A073102 -id:A073102 - cp's OEIS frontend

A140841 Primes of the form 210n + 13.

13, 223, 433, 643, 853, 1063, 1483, 1693, 2113, 2953, 3163, 3373, 3583, 3793, 4003, 4423, 5683, 6733, 7573, 7993, 8623, 9043, 9463, 9883, 10093, 10303, 10513, 10723, 11353, 12613, 12823, 13033, 13873, 14083, 14293, 14503, 14713, 14923, 15973

Offset: 1

Juri-Stepan Gerasimov, Jul 04 2008

nonn

Cf. A073102, A075745.

A140841 := proc(n) local a; if n = 1 then 13; else a := nextprime(procname(n-1)) ; while not a mod 210 in {13} do a := nextprime(a) ; end do: return a; end if; end: seq(A140841(n),n=1..80) ; # R. J. Mathar, Oct 22 2009

Select[13+210Range[0,150],PrimeQ] (* Ray Chandler, Apr 29 2010 *)

forprime(p=1,1e4,if(p%210==13,print1(p", "))) \\ Charles R Greathouse IV, Dec 21 2011

More terms from R. J. Mathar, Oct 22 2009

A217587 Primes p of the form 420k + 1 for some k.

Original entry on oeis.org

421, 2521, 3361, 4201, 4621, 5881, 6301, 7561, 8821, 9241, 9661, 10501, 12601, 13441, 14281, 15121, 15541, 16381, 18061, 18481, 20161, 21001, 21841, 24781, 25621, 26041, 26881, 29401, 30241, 30661, 31081, 32341, 33181, 33601, 35281, 36541, 39901, 41161

Offset: 1

Joshua S.M. Weiner, Oct 07 2012

nonn

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

Subsequence of A073102.

Select[Prime[Range[5000]], Mod[#, 420] == 1 &] (* T. D. Noe, Oct 08 2012 *)
Select[420*Range[100]+1,PrimeQ] (* Harvey P. Dale, Jun 06 2013 *)

A359262 a(n) is the largest number m such that prime(n)^m is in A359260.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 3, 1, 1, 5, 3, 1, 3, 1, 1, 1, 5, 3, 1, 3, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 5, 3, 3, 1, 1, 1, 5, 1, 3, 1, 3, 9, 3, 1, 3, 1, 1, 5, 1, 1, 1, 1, 5, 3, 1, 3, 1, 3, 1, 3, 1, 5, 3, 1, 3, 1, 1, 3, 3, 3, 1, 1, 3, 1, 3, 1, 9, 1, 3, 3, 1, 1

Offset: 1

Amiram Eldar, Dec 23 2022

a(n) is the largest number m such that the arithmetic mean of {1, p, p^2, ..., p^k} is an integer for all k in 1..m.
Apparently, all the terms are of the form prime(k)-2 (A040976). Conjecture: The asymptotic density of the occurrences of prime(k)-2 is (1/s(k-1)-1/s(k)), where s(k) = A005867(k) = phi(prime(k)#), and prime(k)# is the k-th primorial number (A002110).
The sums of the first 10^k terms, for k = 1, 2, ..., are 15, 221, 2291, 23287, 233641, 2337007, 23379901, 233814475, 2338211029, 23382168187, ... . If the mentioned above conjecture is correct, then the asymptotic mean of this sequence is Sum_{k>=1} (prime(k)-2)*(1/s(k-1)-1/s(k)) = 2.33821872365981424748... .
Apparently, the indices of records after n = 1 occur at A000720(A073917(n)) (verified for the first 12 terms of A073917) with record values a(A000720(A073917(n))) = prime(n+1) - 2 (verified for the first 150 terms of A073917).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A002110, A040976, A073917, A005867, A359260, A359261.

Cf. A002476, A073102, A132230.

a[n_] := Module[{p = Prime[n], k = 1, r = s = 1}, While[Divisible[s, k], k++; r *= p; s += r]; k - 2]; Array[a, 100]

a(n) = {my(p = prime(n), k = 1, r = s = 1); while(!(s%k), k++; r *= p; s += r); k - 2; }

a(n) >= 1 for n >= 2.
a(n) >= 3 iff prime(n) == 1 (mod 6) (prime(n) is in A002476).
Conjectures:
a(n) >= 5 iff prime(n) == 1 (mod 30) (prime(n) is in A132230).
a(n) >= 9 iff prime(n) == 1 (mod 210) (prime(n) is in A073102).
a(n) >= prime(k) - 2 iff prime(n) == 1 (mod A002110(k-1)).

A073085 Numbers n such that 210*n+1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 16, 17, 20, 22, 23, 28, 29, 30, 35, 36, 39, 42, 44, 46, 47, 50, 51, 53, 55, 57, 59, 60, 64, 67, 68, 72, 73, 74, 78, 81, 83, 85, 86, 88, 89, 93, 96, 100, 101, 104, 105, 111, 115, 117, 118, 121, 122, 124, 125, 128, 129, 135, 137, 139, 140, 141

Offset: 1

Zak Seidov, Oct 08 2002

			1 is a member because 210*1+1=211 is prime; 100 is a member because 210*100+1=21001 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A073102.

[n: n in [1..200] | IsPrime(210*n + 1)]; Vincenzo Librandi, Sep 30 2012

Flatten[Position[PrimeQ[210Range[100]+1], True]]
Select[Range[150], PrimeQ[(210*# + 1)] &] (* Vincenzo Librandi, Sep 30 2012 *)

is(n)=isprime(210*n+1) \\ Charles R Greathouse IV, May 22 2017

A140848 Primes of the form 210k + 41.

Original entry on oeis.org

41, 251, 461, 881, 1091, 1301, 1511, 1721, 1931, 2141, 2351, 3191, 3821, 4241, 4451, 4871, 5081, 5501, 5711, 6131, 6551, 6761, 6971, 8231, 8861, 9281, 9491, 10331, 11171, 11801, 12011, 12641, 13691, 13901, 14321, 14741, 14951, 15161, 15581

Offset: 1

Juri-Stepan Gerasimov, Jul 04 2008

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A073102.

[ a: n in [0..900] | IsPrime(a) where a is 210*n+41] // Vincenzo Librandi, Nov 24 2010

Select[41+210Range[0,150],PrimeQ] (* Ray Chandler, Apr 29 2010 *)

More terms from Vincenzo Librandi, Apr 28 2010

A140840 Primes of the form 210n+11.

Original entry on oeis.org

11, 431, 641, 1061, 1481, 1901, 2111, 2531, 2741, 3371, 3581, 4001, 4211, 4421, 5051, 5261, 5471, 6101, 6311, 6521, 7151, 8831, 9041, 9461, 10091, 10301, 11351, 11981, 12401, 12611, 12821, 13241, 13451, 14081, 15131, 15551, 15761, 15971, 16811, 17021

Offset: 1

Juri-Stepan Gerasimov, Jul 04 2008

nonn

Cf. A073102, A076355.

Select[210 Range[0, 100] + 11, PrimeQ@ # &]
Select[Range[11,10^4,210],PrimeQ] (* Zak Seidov, Jan 14 2014 *)

a(n) = 210*A076355(n) + 11. - Zak Seidov, Jan 14 2014

I changed the Mathematica coding and extended the sequence. - Robert G. Wilson v, Sep 22 2008

A140842 Primes of the form 210k + 17.

Original entry on oeis.org

17, 227, 647, 857, 1277, 1487, 1697, 1907, 2957, 3167, 3797, 4007, 4217, 4637, 5477, 5897, 6317, 6737, 6947, 7577, 8627, 8837, 9257, 9467, 9677, 9887, 10937, 11777, 11987, 12197, 13037, 13457, 13877, 14087, 14717, 15137, 15767, 16187, 16607

Offset: 1

Juri-Stepan Gerasimov, Jul 04 2008

nonn

Cf. A073102, A075747.

[ a: n in [0..900] | IsPrime(a) where a is 210*n+17] // Vincenzo Librandi, Nov 24 2010

Select[17 + 210 Range[0, 100], PrimeQ] (* T. D. Noe, Apr 27 2010 *)

Extended by several contributors, Apr 29 2010

A140844 Primes of the form 210k + 23.

Original entry on oeis.org

23, 233, 443, 653, 863, 1283, 1493, 1913, 2333, 2543, 2753, 2963, 3593, 3803, 4013, 4643, 5273, 5483, 5693, 5903, 6113, 6323, 7583, 7793, 8423, 9473, 10103, 10313, 10733, 11783, 12203, 12413, 13043, 13463, 13883, 14303, 14723, 15773, 16193

Offset: 1

Juri-Stepan Gerasimov, Jul 04 2008

nonn

Cf. A073102.

[a: n in [0..900]|IsPrime(a) where a is 210*n+23]; // Vincenzo Librandi, Nov 24 2010

Select[23+210Range[0,150],PrimeQ] (* Ray Chandler, Apr 29 2010 *)

Extended by several authors, Apr 29 2010

A140847 Primes of the form 210k + 37.

Original entry on oeis.org

37, 457, 877, 1087, 1297, 2137, 2347, 2557, 2767, 3187, 3607, 4027, 4447, 4657, 5077, 6337, 6547, 6967, 7177, 8017, 8647, 9067, 9277, 9697, 9907, 10957, 11587, 12007, 12637, 13267, 13477, 13687, 14107, 14737, 14947, 15787, 16417, 17047, 17257

Offset: 1

Juri-Stepan Gerasimov, Jul 04 2008

nonn

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

Cf. A073102.

[ a: n in [0..90] | IsPrime(a) where a is 210*n+37 ]; // Vincenzo Librandi, Nov 24 2010

Select[37+210Range[0,150],PrimeQ] (* Ray Chandler, Apr 29 2010 *)

Extended by Ray Chandler, Apr 29 2010

More terms from Vincenzo Librandi, Apr 28 2010

A140856 Primes of the form 210n+71.

Original entry on oeis.org

71, 281, 491, 701, 911, 2381, 2591, 2801, 3011, 3221, 3851, 4271, 4481, 4691, 5531, 5741, 6581, 6791, 7001, 7211, 7841, 8681, 9311, 9521, 9941, 10151, 10781, 11411, 11621, 11831, 12041, 12251, 12671, 13721, 13931, 14561, 14771, 15401, 16451

Offset: 1

Juri-Stepan Gerasimov, Jul 04 2008

nonn

Cf. A073102.

Select[71+210Range[0,150],PrimeQ] (* Ray Chandler, Apr 29 2010 *)

Corrected and extended by D. S. McNeil, Dec 10 2009

cp's OEIS Frontend

A140841 Primes of the form 210n + 13.

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A217587 Primes p of the form 420k + 1 for some k.

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A359262 a(n) is the largest number m such that prime(n)^m is in A359260.

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A073085 Numbers n such that 210*n+1 is prime.

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Magma

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A140848 Primes of the form 210k + 41.

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Magma

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A140840 Primes of the form 210n+11.

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A140842 Primes of the form 210k + 17.

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Magma

Mathematica

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A140844 Primes of the form 210k + 23.

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Magma

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A140847 Primes of the form 210k + 37.

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