cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A216310 The prime ending in 3 is the only prime in a decade.

Original entry on oeis.org

113, 293, 683, 743, 773, 863, 953, 983, 1163, 1193, 1373, 1523, 1583, 1733, 1823, 1913, 2003, 2053, 2153, 2213, 2243, 2273, 2423, 2503, 2633, 2663, 2753, 2843, 3023, 3413, 3433, 3593, 3623, 3643, 3803, 3833, 3863, 4363, 4373, 4463, 4493, 4523, 4583, 4603
Offset: 1

Views

Author

V. Raman, Sep 03 2012

Keywords

Comments

Primes of the form 10n+3 such that 10n+1, 10n+7, and 10n+9 are composite. - Charles R Greathouse IV, Sep 06 2012

Links

Crossrefs

Subsequence of A030431. Cf. A032352, A007811, A078494.

Programs

  • Mathematica
    t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[ps == {10*n + 3}, AppendTo[t, ps[[1]]]], {n, 0, 595}]; t (* T. D. Noe, Sep 04 2012 *)
Select[10*Range[500]+3,PrimeQ[#]&&AllTrue[#+{-2,4,6},CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 27 2016 *)

Formula

a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012

A216311 The prime ending in 7 is the only prime in a decade.

Original entry on oeis.org

97, 127, 307, 367, 397, 457, 487, 557, 587, 727, 787, 797, 877, 907, 937, 967, 1087, 1117, 1447, 1567, 1597, 1637, 1657, 1777, 1847, 1987, 2437, 2467, 2617, 2647, 2707, 2767, 2777, 2887, 2897, 2917, 2927, 3037, 3137, 3217, 3407, 3457, 3607, 3727, 3847
Offset: 1

Views

Author

V. Raman, Sep 03 2012

Keywords

Comments

Primes of the form 10n+7 such that 10n+1, 10n+3, and 10n+9 are composite. - Charles R Greathouse IV, Sep 06 2012

Links

Crossrefs

Subsequence of A030432.
Cf. A032352, A007811, A078494.

Programs

  • Magma
    [p: p in PrimesUpTo(4000) | p mod 10 eq 7 and IsOne(#PrimesInInterval(10*t+1, 10*t+9)) where t is Floor(p/10)]; // Bruno Berselli, Sep 14 2012
  • Mathematica
    t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[ps == {10*n + 7}, AppendTo[t, ps[[1]]]], {n, 0, 529}]; t (* T. D. Noe, Sep 04 2012 *)

Formula

a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012

A216312 The prime ending in 9 is the only prime in a decade.

Original entry on oeis.org

149, 419, 479, 719, 809, 839, 929, 1009, 1049, 1249, 1259, 1319, 1399, 1409, 1709, 1889, 1949, 2039, 2099, 2129, 2179, 2309, 2459, 2579, 2609, 2729, 2789, 2819, 2879, 2939, 2999, 3079, 3109, 3119, 3299, 3359, 3389, 3449, 3659, 3719, 3779, 3989, 4049, 4229
Offset: 1

Views

Author

V. Raman, Sep 03 2012

Keywords

Comments

Primes of the form 10n+9 such that 10n+1, 10n+3, and 10n+7 are composite. - Charles R Greathouse IV, Sep 06 2012

Examples

			149 is prime but 141, 143 and 147 are all composite (being 3 * 47, 11 * 13 and 3 * 7^2 respectively), thus 149 is in the sequence.

Links

Crossrefs

Subsequence of A030433. Cf. A032352, A007811, A078494, A030433.

Programs

  • Magma
    [p: p in PrimesUpTo(4300) | p mod 10 eq 9 and IsOne(#PrimesInInterval(10*t+1, 10*t+9)) where t is Floor(p/10)]; // Bruno Berselli, Sep 14 2012
  • Mathematica
    Select[Prime[Range[700]], Mod[#, 10] == 9 && Union[PrimeQ[{# - 8, # - 6, # - 2}]] == {False} &] (* Alonso del Arte, Sep 03 2012 *)
Select[Table[10n+{1,3,7,9},{n,450}],Boole[PrimeQ[#]]=={0,0,0,1}&][[;;,4]] (* Harvey P. Dale, Mar 08 2023 *)

Formula

a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012

A316630 Numbers k such that 10k+1, 10k+3, 10k+7, and 10k+9 are all composite, and k == 1 (mod 3).

Original entry on oeis.org

133, 196, 232, 256, 298, 328, 397, 403, 406, 418, 430, 457, 484, 640, 643, 664, 709, 727, 742, 802, 847, 865, 898, 907, 970, 991, 1012, 1054, 1057, 1081, 1087, 1096, 1120, 1153, 1156, 1213, 1231, 1246, 1327, 1354, 1360, 1381, 1411, 1423, 1426, 1435, 1480, 1504
Offset: 1

Views

Author

Patrick A. Thomas, Jul 09 2018

Keywords

Comments

The sequence contains all numbers of the form 50151*m + 42175. - Michael B. Porter, Jul 16 2018
If m is in the sequence, then so is m + 3*(10*m+1)*(10*m+3)*(10*m+7)*(10*m+9)*k for all k. - Robert Israel, Aug 08 2018

Examples

			1331 = 11^3, 1333 = 31*43, 1337 = 7*191, 1339 = 13*103, and 133 == 1 (mod 3), so 133 is a sequence member.

Links

Crossrefs

Cf. A007811, A032352.

Programs

  • MATLAB
    m=1; for s=1:510 v=[30*s+11,30*s+13,30*s+17,30*s+19]; if isprime(v)==0  sol(m)=3*s+1; m=m+1;end; end; sol % Marius A. Burtea, Sep 17 2019
  • Magma
    [3*s+1: s in [0..510] | forall{30*s+k: k in [11, 13, 17, 19] | not IsPrime(30*s+k)}]; // Marius A. Burtea, Sep 17 2019
  • Maple
    remove(t -> ormap(isprime, [10*t+1,10*t+3,10*t+7,10*t+9]), [seq(k,k=1..2000,3)]); # Robert Israel, Aug 08 2018
  • Mathematica
    Select[1 + 3 Range@510, Union[ PrimeQ[10 # + {1, 3, 7, 9}]] == {False} &] (* Robert G. Wilson v, Jul 16 2018 *)
Select[Range[1,2000,3],AllTrue[10#+{1,3,7,9},CompositeQ]&]  (* Harvey P. Dale, Feb 24 2024 *)
  • PARI
    ok(k)={if(k%3==1, for(i=0, 4, if(isprime(10*k+2*i+1), return(0))); 1, 0)} \\ Andrew Howroyd, Jul 10 2018

A342814 Numbers k such that k - 1 and floor(k/5) are both prime.

Original entry on oeis.org

12, 14, 18, 38, 68, 98, 158, 308, 338, 368, 398, 488, 548, 758, 788, 908, 968, 998, 1118, 1568, 1658, 1748, 1868, 1988, 2288, 2438, 2618, 2708, 2858, 2888, 3038, 3068, 3218, 3308, 3458, 3548, 3638, 3698, 3848, 4058
Offset: 1

Views

Author

Claude H. R. Dequatre, Mar 22 2021

Keywords

Comments

Except for a(1) and a(2), all terms == 8 (mod 10).
The first three absolute differences (d) between two consecutive floor(k/5) are respectively equal to 0, 1 and 4 and all the others to 6 or a multiple of 6.
Subsequence of A008864, by definition. - Michel Marcus, Mar 22 2021
For n >= 3, a(n) = 5*A023217(n-2) + 3. Higher terms also coincide with A265767 + 1. - Hugo Pfoertner, Mar 22 2021

Examples

			12 is a term because 12 - 1 = 11 and 11 is prime and 12/5 = 2.4 whose floor value is 2 and 2 is also prime.
97 is not a term because 97 - 1 = 96 and 96 is not prime although floor(97/5) = 19 is prime.
Initial terms, associated primes and d:
          k       k - 1     floor(k/5)     d
a(1)     12        11          2
a(2)     14        13          2           0
a(3)     18        17          3           1
a(4)     38        37          7           4
a(5)     68        67         13           6
a(6)     98        97         19           6
a(7)    158       157         31          12
a(8)    308       307         61          30
a(9)    338       337         67           6
a(10)   368       367         73           6

Links

Crossrefs

Cf. A000040, A007530, A007811, A014561, A259645.
Cf. A008864, A023217, A265767.

Programs

  • Maple
    R:= NULL:
p:= 1: count:= 0:
while count < 100 do
  p:= nextprime(p);
  if isprime(floor((p+1)/5)) then
     R:= R,p+1; count:= count+1
  fi
od:
R; # Robert Israel, May 22 2024
  • Mathematica
    Select[Range[2,5000,2],And@@PrimeQ[{#-1,Floor[#/5]}]&] (* Giorgos Kalogeropoulos, Apr 01 2021 *)
  • PARI
    for(k = 1,10000,if(isprime(k - 1) && isprime(k\5),print1(k", ")))

A064964 100000n+1, 100000n+3, 100000n+7, 100000n+9 are all primes.

Original entry on oeis.org

283, 1864, 3145, 3772, 4153, 11902, 18829, 27736, 28129, 33739, 36469, 40207, 47533, 55996, 60871, 63184, 63244, 80839, 91174, 92683, 93379, 103672, 107236, 117337, 117589, 136765, 143110, 146590, 161986, 183889, 189118, 206419, 207055
Offset: 1

Views

Author

Robert G. Wilson v, Oct 29 2001

Keywords

Crossrefs

Cf. A007811 and A064687.

Programs

  • Mathematica
    Select[Range[10^6/4], PrimeQ[10^5# + 1] && PrimeQ[10^5# + 3] && PrimeQ[10^5# + 7] && PrimeQ[10^5# + 9] &]

A064965 1000000n+1, 1000000n+3, 1000000n+7, 1000000n+9 are all primes.

Original entry on oeis.org

14311, 14659, 23299, 40861, 43303, 46405, 62239, 67327, 77071, 94237, 102796, 115201, 120220, 134968, 138721, 152980, 252715, 260947, 272365, 274534, 285244, 298342, 304489, 305713, 311032, 318802, 324025, 325321, 338908, 343885, 352621
Offset: 1

Views

Author

Robert G. Wilson v, Oct 29 2001

Keywords

Crossrefs

Cf. A007811 and A064687.

Programs

  • Mathematica
    Select[Range[10^6/2], PrimeQ[10^6# + 1] && PrimeQ[10^6# + 3] && PrimeQ[10^6# + 7] && PrimeQ[10^6# + 9] &]
Select[Range[400000],AllTrue[10^6*#+{1,3,7,9},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 11 2015 *)

A064966 Numbers n such that 10000000n+1, 10000000n+3, 10000000n+7, 10000000n+9 are all primes.

Original entry on oeis.org

12022, 15298, 44413, 61507, 72199, 87463, 96538, 108862, 112129, 117694, 122176, 125716, 175078, 185746, 201493, 227221, 250414, 267844, 273460, 371194, 387028, 391765, 397066, 397792, 454921, 581365, 601177, 621010, 642199, 659788, 677206
Offset: 1

Views

Author

Robert G. Wilson v, Oct 29 2001

Keywords

Crossrefs

Cf. A007811 and A064687.

Programs

  • Mathematica
    Select[Range[10^6/2], PrimeQ[10^7# + 1] && PrimeQ[10^7# + 3] && PrimeQ[10^7# + 7] && PrimeQ[10^7# + 9] &]
Select[Range[678000],AllTrue[# 10^7+{1,3,7,9},PrimeQ]&] (* Harvey P. Dale, Nov 13 2022 *)

A064967 100000000n+1, 100000000n+3, 100000000n+7, 100000000n+9 are all primes.

Original entry on oeis.org

27346, 62101, 149650, 168130, 207670, 230830, 242443, 249439, 257227, 278521, 300028, 329389, 342700, 401980, 436315, 452281, 456985, 523972, 528946, 530671, 535918, 612595, 642832, 657151, 732799, 733783, 746848, 758857, 857662, 866608
Offset: 1

Views

Author

Robert G. Wilson v, Oct 29 2001

Keywords

Crossrefs

Cf. A007811 and A064687.

Programs

  • Mathematica
    Select[Range[10^6], PrimeQ[10^8# + 1] && PrimeQ[10^8# + 3] && PrimeQ[10^8# + 7] && PrimeQ[10^8# + 9] &]
Select[Range[900000],AllTrue[100000000#+{1,3,7,9},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 14 2016 *)

A178083 Numbers k for which 10k+1, 10k+3, 10k+7, 10k+9, 10k+13 and 10k+19 are primes.

Original entry on oeis.org

1, 148, 16570, 32614, 66109, 102250, 106870, 124636, 146140, 191773, 305887, 415591, 421828, 552700, 834415, 1013440, 1176505, 1190050, 1306432, 1572082, 1576009, 1850437, 1873684, 1912954, 1921144, 2004997, 2103613, 2376340
Offset: 1

Views

Author

Roger L. Bagula, May 19 2010

Keywords

Comments

Subsequence of A178084 and of A007811.

Examples

			Associated prime 6-plets are:
11, 13, 17, 19, 23, 29;   # from k=1
1481, 1483, 1487, 1489, 1493, 1499; # from k=148
165701, 165703, 165707, 165709, 165713, 165719; # from k=16570
326141, 326143, 326147, 326149, 326153, 326159; # from k=32614

Crossrefs

Cf. A007811.

Programs

  • Magma
    [n: n in [0..1000]| IsPrime(10*n+1) and IsPrime(10*n+3) and IsPrime(10*n+7) and IsPrime(10*n+9) and IsPrime(10*n+13) and IsPrime(10*n+19)] // Vincenzo Librandi, Nov 30 2010
  • Mathematica
    Flatten[Table[If[PrimeQ[10* n + 1] && PrimeQ[10*n + 3] && PrimeQ[10*n + 7] && PrimeQ[10*n + 9] && PrimeQ[10*(n + 1) + 3] && PrimeQ[10*(n + 1) + 9], n, {}], {n, 0, 200000}]]

Extensions

More terms from Zak Seidov, D. S. McNeil and Vincenzo Librandi, May 22 2010
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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