A157711 -id:A157711 - cp's OEIS frontend

A383675 Number of n-digit terms in A157711.

0, 0, 0, 0, 1, 0, 2, 2, 1, 4, 9, 5, 8, 3, 9, 9, 12, 6, 14, 4, 5, 9, 8, 10, 13, 10, 8, 19, 17, 15, 20, 16, 27, 16, 26, 14, 23, 18, 26, 22, 40, 23, 21, 18, 32, 24, 29, 15, 33, 21, 25, 33, 34, 25, 25, 22, 47, 30, 40, 25, 37, 29, 38, 33, 47, 30, 41, 37, 45, 41, 46, 33, 42, 36, 52, 39, 48, 28, 49, 37

Offset: 1

Hans Havermann, May 29 2025

The b-file terms are based on probable prime counts that have been verified correct (by counting proven primes) to index 200. Subsequent terms (as they get larger) seemingly face an increasing probability of counting a probable prime that is actually a composite but it is unknown if such probability is ever large enough to impact the intended proven prime count.

			There are two A157711 terms (1011001, 1100101) containing 7 digits, so a(7) = 2.

Hans Havermann, Table of n, a(n) for n = 1..2500
Hans Havermann, Combined point/points-joined plot of 2500 terms

Cf. A157711.

A038446 Sums of 4 distinct powers of 10.

Original entry on oeis.org

1111, 10111, 11011, 11101, 11110, 100111, 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100, 1000111, 1001011, 1001101, 1001110, 1010011, 1010101, 1010110, 1011001, 1011010, 1011100, 1100011, 1100101, 1100110, 1101001, 1101010, 1101100, 1110001

Offset: 1

Olivier Gérard

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

Cf. A011557, A157711 (primes subsequence).

Cf. A038444, A038445, A038447, A038448, A038449, A038450, A038451, A038452, A038453, A038454.

Total/@Subsets[10^Range[0,6],{4}]//Union (* Harvey P. Dale, Nov 07 2021 *)

from itertools import islice
def A038446_gen(): # generator of terms
    yield int(bin(n:=15)[2:])
    while True: yield int(bin((n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b))[2:])
A038446_list = list(islice(A038446_gen(),20)) # Chai Wah Wu, Mar 11 2025

Offset corrected by Amiram Eldar, Jul 12 2022

A165508 Numbers k such that 10^k + 111 is prime.

Original entry on oeis.org

2, 4, 184, 460, 784, 3248, 5194, 92386, 156428, 228208

Offset: 1

Rick L. Shepherd, Sep 21 2009

Terms must be congruent to 2 or 4 mod 6. Other than the first term, which produces 10^2 + 111 = 211, these terms produce primes whose decimal representation is 1  111 concatenated. These are only known to be highly probable primes for 184 and beyond. No more terms up to 15000.
a(8) > 55000. - Tyler NeSmith, Jul 10 2021
The corresponding primes have digit sum 4 (A062339). - Jeppe Stig Nielsen, Feb 10 2023
a(9) > 10^5. - Jeppe Stig Nielsen, Feb 11 2023
a(11) > 6.6*10^5. - Boyan Hu, Nov 14 2024

			As 10111 = 10^4 + 111 is a prime, 4 is a term.

Cf. A096912, A157711, A062339, A020449, A036929, A161786.

Select[Range[5200], PrimeQ[10^# + 111] &] (* G. C. Greubel, Oct 21 2018 *)

is(k)=ispseudoprime(10^k+111) \\ Charles R Greathouse IV, Jun 13 2017

a(8) from Jeppe Stig Nielsen, Feb 10 2023

a(9)-a(10) from Boyan Hu, Oct 23 2024

A222962 Primes of the form kk*k+k+1, where kk is the concatenation of k with itself.

Original entry on oeis.org

13, 47, 103, 181, 281, 547, 10111, 14557, 22741, 25873, 29207, 44563, 48907, 53453, 90931, 103457, 110023, 116791, 161641, 169823, 178207, 186793, 195581, 232753, 242551, 273157, 283763, 305581, 316793, 440023, 523657, 538303, 568201, 614563, 662743

Offset: 1

Vincenzo Librandi, Mar 18 2013

Corresponding values of k are: 1, 2, 3, 4, 5, 7, 10, 12, 15, 16, 17, 21, 22, 23, 30, 32, 33, 34, 40, 41, 42, 43, 44, 48, 49, 52, 53, 55, 56, 66, 72, 73, 75, 78, 81, 82, 83, 92,...

a(7), a(43) and a(204) (see b-file) have the form 10^(3n+1)+10^(2n)+10^n+1 = (10^(n+1)*10^n+10^n)*10^n+10^n+1. The next term of this type is 10^247+10^164+10^82+1.

			22741 is in the sequence because it is prime and 22741=1515*15+15+1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A020338, A157711.

[p: n in [0..100] | IsPrime(p) where p is Seqint(Intseq(n) cat Intseq(n))*n+n+1]; // Bruno Berselli, Mar 21 2013

f[n_] := FromDigits@Flatten@IntegerDigits[{n, n}] n + n + 1; Select[Table[f[n], {n, 100}], PrimeQ] (* Bruno Berselli, Mar 21 2013 *)
Select[Table[n(n*10^IntegerLength[n]+n)+n+1,{n,100}],PrimeQ] (* Harvey P. Dale, Oct 29 2023 *)

Edited by Bruno Berselli, Mar 22 2013

A383918 Primes made up of 0's and five 1's only.

Original entry on oeis.org

101111, 10011101, 10101101, 10110011, 10111001, 11000111, 11100101, 100100111, 100111001, 101001011, 101100011, 110010101, 110101001, 111000101, 111001001, 1000011011, 1000110101, 1001000111, 1001001011, 1001010011, 1010000111, 1010001101, 1010010011, 1010100011, 1010110001

Offset: 1

René-Louis Clerc, May 15 2025

Expression of the primes that are 0-successors of the preprime 11111 (= 41*271); they constitute the infinite set of secondary primes with five 1's and zeros denoted {11111} (Definitions 1, 2, 3, 4 of Clerc).

Robert Israel, Table of n, a(n) for n = 1..10000
René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires, 2025.

Cf. A157711, A038447, A020449.

f:= proc(n) local R,c,i;
 sort(select(isprime, [seq(1+10^(n-1) + add(10^i,i=c), c=combinat:-choose(n-2,3))]))
end proc:
map(op,[seq(f(i),i=6..10)]); # Robert Israel, May 29 2025

list(M) = for(i=3, M, for(j=2, i-1, for(k=1, j-1, for(r=1, k-1, my(p=10^i+10^j+10^k+10^r+1); isprime(p) && print1(p, ", ")))))

from itertools import count, islice
from sympy import isprime
def A383918_gen(): # generator of terms
    for a in count(4):
        for b in range(3,a):
            for c in range(2,b):
                for d in range(1,c):
                    if isprime(p:=10**a+10**b+10**c+10**d|1):
                        yield(p)
A383918_list = list(islice(A383918_gen(),30)) # Chai Wah Wu, May 29 2025

A383919 Primes made up of 0's and seven 1's only.

Original entry on oeis.org

11110111, 11111101, 101101111, 101111011, 110111011, 111010111, 1001110111, 1010011111, 1011110011, 1100101111, 1101010111, 1101110011, 1110011101, 1110110011, 1111100101, 1111110001, 10010110111, 10011101011, 10011110101, 10100111101, 10111001011, 10111110001, 11001011101

Offset: 1

René-Louis Clerc, May 15 2025

Expression of the primes that are 0-successors of the preprime 1111111 (= 239*4649); they constitute the infinite set of secondary primes with seven 1's and zeros denoted {1111111} (Definitions 1, 2, 3, 4 of Clerc).

Alois P. Heinz, Table of n, a(n) for n = 1..12277
René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires, 2025.

Intersection of A020449 and A062337.

Cf. A157711, A038447, A020449.

list(M) = for(i=3, M, for(j=2, i-1, for(k=1, j-1, for(r=1, k-1, for(l=1, r-1, for(m=1, l-1, my(p=10^i+10^j+10^k+10^r+10^l+10^m+1); isprime(p) && print1(p, ", ")))))))

from itertools import count, islice
from sympy import isprime
def A383919_gen(): # generator of terms
    for a in count(6):
        for b in range(5,a):
            for c in range(4,b):
                for d in range(3,c):
                    for e in range(2,d):
                        for f in range(1,e):
                            if isprime(p:=10**a+10**b+10**c+10**d+10**e+10**f|1):
                                yield(p)
A383919_list = list(islice(A383919_gen(),23)) # Chai Wah Wu, May 28 2025

cp's OEIS Frontend

A383675 Number of n-digit terms in A157711.

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A038446 Sums of 4 distinct powers of 10.

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A165508 Numbers k such that 10^k + 111 is prime.

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A222962 Primes of the form kk*k+k+1, where kk is the concatenation of k with itself.

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Magma

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A383918 Primes made up of 0's and five 1's only.

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Maple

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A383919 Primes made up of 0's and seven 1's only.

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PARI

Python