) contains primality information for the numbers in the interval (k30,...,(k+1)30) packed into one byte using the fact that only numbers == 1, 7, 11, 13, 17, 19, 23, 29 mod 30 can be prime.