A079397 -id:A079397 - cp's OEIS frontend

A179914 Primes with six embedded primes.

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 &]

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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 8.

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

import Data.List (elemIndices)
a179916 n = a179916_list !! (n-1)
a179916_list = map (a000040 . (+ 1)) $ elemIndices 8 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@ 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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 9.

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

import Data.List (elemIndices)
a179917 n = a179917_list !! (n-1)
a179917_list = map (a000040 . (+ 1)) $ elemIndices 9 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@ 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

Robert G. Wilson v, Aug 01 2010

A079066(a(n)) = 10.

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

import Data.List (elemIndices)
a179918 n = a179918_list !! (n-1)
a179918_list = map (a000040 . (+ 1)) $ elemIndices 10 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@ 12280, f@# == 11 &]

A217104 Minimal number (in decimal representation) with n nonprime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 5, 4, 19, 17, 16, 75, 67, 66, 64, 269, 263, 266, 257, 256, 1053, 1031, 1035, 1029, 1026, 1024, 4125, 4119, 4123, 4107, 4099, 4098, 4096, 16479, 16427, 16431, 16407, 16395, 16391, 16386, 16384, 65709, 65629, 65579, 65581, 65559, 65543, 65539, 65537, 65536

Offset: 0

Hieronymus Fischer, Dec 12 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 4^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,... the m(n) in base-4 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i = (k*(k+1)/2)+i = n, which proves the statement.

If p is a number with k prime substrings and d digits (in base-4 representation), m>=d, than b := p*4^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_4 is the least number with zero nonprime substrings in base-4 representation.
a(1) = 1, since 1 = 1_4 is the least number with 1 nonprime substring in base-4 representation.
a(2) = 5, since 5 = 11_4 is the least number with 2 nonprime substrings in base-4 representation (these are 2-times 1).
a(3) = 4, since 4 = 10_4 is the least number with 3 nonprime substrings in base-4 representation (these are 0, 1 and 10).
a(4) = 19, since 19 = 103_4 is the least number with 4 nonprime substrings in base-4 representation, these are 0, 1, 10, and 03 (remember, that substrings with leading zeros are considered to be nonprime).
a(7) = 75, since 75 = 1023_4 is the least number with 7 nonprime substrings in base-4 representation, these are 0, 1, 10, 02, 023, 102 and 1023 (remember, that substrings with leading zeros are considered to be nonprime: 2_4 = 2, 3_4 = 3 and 23_4 = 11 are the only base-4 prime substrings of 75).

Hieronymus Fischer, Table of n, a(n) for n = 0..528

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217102-A217109.
Cf. A217302-A217309.

a(n) >= 4^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
a(A000217(n)) = 4^(n-1), n>0.
a(A000217(n)-k) >= 4^(n-1) + k, 0<=k0.
a(A000217(n)-k) = 4^(n-1) + p, where p is the minimal number >= 0 such that 4^(n-1) + p, has k prime substrings in base-4 representation, 0<=k0.

A217105 Minimal number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 5, 6, 27, 25, 34, 127, 128, 125, 170, 636, 632, 627, 625, 850, 3162, 3137, 3132, 3127, 3125, 4250, 15686, 15661, 15638, 15632, 15627, 15625, 21250, 78192, 78163, 78162, 78137, 78132, 78127, 78125, 106250, 390818, 390692, 390686, 390662, 390638, 390632

Offset: 0

Hieronymus Fischer, Dec 12 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 5^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,... the m(n) in base-5 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s, thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i = (k*(k+1)/2)+i = n, which proves the statement.

If p is a number with k prime substrings and d digits (in base-5 representation), p != 1 (mod 5), m>=d, than b := p*5^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_5 is the least number with zero nonprime substrings in base-4 representation.
a(1) = 1, since 1 = 1_5 is the least number with 1 nonprime substring in base-5 representation.
a(2) = 5, since 5 = 10_5 is the least number with 2 nonprime substrings in base-5 representation (0 and 1).
a(3) = 6, since 6 = 11_5 is the least number with 3 nonprime substrings in base-5 representation (2-times 1 and 11).
a(4) = 27, since 27 = 102_5 is the least number with 4 nonprime substrings in base-5 representation, these are 0, 1, 02, and 102 (remember, that substrings with leading zeros are considered to be nonprime).
a(6) = 34, since 34 = 114_5 is the least number with 6 nonprime substrings in base-5 representation, these are 1, 1, 4, 11, 14, and 114.

Hieronymus Fischer, Table of n, a(n) for n = 0..351

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397.
Cf. A217102-A217109, A217112-A217119.
Cf. A217302-A217309.

a(n) >= 5^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n=1 or n+1 is a triangular number (cf. A000217).
a(A000217(n)-1) = 5^(n-1), n>1.
a(A000217(n)) = floor(34 * 5^(n-3)), n>0.
a(A000217(n)) = 114000...000_5 (with n digits), n>0.
a(A000217(n)-k) >= 5^(n-1) + k-1, 1<=k<=n, n>1.
a(A000217(n)-k) = 5^(n-1) + p, where p is the minimal number >= 0 such that 5^(n-1) + p, has k prime substrings in base-5 representation, 1<=k<=n, n>1.

A217106 Minimal number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 7, 6, 41, 37, 36, 223, 224, 218, 216, 1319, 1307, 1301, 1297, 1296, 7829, 7793, 7787, 7783, 7778, 7776, 46703, 46709, 46679, 46673, 46663, 46658, 46656, 280205, 280075, 279983, 279979, 279949, 279941, 279938, 279936, 1679879, 1679807, 1679699, 1679669

Offset: 0

Hieronymus Fischer, Dec 12 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 6^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,... the m(n) in base-6 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s, thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i = (k*(k+1)/2)+i = n, which proves the statement.

If p is a number with k prime substrings and d digits (in base-6 representation), m>=d, than b := p*6^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_6 is the least number with zero nonprime substrings in base-6 representation.
a(1) = 1, since 1 = 1_6 is the least number with 1 nonprime substring in base-6 representation.
a(2) = 7, since 7 = 11_6 is the least number with 2 nonprime substrings in base-6 representation (1 and 1).
a(3) = 6, since 6 = 10_6 is the least number with 3 nonprime substrings in base-6 representation (0, 1 and 10).
a(4) = 41, since 41 = 105_6 is the least number with 4 nonprime substrings in base-6 representation, these are 0, 1, 10, and 05 (remember, that substrings with leading zeros are considered to be nonprime).

Hieronymus Fischer, Table of n, a(n) for n = 0..300

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217102-A217109.
Cf. A217302-A217309.

a(n) >= 6^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
a(A000217(n)) = 6^(n-1), n>0.
a(A000217(n)-k) >= 6^(n-1) + k, 0<=k0.
a(A000217(n)-k) = 6^(n-1) + p, where p is the minimal number >= 0 such that 6^(n-1) + p, has k prime substrings in base-6 representation, 0<=k0.

A217107 Minimal number (in decimal representation) with n nonprime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 7, 8, 51, 49, 57, 353, 345, 343, 400, 2417, 2411, 2403, 2401, 9604, 16880, 16823, 16829, 16809, 16807, 67228, 117763, 117721, 117666, 117659, 117651, 117649, 470596, 823709, 823664, 823615, 823560, 823553, 823545, 823543, 3294172, 5765310, 5765063

Offset: 0

Hieronymus Fischer, Dec 12 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 7^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,... the m(n) in base-7 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s, thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i = (k*(k+1)/2)+i = n, which proves the statement.

If p is a number with k prime substrings and d digits (in base-7 representation), p != 1 (mod 7), m>=d, than b := p*7^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_7 is the least number with zero nonprime substrings in base-7 representation.
a(1) = 1, since 1 = 1_7 is the least number with 1 nonprime substring in base-7 representation.
a(2) = 7, since 7 = 10_7 is the least number with 2 nonprime substrings in base-7 representation (these are 0 and 1).
a(3) = 8, since 8 = 11_7 is the least number with 3 nonprime substrings in base-7 representation (1, 1 and 11).
a(4) = 51, since 51 = 102_7 is the least number with 4 nonprime substrings in base-7 representation, these are 0, 1, 02, and 102 (remember, that substrings with leading zeros are considered to be nonprime).

Hieronymus Fischer, Table of n, a(n) for n = 0..210

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217102-A217109.
Cf. A217302-A217309.

a(n) >= 7^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n=1 or n+1 is a triangular number (cf. A000217).
a(A000217(n)-1) = 7^(n-1), n>1.
a(A000217(n)) = floor(400 * 7^(n-4)), n>0.
a(A000217(n)) = 111…111_7 (with n digits), n>0.
a(A000217(n)-k) >= 7^(n-1) + k-1, 1<=k<=n, n>1.
a(A000217(n)-k) = 7^(n-1) + p, where p is the minimal number >= 0 such that 7^(n-1) + p, has k prime substrings in base-7 representation, 1<=k<=n, n>1.

A217108 Minimal number (in decimal representation) with n nonprime substrings in base-8 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

2, 1, 10, 8, 67, 66, 64, 523, 525, 514, 512, 4127, 4115, 4099, 4098, 4096, 32797, 32799, 32779, 32771, 32770, 32768, 262237, 262239, 262173, 262163, 262147, 262146, 262144, 2097391, 2097259, 2097211, 2097181, 2097169, 2097163, 2097154, 2097152, 16777695

Offset: 0

Hieronymus Fischer, Dec 12 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 8^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,... the m(n) in base-8 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s, thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i = (k*(k+1)/2)+i = n, which proves the statement.

If p is a number with k prime substrings and d digits (in base-8 representation), m>=d, than b := p*8^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

			a(0) = 2, since 2 = 2_8 is the least number with zero nonprime substrings in base-8 representation.
a(1) = 1, since 1 = 1_8 is the least number with 1 nonprime substring in base-8 representation.
a(2) = 10, since 10 = 12_8 is the least number with 2 nonprime substrings in base-8 representation (1 and 12).
a(3) = 8, since 8 = 10_8 is the least number with 3 nonprime substrings in base-8 representation (0, 1 and 10).
a(4) = 67, since 67 = 103_8 is the least number with 4 nonprime substrings in base-8 representation, these are 0, 1, 10, and 03 (remember, that substrings with leading zeros are considered to be nonprime).

Hieronymus Fischer, Table of n, a(n) for n = 0..210

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300-A213321.
Cf. A217102-A217109.
Cf. A217302-A217309.

a(n) >= 8^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).
a(A000217(n)) = 8^(n-1), n>0.
a(A000217(n)-k) >= 8^(n-1) + k, 0<=k0.
a(A000217(n)-k) = 8^(n-1) + p, where p is the minimal number >= 0 such that 8^(n-1) + p, has k prime substrings in base-8 representation, 0<=k0.

cp's OEIS Frontend

A179914 Primes with six embedded primes.

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A179915 Primes with seven embedded primes.

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A179916 Primes with eight embedded primes.

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A179917 Primes with nine embedded primes.

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A179918 Primes with ten embedded primes.

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A217104 Minimal number (in decimal representation) with n nonprime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

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A217105 Minimal number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

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A217106 Minimal number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

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A217107 Minimal number (in decimal representation) with n nonprime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

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