A024906 -id:A024906 - cp's OEIS frontend

A024911 Duplicate of A024906.

2, 4, 8, 12, 14, 18, 20, 22, 30, 34, 42, 44, 48, 54, 58, 60, 64, 68, 70, 82, 84, 90, 92, 98, 102

Offset: 1

dead

A111094 Numbers k such that 18*k + 1 is prime.

1, 2, 4, 6, 7, 9, 10, 11, 15, 17, 21, 22, 24, 27, 29, 30, 32, 34, 35, 41, 42, 45, 46, 49, 51, 52, 55, 56, 59, 62, 64, 65, 71, 72, 79, 81, 85, 86, 87, 90, 92, 94, 97, 99, 100, 104, 111, 112, 114, 116, 119, 120, 121, 125, 126, 127, 130, 132, 137, 139, 140, 141, 142, 144

Offset: 1

Parthasarathy Nambi, Oct 20 2005

a(n) = A024906(n)/2. - Zak Seidov, Feb 02 2010

			If k=99 then 18*k + 1 = 1783 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A024906.

[ n: n in [0..1500] | IsPrime(18*n + 1) ]; // Vincenzo Librandi, Jan 31 2011

Select[Range[200],PrimeQ[18#+1]&]  (* Harvey P. Dale, Mar 14 2011 *)

is(n)=isprime(18*n+1) \\ Charles R Greathouse IV, Feb 20 2017

More terms from Zak Seidov, Feb 02 2010

A211821 Numbers with all divisors with additive digital root of 1.

Original entry on oeis.org

1, 19, 37, 73, 109, 127, 163, 181, 199, 271, 307, 361, 379, 397, 433, 487, 523, 541, 577, 613, 631, 703, 739, 757, 811, 829, 883, 919, 937, 991, 1009, 1063, 1117, 1153, 1171, 1279, 1297, 1369, 1387, 1423, 1459, 1531, 1549, 1567, 1621, 1657, 1693, 1747, 1783

Offset: 1

Jaroslav Krizek, Apr 26 2012

All divisors of numbers from this sequence are in this sequence. Likewise, the product of any terms in this sequence is a number that is also in this sequence.
Union of A061237 (prime numbers == 1 (mod 9)) and nonprime numbers A211822.
Subsequence of A017173 (numbers of form 9n+1). - Jaroslav Krizek
For prime numbers, it is enough to verify that the number itself is congruent to 1 mod 9. The first composite term is 361, which is the square of the first prime in this sequence. - Alonso del Arte, May 02 2012

			Number 703 with divisors 1, 19, 37, 703 is in sequence because all divisors have additive digital root of 1.

Cf. A211822, A211823, A024906, A061237, A017173.

digitalRoot[n_, b_:10] := FixedPoint[Plus@@IntegerDigits[#, b] &,  n]; A211821 = Select[Range[1, 1999, 9], Union[digitalRoot[Divisors[#]]] == {1} &] (* Alonso del Arte, May 02 2012 *)

a(n) = 9*k(n) + 1 for k(n) = A211823(n).

A211822 Nonprime numbers with all divisors with additive digital root of 1.

Original entry on oeis.org

1, 361, 703, 1369, 1387, 2071, 2413, 2701, 3097, 3439, 3781, 4033, 4699, 5149, 5329, 5833, 6031, 6697, 6859, 7201, 7363, 7543, 7957, 8227, 9253, 9271, 9937, 10027, 10279, 10963, 11359, 11647, 11881, 11899, 11989, 13213, 13357

Offset: 1

Jaroslav Krizek, Apr 26 2012

Complement of A061237 (prime numbers == 1 (mod 9)) with respect to A211821.

			Number 6859 with divisors 1, 19, 361, 6859 is in sequence because all divisors have additive digital root of 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A211821, A024906, A061237, A017173, A211823.

filter:= n -> not isprime(n) and numtheory:-factorset(n) mod 9 = {1}:
filter(1):= true:
select(filter, [seq(i,i=1..20000,9)]); # Robert Israel, May 10 2020

(* First run the program for A211821 *) Select[A211821, Not[PrimeQ[#]] &] (* Alonso del Arte, May 02 2012 *)

A211823 Numbers k such that 9*k+1 are numbers with all divisors with additive digital root = 1.

Original entry on oeis.org

0, 2, 4, 8, 12, 14, 18, 20, 22, 30, 34, 40, 42, 44, 48, 54, 58, 60, 64, 68, 70, 78, 82, 84, 90, 92, 98, 102, 104, 110, 112, 118, 124, 128, 130, 142, 144, 152, 154, 158, 162, 170, 172, 174, 180, 184, 188, 194, 198, 200, 208, 222, 224, 228, 230, 232, 238, 240, 242

Offset: 1

Jaroslav Krizek, Apr 26 2012

Numbers of form 9*k+1 with all divisors with digital root = 1 is in A211821.

Supersequence of A024906 (numbers n such that 9*n+1 is prime).

			Number k = 40 is in sequence because number 9*40 + 1 = 361 is number with all divisors (1, 19, 361) with additive digital root = 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A211821, A024906, A061237, A017173, A211822

adrQ[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]==1; Select[Range[ 0,250],AllTrue[Divisors[9#+1],adrQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2020 *)

Corrected (230 inserted) by Harvey P. Dale, Aug 27 2020

A224467 Numbers n such that 27*n+1 is prime.

Original entry on oeis.org

4, 6, 10, 14, 16, 18, 20, 28, 30, 34, 48, 54, 58, 60, 66, 74, 76, 80, 84, 88, 94, 96, 98, 108, 110, 114, 118, 128, 130, 136, 138, 144, 146, 150, 154, 166, 170, 180, 184, 186, 188, 198, 206, 214, 230, 236, 238, 240, 258, 264, 268, 278, 280, 284, 286, 296, 300

Offset: 1

K. D. Bajpai, Jul 20 2013

			27*6 + 1 = 163 which is prime. Hence 6 is in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..5736

Cf. A005123, A024906.

isA224467 := proc(n)
    isprime(27*n+1) ;
end proc:
for n from 1 to 300 do
    if isA224467(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 20 2013

Select[Range[300],PrimeQ[27#+1]&] (* Harvey P. Dale, Apr 14 2017 *)

is(n)=isprime(27*n+1) \\ Charles R Greathouse IV, Jun 06 2017

A368201 Least k such that 9nk+1 is a prime.

Original entry on oeis.org

2, 1, 4, 1, 4, 2, 2, 1, 2, 2, 2, 1, 8, 1, 2, 3, 2, 1, 12, 1, 2, 1, 4, 2, 8, 4, 2, 3, 2, 1, 4, 2, 6, 1, 2, 4, 6, 6, 10, 5, 2, 1, 4, 1, 2, 2, 4, 1, 2, 4, 2, 2, 14, 1, 2, 2, 4, 1, 2, 1, 14, 2, 4, 1, 2, 3, 12, 1, 6, 1, 2, 2, 6, 3, 6, 3, 6, 5, 2, 3, 2, 1, 6, 1, 2

Offset: 1

Robert Price, Dec 16 2023

nonn

Robert Price, Table of n, a(n) for n = 1..5000

Cf. A016014, A024906.

A070853 lists the corresponding primes.

A368201 = {};
Do[k=1; While[!PrimeQ[9 n k+1], k++]; AppendTo[A368201,k], {n,85}];
A368201

a(n) = my(k=1); while (!isprime(9*n*k+1), k++); k; \\ Michel Marcus, Dec 16 2023

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A024911 Duplicate of A024906.

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A111094 Numbers k such that 18*k + 1 is prime.

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A211821 Numbers with all divisors with additive digital root of 1.

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A211822 Nonprime numbers with all divisors with additive digital root of 1.

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A211823 Numbers k such that 9*k+1 are numbers with all divisors with additive digital root = 1.

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A224467 Numbers n such that 27*n+1 is prime.

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A368201 Least k such that 9*n*k+1 is a prime.

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A368201 Least k such that 9nk+1 is a prime.