A179909 -id:A179909 - cp's OEIS frontend

A033274 Primes that do not contain any other prime as a proper substring.

2, 3, 5, 7, 11, 19, 41, 61, 89, 101, 109, 149, 181, 401, 409, 449, 491, 499, 601, 691, 809, 881, 991, 1009, 1049, 1069, 1481, 1609, 1669, 1699, 1801, 4001, 4049, 4481, 4649, 4801, 4909, 4969, 6091, 6469, 6481, 6869, 6949, 8009, 8069, 8081, 8609, 8669, 8681

Offset: 1

Michael Kleber

If there is more than one digit, all digits must be nonprime numbers.
A179335(n) = prime(n) iff prime(n) is in this sequence. For n > 4, prime(n) is in this sequence iff A109066(n) = 0. - Reinhard Zumkeller, Jul 11 2010, corrected by M. F. Hasler, Aug 27 2012
A079066(n) = 0 iff prime(n) is in this sequence. [Corrected by M. F. Hasler, Aug 27 2012]
What are the asymptotics of this sequence? - Charles R Greathouse IV, Aug 27 2012

			149 is a term as 1, 4, 9, 14, 49 are all nonprimes.
199 is not a term as 19 is a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Cf. A089768, A089770, A039996, A079397, A033274, A034844, A179909-A179919.

import Data.List (elemIndices)
a033274 n = a033274_list !! (n-1)
a033274_list = map (a000040 . (+ 1)) $ elemIndices 0 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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@ 1100, f@# == 1 &] (* Robert G. Wilson v, Aug 01 2010 *)
Select[Prime[Range[1100]],NoneTrue[Flatten[Table[FromDigits/@Partition[IntegerDigits[#],d,1],{d,IntegerLength[#]-1}]],PrimeQ]&] (* Harvey P. Dale, Apr 19 2025 *)

from sympy import isprime
def ok(n):
    if n in {2, 3, 5, 7}: return True
    s = str(n)
    if set(s) & {"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)
print([k for k in range(9000) if ok(k)]) # Michael S. Branicky, Jun 29 2022

Edited by N. J. A. Sloane at the suggestion of Luca Colucci, Apr 03 2008

A079066 "Memory" of prime(n): the number of distinct (previous) primes contained as substrings in prime(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 0, 3, 2, 3, 4, 2, 0, 1, 2, 1, 2, 4, 3, 0, 1, 2, 3, 1, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 1, 3, 2, 1, 3, 3, 2, 3, 3, 4, 2, 3, 3, 1, 3, 3, 3, 4, 4, 2, 2, 3, 0, 0, 2, 1, 3, 2, 2, 2, 0, 2, 1, 1, 2, 3, 1, 0, 0, 2, 1, 2, 4, 2, 3, 2, 2, 1, 3

Offset: 1

Joseph L. Pe, Feb 02 2003

			The primes contained as substrings in prime(3) = 113 are 3, 11, 13. Hence a(30) = 3. 113 is the smallest prime with memory = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A079075, A035232, A039996, A039997.

Cf. A033274, A179909, A179910, A179911, A179912, A179913, A179914, A179915, A179916, A179917, A179918, A179919, A179922.

import Data.List (isInfixOf)
a079066 n =
   length $ filter (`isInfixOf` (primesDec !! n)) $ take n primesDec
primesDec = "_" : map show a000040_list
-- Reinhard Zumkeller, Jul 19 2011

a:= n-> (s-> -1+nops(select(t -> t[1]<>"0" and isprime(parse(t)),
        {seq(seq(s[i..j], i=1..j), j=1..length(s))})))(""||(ithprime(n))):
seq(a(n), n=1..105);  # Alois P. Heinz, Jul 29 2025

ub = 105; tprime = Table[ToString[Prime[i]], {i, 1, ub}]; a = {}; For[i = 1, i <= ub, i++, m = 0; For[j = 1, j < i, j++, If[Length[StringPosition[tprime[[i]], tprime[[j]]]] > 0, m = m + 1]]; a = Append[a, m]]; a

a(n) = A039997(prime(n)) - 1.

a(n) = A039996(n) - 1. - Alois P. Heinz, Jul 29 2025

Edited by Robert G. Wilson v, Feb 25 2003

Name clarified by Sean A. Irvine, Jul 29 2025

A179919 Primes with eleven embedded primes.

Original entry on oeis.org

111317, 113177, 113537, 113719, 113731, 117193, 117331, 121379, 123733, 129719, 131797, 132173, 132971, 136733, 136739, 137197, 137321, 137339, 137353, 137359, 137393, 137573, 152311, 172313, 173137, 173359, 174311, 193373, 211319, 213799

Offset: 1

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 11.

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

import Data.List (elemIndices)
a179919 n = a179919_list !! (n-1)
a179919_list = map (a000040 . (+ 1)) $ elemIndices 11 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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@ 19110, f@# == 12 &]

A179922 Primes with twelve embedded primes.

Original entry on oeis.org

113171, 113173, 113797, 123719, 153137, 179719, 199739, 213173, 229373, 231197, 233113, 233713, 236779, 237331, 237619, 237971, 241973, 259397, 317971, 327193, 343373, 353173, 361373, 372719, 373379, 382373, 432713, 519733, 521137, 521317

Offset: 1

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 12.

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

import Data.List (elemIndices)
a179922 n = a179922_list !! (n-1)
a179922_list = map (a000040 . (+ 1)) $ elemIndices 12 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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@ 43150, f@# == 13 &]

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

Robert G. Wilson v, Aug 01 2010

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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

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

import Data.List (elemIndices)
a179910 n = a179910_list !! (n-1)
a179910_list = map (a000040 . (+ 1)) $ elemIndices 2 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

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

import Data.List (elemIndices)
a179911 n = a179911_list !! (n-1)
a179911_list = map (a000040 . (+ 1)) $ elemIndices 3 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 4.

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

import Data.List (elemIndices)
a179912 n = a179912_list !! (n-1)
a179912_list = map (a000040 . (+ 1)) $ elemIndices 4 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 5.

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

import Data.List (elemIndices)
a179913 n = a179913_list !! (n-1)
a179913_list = map (a000040 . (+ 1)) $ elemIndices 5 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 6.

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

import Data.List (elemIndices)
a179914 n = a179914_list !! (n-1)
a179914_list = map (a000040 . (+ 1)) $ elemIndices 6 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 7.

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

import Data.List (elemIndices)
a179915 n = a179915_list !! (n-1)
a179915_list = map (a000040 . (+ 1)) $ elemIndices 7 a079066_list
-- Reinhard Zumkeller, Jul 19 2011

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 &]

cp's OEIS Frontend

A033274 Primes that do not contain any other prime as a proper substring.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Extensions

A079066 "Memory" of prime(n): the number of distinct (previous) primes contained as substrings in prime(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A179919 Primes with eleven embedded primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Haskell

Mathematica

A179922 Primes with twelve embedded primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Haskell

Mathematica

A179910 Primes with two embedded primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A179911 Primes with three embedded primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A179912 Primes with four embedded primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Haskell

Mathematica

A179913 Primes with five embedded primes.

Views

Author

Keywords

Comments

Crossrefs