A123986 -id:A123986 - cp's OEIS frontend

A122565 Twin primes of form 4k+1, 4k+3.

5, 7, 17, 19, 29, 31, 41, 43, 101, 103, 137, 139, 149, 151, 197, 199, 269, 271, 281, 283, 461, 463, 521, 523, 569, 571, 617, 619, 641, 643, 809, 811, 821, 823, 857, 859, 881, 883, 1049, 1051, 1061, 1063, 1229, 1231, 1277, 1279, 1289, 1291, 1301, 1303, 1481

Offset: 1

Miklos Kristof, Sep 21 2006

nonn

Twin primes with Hamming distance 1 between them. - Vladimir Shevelev, Jan 29 2012

			a(3) = 17 = 4*4 + 1, a(4) = 19 = 4*4 + 3.

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

Cf. A001097, A123986, A205649.

i:=1:for k from 1 to 600 do if isprime(4*k+1) and isprime(4*k+3) then a[i]:=4*k+1:a[i+1]:=4*k+3: i:=i+2 fi od: seq(a[n],n=1..i-1);

Flatten[Select[Table[4k+{1,3},{k,400}],And@@PrimeQ[#] &]] (* Jayanta Basu, May 26 2013 *)

A364678 Maximum number of primes between consecutive multiples of n, as permitted by divisibility considerations.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 5, 7, 7, 6, 7, 7, 7, 7, 8, 7, 8, 9, 8, 10, 8, 10, 10, 10, 11, 11, 11, 10, 11, 11, 11, 12, 12, 12, 12, 13, 12, 13, 14, 13, 13, 14, 14, 15, 15, 14, 15, 15, 15, 16, 15, 15, 16, 16, 17, 16, 17, 18, 18, 18, 18, 18, 17, 19, 19, 19, 19, 20, 20, 19, 19, 20, 21, 21

Offset: 1

Brian Kehrig, Aug 24 2023

nonn

Alternatively: a(n) = the maximum number of elements of an admissible k-tuple strictly contained in (0,n) such that all elements are relatively prime to n. Recall that an admissible tuple is defined as a tuple of integers with the property that all primes p have at least one residue class that has no intersection with the tuple.
For n > 1, we have a(n) <= A023193(n-1), with equality if (but not only if) n is prime or a power of 2. The smallest n for which it is not an equality is n=14.
Conjecture 1: Every nonnegative integer appears in this sequence.
Conjecture 2: For all n, there is an infinitude of k's such that there are a(n) primes between n*k and n*(k+1).
Conjecture 2 resembles the k-tuples conjecture a.k.a. the first Hardy-Littlewood conjecture, although it is not the same.
A notable value is a(35) = 8. Compare with A000010(210) = 48. This says that between any two consecutive multiples of 210 the 48 values that are not divisible by 2, 3, 5 or 7 are equally distributed between 6 equal divisions of 210; that is, 8 are in the interval [0, 34], 8 in the interval [35, 69], etc. - Peter Munn, Feb 16 2024

			Between two multiples of 15 (n and n+15), only n+1, n+2, n+4, n+7, n+8, n+11, n+13, and n+14 could possibly be prime based on divisibility by 3 and 5. However, 4 of these are even and 4 are odd, so at most 4 of them can be prime. Thus, a(15)=4.

Brian Kehrig, Table of n, a(n) for n = 1..1000
Brian Kehrig, Python code for sequence

Cf. A000010, A023193.

Multiples of n following which the maximum number of primes occur for particular n: A005097 (2), A144769 (3), A123986 (4), A056956 (6), A007811 (10), A123985 (12), A309871 (18).

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A309871 Numbers n for which 18n+1, 18n+5, 18n+7, 18n+11, 18n+13 and 18n+17 are primes.

Original entry on oeis.org

892, 2432, 156817, 806697, 822937, 1377022, 1389412, 1418007, 1619642, 1753552, 2017732, 2058647, 2329302, 2554142, 2703347, 3058772, 3135107, 3326522, 3391797, 3723457, 4126867, 4132782, 4171422, 4411837, 4610252, 6378487, 6440087, 6878987, 6897782, 6991547

Offset: 1

Ely Golden, Aug 21 2019

nonn

Ely Golden, Table of n, a(n) for n = 1..1001

Cf. A123986, A056956, A007811, A123985.

tot[n_] := Select[Range[n], CoprimeQ[#, n] &]; m = 18; t = tot[m]; aQ[n_] := AllTrue[m * n + t, PrimeQ]; Select[Range[10^6], aQ] (* Amiram Eldar, Aug 22 2019 *)

x = 1
for i in range(5000000):
    if (18*i+1 in Primes()
    and 18*i+5 in Primes()
    and 18*i+7 in Primes()
    and 18*i+11 in Primes()
    and 18*i+13 in Primes()
    and 18*i+17 in Primes()):
        print(str(x)+" "+str(i))
        x += 1

A254288 Numbers k such that 4*k + {1, 3, 7, 9, 13, 19} are all prime.

Original entry on oeis.org

1, 370, 41425, 81535, 255625, 267175, 311590, 365350, 1054570, 1381750, 2533600, 2975125, 3266080, 3930205, 4684210, 4782385, 4802860, 5940850, 6414610, 7986565, 8429245, 8570470, 8636305, 8810080, 9270715, 9857980, 10459525, 13708225, 13917490, 15127720, 15252460

Offset: 1

K. D. Bajpai, Jan 27 2015

nonn

All terms in this sequence are congruent to 1 mod 3.

Subsequence of A123986.

			a(2) = 370;
4*370 +  1 = 1481;
4*370 +  3 = 1483;
4*370 +  7 = 1487;
4*370 +  9 = 1489;
4*370 + 13 = 1493;
4*370 + 19 = 1499;
All six are prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 726 terms from K. D. Bajpai)

Cf. A000040, A005098, A095278, A123986.

[n: n in [0..10^8] | forall{4*n+i: i in [1, 3, 7, 9, 13, 19] |  IsPrime(4*n+i)}]; // Vincenzo Librandi, Mar 12 2015

Select[Range[5*10^7], PrimeQ[4*# + 1] && PrimeQ[4*# + 3] && PrimeQ[4*# + 7] && PrimeQ[4*# + 9] && PrimeQ[4*# + 13] && PrimeQ[4*# + 19] &]
Select[Range[5*10^6], And @@ PrimeQ /@ ({1, 3, 7, 9, 13, 19} + 4 #) &]

for(n=1,10^7, if( isprime(4*n + 1) && isprime(4*n + 3) &&isprime(4*n + 7) &&isprime(4*n + 9) &&isprime(4*n + 13) &&isprime(4*n + 19) , print1(n,", ")))

A254376 Numbers n such that 4n+1, 4n+3, 4n+7, 4n+9 and 4n+13 are prime.

Original entry on oeis.org

1, 25, 370, 4015, 4855, 10945, 36040, 41425, 41710, 50455, 56335, 61900, 81535, 86995, 116290, 129700, 134110, 158365, 207430, 239635, 255625, 265990, 267175, 272815, 293395, 311590, 335080, 337810, 339700, 342115, 365350, 393385, 403960, 481345, 488590, 550990

Offset: 1

K. D. Bajpai, Jan 29 2015

nonn

All terms in this sequence are 1 mod 3.
Each term yields a set of five consecutive primes.
Alternatively: Numbers n such that 4n+k forms a set of five consecutive primes for k = {1,3,7,9,13}.
Subsequence of A123986.

			25 is in the list because 4*25 + 1 = 101, 4*25 + 3 = 103, 4*25 + 7 = 107, 4*25 + 9 = 109 and 4*25 + 13 = 113 are all prime.

K. D. Bajpai, Table of n, a(n) for n = 1..8458

Cf. A000040, A005098, A095278, A123986.

[n: n in [0..10^6] | forall{4*n+r: r in [1,3,7,9,13] | IsPrime(4*n+r)}]; // Vincenzo Librandi, Feb 16 2015

Select[Range[1, 500000], PrimeQ[4*# + 1] && PrimeQ[4*# + 3] && PrimeQ[4*# + 7] && PrimeQ[4*# + 9] && PrimeQ[4*# + 13] &]
Select[Range[5*10^6], And @@ PrimeQ /@ ({1, 3, 7, 9, 13} + 4 #) &]

for(n=1,10^7, if( isprime(4*n + 1) && isprime(4*n + 3) &&isprime(4*n + 7) &&isprime(4*n + 9) &&isprime(4*n + 13), print1(n,", ")))

A174235 The absolute values of (n-th number k such that k4-+1 is twin prime pair minus n-th number m such that m4+2-+1 is twin prime pair).

Original entry on oeis.org

0, 1, 8, 8, 2, 11, 11, 8, 7, 8, 28, 25, 34, 4, 5, 5, 50, 44, 53, 26, 65, 50, 44, 50, 77, 35, 7, 17, 1, 4, 8, 2, 4, 32, 26, 71, 77, 41, 86, 98, 95, 113, 83, 26, 11, 20, 17, 4, 23, 13, 16, 23, 23, 20, 26, 14, 80, 137, 116, 128, 122, 146, 149, 158, 200, 158, 158, 113, 110, 164, 200

Offset: 1

Juri-Stepan Gerasimov, Mar 13 2010

nonn

Cf. A045753, A123986.

a(n)=Abs(A045753(n)-A123986(n)).

Values from a(63) onwards corrected by R. J. Mathar, Apr 16 2010

A174236 n-th number k such that k4-+1 is twin prime pair plus n-th number m such that m4+2-+1 is twin prime pair.

Original entry on oeis.org

2, 7, 22, 28, 52, 79, 85, 106, 127, 148, 202, 235, 250, 304, 325, 409, 460, 472, 493, 550, 595, 664, 682, 694, 727, 775, 841, 877, 937, 970, 1006, 1042, 1060, 1102, 1144, 1225, 1267, 1315, 1414, 1462, 1489, 1513, 1567, 1654, 1675, 1714, 1747, 1774, 1813, 1897

Offset: 1

Juri-Stepan Gerasimov, Mar 13 2010

nonn

Cf. A045753, A123986.

a(n)=A045753(n)+A123986(n).

Corrected (108 replaced by 106) by R. J. Mathar, Apr 16 2010

A201541 Numbers n such that 12n+5 and 12n+7 are primes.

Original entry on oeis.org

0, 1, 2, 3, 8, 11, 12, 16, 22, 23, 38, 43, 47, 51, 53, 67, 68, 71, 73, 87, 88, 102, 106, 107, 108, 123, 141, 143, 156, 162, 166, 173, 177, 178, 186, 192, 198, 212, 221, 227, 232, 233, 247, 271, 277, 282, 288, 296, 298, 318, 326, 327, 333, 337, 346, 351, 352

Offset: 0

Zak Seidov, Dec 02 2011

nonn

A123979 and A123985 are subsequences.

Cf. A123985, A123986, A123979.

Select[Range[0, 352], PrimeQ[12 # + 5] && PrimeQ[12 # + 7] &] (* T. D. Noe, Dec 05 2011 *)

a(n)=(A123986(n)-1)/3.

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A122565 Twin primes of form 4k+1, 4k+3.

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A364678 Maximum number of primes between consecutive multiples of n, as permitted by divisibility considerations.

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A309871 Numbers n for which 18n+1, 18n+5, 18n+7, 18n+11, 18n+13 and 18n+17 are primes.

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A254288 Numbers k such that 4*k + {1, 3, 7, 9, 13, 19} are all prime.

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A254376 Numbers n such that 4n+1, 4n+3, 4n+7, 4n+9 and 4n+13 are prime.

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A174235 The absolute values of (n-th number k such that k*4-+1 is twin prime pair minus n-th number m such that m*4+2-+1 is twin prime pair).

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A174236 n-th number k such that k*4-+1 is twin prime pair plus n-th number m such that m*4+2-+1 is twin prime pair.

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A201541 Numbers n such that 12n+5 and 12n+7 are primes.

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A174235 The absolute values of (n-th number k such that k4-+1 is twin prime pair minus n-th number m such that m4+2-+1 is twin prime pair).

A174236 n-th number k such that k4-+1 is twin prime pair plus n-th number m such that m4+2-+1 is twin prime pair.