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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A179910 Primes with two embedded primes.

Original entry on oeis.org

23, 37, 53, 73, 127, 139, 157, 167, 193, 211, 227, 229, 241, 251, 263, 277, 307, 331, 383, 389, 419, 433, 439, 443, 457, 467, 503, 521, 541, 557, 563, 577, 587, 599, 619, 631, 643, 647, 659, 677, 683, 727, 751, 757, 761, 827, 829, 839, 857, 859, 883, 929
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

It appears that p having n embedded primes means that the set of prime integers generated by contiguous proper substrings of p has size n.
A079066(a(n)) = 2.

Links

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179910 n = a179910_list !! (n-1)
a179910_list = map (a000040 . (+ 1)) $ elemIndices 2 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 160, f@# == 3 &]
Select[ Prime@ Range@ 160, Function[ n, Length@ Select[ Union[ FromDigits /@ (Flatten[ Table[ Partition[#, k, 1], {k, Length@ # - 1}], 1] &)@ IntegerDigits@ n], PrimeQ]]@ # == 2 &] (* Michael Somos, Jan 13 2011 *)

A179911 Primes with three embedded primes.

Original entry on oeis.org

113, 131, 179, 197, 223, 233, 239, 257, 271, 283, 293, 311, 313, 337, 347, 353, 359, 367, 397, 431, 479, 547, 571, 593, 613, 617, 653, 719, 733, 739, 743, 773, 797, 823, 853, 937, 953, 971, 1013, 1031, 1097, 1103, 1117, 1129, 1151, 1163, 1213, 1217, 1229
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 3.

Links

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179911 n = a179911_list !! (n-1)
a179911_list = map (a000040 . (+ 1)) $ elemIndices 3 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 210, f@# == 4 &]

A179912 Primes with four embedded primes.

Original entry on oeis.org

137, 173, 317, 373, 379, 523, 673, 1123, 1153, 1171, 1193, 1223, 1231, 1277, 1279, 1283, 1297, 1307, 1327, 1531, 1579, 1597, 1613, 1637, 1759, 1783, 1823, 1913, 1931, 2053, 2153, 2333, 2339, 2341, 2351, 2393, 2399, 2411, 2467, 2503, 2539, 2543, 2557
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 4.

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179912 n = a179912_list !! (n-1)
a179912_list = map (a000040 . (+ 1)) $ elemIndices 4 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 380, f@# == 5 &]

A179913 Primes with five embedded primes.

Original entry on oeis.org

1237, 1319, 1367, 1523, 1571, 1723, 1753, 1979, 1997, 2131, 2179, 2239, 2273, 2293, 2297, 2357, 2377, 2383, 2389, 2417, 2437, 2473, 2531, 2579, 2593, 2617, 2711, 2731, 2753, 2797, 3119, 3167, 3257, 3271, 3313, 3371, 3547, 3571, 3593, 3617, 3671, 3677
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 5.

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179913 n = a179913_list !! (n-1)
a179913_list = map (a000040 . (+ 1)) $ elemIndices 5 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 510, f@# == 6 &]

A179914 Primes with six embedded primes.

Original entry on oeis.org

1733, 1973, 2113, 2137, 2237, 2311, 2347, 2371, 2713, 2719, 2837, 2953, 2971, 3373, 3673, 3719, 3733, 3739, 4337, 4373, 4397, 4673, 5231, 5233, 5347, 5479, 6131, 6197, 6317, 6733, 6737, 7193, 7331, 7523, 8237, 8317, 8537, 9719, 10313, 10337, 10937
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 6.

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179914 n = a179914_list !! (n-1)
a179914_list = map (a000040 . (+ 1)) $ elemIndices 6 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 1330, f@# == 7 &]

A179915 Primes with seven embedded primes.

Original entry on oeis.org

1373, 3137, 3797, 5237, 6173, 11173, 11311, 11353, 11719, 11731, 11971, 12113, 12239, 12347, 12377, 12953, 12973, 13127, 13177, 13217, 13537, 13597, 13679, 13709, 13711, 13723, 13729, 13751, 13757, 13759, 13799, 13967, 13997, 15137
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 7.

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179915 n = a179915_list !! (n-1)
a179915_list = map (a000040 . (+ 1)) $ elemIndices 7 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 1770, f@# == 8 &]

A179916 Primes with eight embedded primes.

Original entry on oeis.org

12373, 12379, 12713, 13171, 15233, 17333, 17359, 17971, 19373, 19379, 21139, 21319, 22973, 23167, 23197, 23311, 23473, 23537, 23593, 23671, 23677, 23761, 23773, 23977, 24113, 24137, 24179, 24197, 24317, 24337, 24379, 24733, 25237
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 8.

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179916 n = a179916_list !! (n-1)
a179916_list = map (a000040 . (+ 1)) $ elemIndices 8 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 2790, f@# == 9 &]

A179917 Primes with nine embedded primes.

Original entry on oeis.org

11317, 19739, 19973, 21317, 21379, 22397, 22937, 23117, 23173, 23371, 23971, 24373, 26317, 27197, 29173, 29537, 32719, 33739, 33797, 37397, 39719, 51137, 51973, 52313, 53173, 53479, 53719, 57173, 57193, 61379, 61979, 63179, 66173
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 9.

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179917 n = a179917_list !! (n-1)
a179917_list = map (a000040 . (+ 1)) $ elemIndices 9 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 6610, f@# == 10 &]

A179918 Primes with ten embedded primes.

Original entry on oeis.org

23719, 31379, 52379, 111373, 111731, 111733, 112397, 113117, 113167, 113723, 113759, 113761, 115237, 117191, 117431, 121139, 122971, 123113, 123373, 123479, 123731, 124337, 126173, 126317, 127139, 127733, 127739, 127973, 129733, 131171
Offset: 1

Views

Author

Robert G. Wilson v, Aug 01 2010

Keywords

Comments

A079066(a(n)) = 10.

Crossrefs

Cf. A039996, A079397, A033274, A034844, A092621, A179909 - A179919, A033274.

Programs

  • Haskell
    import Data.List (elemIndices)
a179918 n = a179918_list !! (n-1)
a179918_list = map (a000040 . (+ 1)) $ elemIndices 10 a079066_list
-- Reinhard Zumkeller, Jul 19 2011
  • Mathematica
    f[n_] := Block[ {id = IntegerDigits@n}, len = Length@ id - 1; Count[ PrimeQ@ Union[ FromDigits@# & /@ Flatten[ Table[ Partition[ id, k, 1], {k, len}], 1]], True] + 1]; Select[ Prime@ Range@ 12280, f@# == 11 &]

A279366 Primes whose proper substrings of consecutive digits are all composite.

Original entry on oeis.org

89, 449, 499, 4649, 4969, 6469, 6869, 6949, 8669, 8699, 8849, 9649, 9949, 44699, 46649, 48649, 48869, 49669, 64849, 84869, 86969, 88469, 94849, 94949, 98869, 99469, 444469, 444869, 446969, 466649, 468869, 469849, 469969, 494699, 496669, 496849, 498469, 644669
Offset: 1

Views

Author

Rodrigo de O. Leite, Dec 10 2016

Keywords

Comments

All digits are composite. Each term ends with the digit '9'. Since each term is prime, it never serves as the suffix of any subsequent term; e.g., no term beyond 89 ends with the digits '89', so the only remaining allowed two-digit endings are '49', '69', and '99'; no terms beyond 449 and 499 end with '449' or '499' (and '899' is ruled out because of 89), so the only remaining allowed three-digit endings are '469', '649', '669', '699', '849', '869', '949', '969', and '999' (and each of these appears as the ending of at least one four-digit term, except '999', which doesn't appear as the ending of any term until a(75) = 4696999). - Jon E. Schoenfield, Dec 10 2016
Number of terms < 10^k, k=1,2,3,...: 0, 1, 2, 10, 13, 38, 66, 197, 410, 1053, 2542, 7159, 18182, 49388, ..., . Robert G. Wilson v, Jan 15 2017

Examples

			44699 is in the sequence because 4, 6, 9, 44, 46, 69, 99, 446, 469, 669, 4469 and 4699 are composite numbers. However, 846499 is not included because 4649 is prime.

Links

Crossrefs

Subsequence of A051416.
Cf. A033274, A061372, A254754.

Programs

  • Mathematica
    Select[Prime@ Range[5, 10^5], Function[n, Times @@ Boole@ Map[CompositeQ, Flatten@ Map[FromDigits /@ Partition[n, #, 1] &, Range[Length@ n - 1]]] == 1]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 10 2016 *)
Select[Flatten[Table[FromDigits/@Tuples[{4,6,8,9},d],{d,6}]],PrimeQ[#]&&AllTrue[ FromDigits /@ Union[Flatten[Table[Partition[IntegerDigits[#],n,1],{n,IntegerLength[#]-1}],1]], CompositeQ]&] (* Harvey P. Dale, Jul 15 2023 *)
  • Python
    from sympy import isprime
from itertools import count, islice, product
def ok(n):
    s = str(n)
    if set(s) & {"1", "2", "3", "5", "7"} or not isprime(n): return False
    ss2 = set(s[i:i+l] for i in range(len(s)-1) for l in range(2, len(s)))
    return not any(isprime(int(ss)) for ss in ss2)
def agen():
    for d in count(2):
        for p in product("4689", repeat=d-1):
            t = int("".join(p)+"9")
            if ok(t): yield t
print(list(islice(agen(), 38))) # Michael S. Branicky, Oct 07 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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