A067793 -id:A067793 - cp's OEIS frontend

A038509 Composite numbers congruent to +-1 mod 6.

25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169, 175, 185, 187, 203, 205, 209, 215, 217, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, 355, 361, 365, 371, 377, 385

Offset: 1

Jeff Burch

Or, composite numbers with smallest prime factor >= 5.
Or, nonprime numbers n such that binomial(n+3, 3) mod n == 1. - Hieronymus Fischer, Sep 30 2007
Note that the primes > 3 are congruent to +-1 mod 6.
This sequence differs from A067793 (composite n such that phi(n) > 2n/3) starting at 385. Numbers in this sequence but not in A067793 are 385, 455, 595, 665, 805, 1015, 1085, 1925, 2275, 2695, etc. See A069043. - R. J. Mathar, Jun 08 2008 and Zak Seidov, Nov 02 2011
Intersection of A002808 and A007310. - Reinhard Zumkeller, Jun 30 2012
The product (24/25) * (36/35) * (48/49) * (54/55) * (66/65) * (78/77) * (84/85) * (90/91) * ... * ((6*k)/a(n)) * ... = Pi^2/(6*sqrt(3)), where 6*k is the nearest number to a(n), with k in A067611 but not in A002822. (See A258414.) - Dimitris Valianatos, Mar 27 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A171993 (nonprimes of the form 3*k+-1).
Cf. A000040, A133620-A133625, A133630, A133634-A133636.
Cf. A133873, A133883, A133880, A133890, A133900, A133910.
Cf. A069043, A067793 (composite n such that phi(n) > 2n/3).

a038509 n = a038509_list !! (n-1)
a038509_list = [x | x <- a002808_list, gcd x 6 == 1]
-- Reinhard Zumkeller, Aug 05 2014, Jun 30 2012

A038509 := proc(n)
    option remember;
    if n = 1 then
        25;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and modp(a,6) in {1,5} then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A038509(n),n=1..30) ; # R. J. Mathar, Feb 28 2020

Select[Range[1000], FactorInteger[#][[1,1]] >= 5 && ! PrimeQ[#] &] (* Robert G. Wilson v, Dec 19 2009 *)
With[{nn=400},Select[Rest[Complement[Range[nn],Prime[Range[ PrimePi[ nn]]]]], MemberQ[ {1,5},Mod[#,6]]&]] (* Harvey P. Dale, Feb 21 2013 *)
Select[Range[400],CompositeQ[#]&&MemberQ[{1,5},Mod[#,6]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2019 *)

is(n)=gcd(n,6)==1 && !isprime(n) && n>7 \\ Charles R Greathouse IV, Nov 20 2012

a(n) ~ 3n. - Charles R Greathouse IV, Nov 20 2012

More terms from Robert G. Wilson v, Dec 19 2009

Entry revised by N. J. A. Sloane, Dec 31 2011, at the suggestion of Gary Detlefs

A069043 Composite numbers k with no prime factor among (2, 3) (cf. A038509) and such that phi(k) < 2*k/3.

Original entry on oeis.org

385, 455, 595, 665, 805, 1015, 1085, 1925, 2275, 2695, 2975, 3185, 3325, 4025, 4165, 4235, 4655, 5005, 5075, 5425, 5635, 5915, 6545, 7105, 7315, 7595, 7735, 8645, 8855, 9625, 10115, 10465, 11165, 11305, 11375, 11935, 12155, 12635, 13195, 13475

Offset: 1

Benoit Cloitre, Apr 03 2002

Almost all composite numbers k such that phi(k) > 2k/3 have no prime factor among 2 and 3.

Marius A. Burtea, Table of n, a(n) for n = 1..10000

Cf. A000010, A038509, A067793.

[k:k in [1..14000]| not IsPrime(k) and Gcd(6,k) eq 1 and EulerPhi(k) lt 2*k/3]; // Marius A. Burtea, Oct 01 2019

Select[Flatten @ Table[6*n + {1, 5}, {n, 0, 2500}], CompositeQ[#] && EulerPhi[#] < 2*#/3 &] (* Amiram Eldar, Jun 09 2022 *)

lista(nn) = {forcomposite(n=1, nn, if ((gcd(n, 6) == 1) && (n/eulerphi(n) > 3/2), print1(n, ", ")););} \\ Michel Marcus, Jul 05 2015

A038509 SETMINUS A067793: numbers in A038509 but not in A067793.

A054550 Composite numbers whose least prime factor is either 5 or 7.

Original entry on oeis.org

25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 125, 133, 145, 155, 161, 175, 185, 203, 205, 215, 217, 235, 245, 259, 265, 275, 287, 295, 301, 305, 325, 329, 335, 343, 355, 365, 371, 385, 395, 413, 415, 425, 427, 445, 455, 469, 475, 485, 497, 505, 511, 515, 535

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), May 15 2000

Original definition: Union of 4 AP's: 25+30n, 35+30n, 49+42n, 77+42n.

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

Cf. A002808, A038509, A038511, A047229, A067793.

Union[Flatten[Table[{30n+{25,35},42n+{49,77}},{n,0,20}]]] (* Harvey P. Dale, Feb 19 2016 *)

select( is_A054550(n)=vecsum((n=factor(n,0))[,2])>1&&n[1,1]>=5, [0..550]) \\ M. F. Hasler, Nov 18 2018

Vec(x*(25 + 10*x + 14*x^2 + 6*x^3 + 10*x^4 + 12*x^5 + 8*x^6 + 6*x^7 + 4*x^8 + 20*x^9 + 4*x^10 + 6*x^11 + 8*x^12 + 12*x^13 + 10*x^14 + 6*x^15 + 14*x^16 + 10*x^17 + 18*x^18 + 2*x^19 + 10*x^20 + 2*x^21 - 7*x^22) / ((1 - x)^2*(1 + x)*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)) + O(x^60)) \\ Colin Barker, Nov 19 2018

a(n) = a(n-1) + a(n-22) - a(n-23). - Charles R Greathouse IV, Jun 01 2018

G.f.: x*(25 + 10*x + 14*x^2 + 6*x^3 + 10*x^4 + 12*x^5 + 8*x^6 + 6*x^7 + 4*x^8 + 20*x^9 + 4*x^10 + 6*x^11 + 8*x^12 + 12*x^13 + 10*x^14 + 6*x^15 + 14*x^16 + 10*x^17 + 18*x^18 + 2*x^19 + 10*x^20 + 2*x^21 - 7*x^22) / ((1 - x)^2*(1 + x)*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)). - Colin Barker, Nov 19 2018

More terms from R. J. Mathar, Sep 30 2008

New name suggested by Andrew Howroyd, Nov 19 2018

A287918 Union of nonprime 1 <= t <= m for m in A036913, with gcd(t,m) = 1.

Original entry on oeis.org

1, 25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155, 161, 169, 185, 187, 203, 205, 209, 215, 217, 221, 235, 247, 253, 259, 265, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 341, 343, 355, 361, 365, 371, 377, 391, 395, 403

Offset: 1

Jamie Morken and Michael De Vlieger, Jun 11 2017

nonn

List of nonprime totatives t of m for m in A036913.
Nonprime 1 is coprime to all numbers, thus a(1) = 1.
The integers {175, 245, 275} are absent, distinguishing this sequence from A038509 and A067793. These terms have factors 5^2 * 7, 5 * 7^2, 5^2 * 11. Only the terms in positions {2, 3, 4, 6, 8, 11, 18} of A036913 (i.e., {6, 12, 18, 42, 66, 126, 462}) are larger and coprime to 5. Of these only 462 is greater than these three terms, however 462 is divisible by 7 and 11. Thus {175, 245, 275} are not terms.
Squared primes q^2 for q >= 5 appear in the sequence at positions {2, 4, 13, 20, 35, 48, 71, 107, 123, 173, ...}. These are coprime to and smaller than {42, 60, 126, 210, 330, 420, ...} at indices {6, 7, 11, 13, 16, 17, 20, 25, 25, 28, 30, 30, 31, 40, 33, 35, ...} in A036913.

			From _Michael De Vlieger_, Jun 14 2017: (Start)
List of nonprime totatives 1 <= t <= m for m <= 210 in A036913:
    m: 1 <= t <= m
    2: 1;
    6: 1;
   12: 1;
   18: 1;
   30: 1;
   42: 1, 25;
   60: 1, 49;
   66: 1, 25, 35, 49, 65;
   90: 1, 49, 77;
  120: 1, 49, 77, 91, 119;
  126: 1, 25, 55, 65, 85, 95, 115, 121, 125;
  150: 1, 49, 77, 91, 119, 121, 133, 143;
  210: 1, 121, 143, 169, 187, 209;
       ...
Indices of A036913 of first and last terms m such that gcd(a(n),m)=1:
   n   a(n)   Freq.  First   Last
  -------------------------------
   1      1     oo       1     oo
   2     25      4       6     18
   3     35      1       8      8
   4     49     14       7     40
   5     55      1      11     11
   6     65      3       8     18
   7     77      8       9     24
   8     85      2      11     18
   9     91     11      10     40
  10     95      2      11     18
  11    115      2      11     18
  12    119      9      10     27
  13    121     75      11    308
  14    125      2      11     18
  15    133     10      12     40
  16    143     36      12    107
  17    145      1      18     18
  18    155      1      18     18
  19    161      8      14     40
  20    169     96      13    248
  ...
Positions of squared primes q^2 in a(n):
        q^2           q
    n   a(n)  sqrt(a(n))     k    m = A036913(k)
  ----------------------------------------------
    2     25          5      6       42
    4     49          7      7       60
   13    121         11     11      126
   20    169         13     13      210
   35    289         17     16      330
   48    361         19     17      420
   71    529         23     20      630
  107    841         29     25     1050
  123    961         31     25     1050
  173   1369         37     28     1470
  210   1681         41     30     1890
  234   1849         43     30     1890
  283   2209         47     31     2310
  303   2401         49     40     5610
  359   2809         53     33     2940
  456   3481         59     35     3570
  486   3721         61     36     3990
  598   4489         67     37     4620
  676   5041         71     39     5460
  721   5329         73     39     5460
  ...
(End)

Cf. A001248, A036913, A038509, A067793, A285784, A287917.

With[{nn = 403, s = Union@FoldList[Max, Values[#][[All, -1]]] &@ KeySort@ PositionIndex@ EulerPhi@ Range[Product[Prime@ i, {i, 8}]]}, Union@ Flatten@ Map[Function[n, Select[Range@ Min[n, nn], And[CoprimeQ[#, n], ! PrimeQ@ #] &]], s]] (* Michael De Vlieger, Jun 14 2017 *)

cp's OEIS Frontend

A038509 Composite numbers congruent to +-1 mod 6.

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A069043 Composite numbers k with no prime factor among (2, 3) (cf. A038509) and such that phi(k) < 2*k/3.

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A054550 Composite numbers whose least prime factor is either 5 or 7.

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A287918 Union of nonprime 1 <= t <= m for m in A036913, with gcd(t,m) = 1.

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