A119289 -id:A119289 - cp's OEIS frontend

A232125 Smallest prime such that the n numbers obtained by removing 1 digit on the right are also prime, while no digit can be added on the right to get another prime.

53, 53, 317, 2393, 23333, 373393, 2399333, 23399339, 1979339333, 103997939939, 4099339193933, 145701173999399393, 2744903797739993993333, 52327811119399399313393, 13302806296379339933399333

Offset: 0

Michel Marcus, Nov 19 2013

Inspired by article on 43 in Archimedes' Lab link.

			a(0)=53 because 53 is the smallest prime such that all numbers obtained by adding a digit to the right are composite.
a(1)=53 because 5 and 53 are primes.
a(2)=317 because 3, 31, 317 are all primes, and 317 has the same property as 53 when adding a digit to the right.

G. A. Sarcone and M. J. Waeber, What's Special About This Number?, Archimedes' Lab website.

Cf. A024770, A119289, A227919, A239747.

a(n) = {n++; v = vector(n); i = 1; ok = 0; until (ok, while ((i>1) && (v[i] == 9), v[i] = 0; i--); if (i == 1, v[i] = nextprime(v[i]+1), v[i] = v[i]+1); curp = sum (j=1, i, v[j]*(10^(i-j))); if (isprime(curp), if (i != n, i++, nbp = 0; for (z=1, 9, if (isprime(10*curp+z), nbp++);); if (nbp == 0, ok = 1);););); sum (j=1, n, v[j]*(10^(n-j)));}

from sympy import isprime, nextprime
def a(n):
    p, oo = 2, float('inf')
    while True:
        extends, reach, r1 = 0, [str(p)], []
        while len(reach) > 0 and extends <= n:
            minnotext = oo
            for s in reach:
                wasextended = False
                for d in "1379":
                    if isprime(int(s+d)): r1.append(s+d); wasextended = True
                if not wasextended: minnotext = min(minnotext, int(s))
            if extends == n and minnotext < oo: return minnotext
            if len(r1) > 0: extends += 1
            reach, r1 = r1, []
        p = nextprime(p)
for n in range(12): print(a(n), end=", ") # Michael S. Branicky, Aug 08 2021

a(12)-a(13) from Michael S. Branicky, Aug 08 2021

a(14) from Michael S. Branicky, Aug 23 2021

A240678 Primes p such that p*10+k is prime for exactly one value of the digit k.

Original entry on oeis.org

11, 29, 41, 47, 71, 79, 83, 131, 137, 139, 151, 163, 173, 181, 191, 227, 257, 263, 277, 281, 293, 307, 311, 313, 359, 383, 449, 491, 503, 509, 557, 563, 569, 577, 587, 593, 601, 617, 647, 659, 661, 677, 683, 719, 739, 743, 751, 809, 821, 827, 857, 877, 881

Offset: 1

Colin Barker, Apr 10 2014

			11 is in the sequence because 113 is prime, but 111, 117 and 119 are not prime.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A067267, A119289, A240679, A240680, A240689.

Select[Prime[Range[200]],Total[Boole[PrimeQ[10 #+{1,3,7,9}]]]==1&] (* Harvey P. Dale, Apr 19 2019 *)

forprime(p=2, 1500, t=0; forstep(k=1, 9, 2, if(isprime(p*10+k), t++)); if(t==1, print1(p, ", ")))

from sympy import isprime, primerange
def ok(p): return sum(1 for k in [1, 3, 7, 9] if isprime(p*10+k)) == 1
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(881)) # Michael S. Branicky, Nov 29 2021

A240679 Primes p such that p*10+k is prime for exactly two values of the digit k.

Original entry on oeis.org

2, 3, 5, 17, 23, 37, 59, 67, 73, 97, 101, 127, 149, 157, 193, 197, 211, 223, 229, 233, 239, 241, 269, 283, 331, 337, 349, 353, 373, 379, 401, 433, 439, 463, 467, 479, 487, 499, 571, 607, 613, 619, 631, 673, 691, 701, 733, 757, 769, 811, 853, 859, 937, 941

Offset: 1

Colin Barker, Apr 10 2014

			2 is in the sequence because 23 and 29 are prime, but 21 and 27 are not prime.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A067267, A119289, A240678, A240680, A240689.

Select[Prime[Range[200]],Count[Table[10 #+k,{k,{1,3,7,9}}],?PrimeQ] == 2&] (* _Harvey P. Dale, Jan 24 2019 *)

forprime(p=2, 1500, t=0; forstep(k=1, 9, 2, if(isprime(p*10+k), t++)); if(t==2, print1(p, ", ")))

A240680 Primes p such that p*10+k is prime for exactly three values of the digit k.

Original entry on oeis.org

7, 13, 31, 43, 61, 103, 109, 199, 271, 367, 409, 421, 523, 541, 547, 787, 823, 829, 883, 1009, 1033, 1117, 1237, 1291, 1669, 1999, 2131, 2161, 2203, 2269, 2437, 2503, 2593, 2671, 2857, 3049, 3253, 3271, 3361, 3559, 3583, 3769, 3823, 4003, 4201, 4339, 4357

Offset: 1

Colin Barker, Apr 10 2014

			7 is in the sequence because 71, 73 and 79 are prime, but 77 is not prime.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A067267, A119289, A240678, A240679, A240689.

[p: p in PrimesUpTo(10^4) | {k: k in [1,3,7,9] | IsPrime(p*10+k)} in Subsets({1,3,7,9},3)]; // Bruno Berselli, Apr 10 2014

Select[Prime[Range[600]],Total[Boole[PrimeQ[10#+{1,3,7,9}]]]==3&] (* Harvey P. Dale, Apr 07 2023 *)

forprime(p=2, 10000, t=0; forstep(k=1, 9, 2, if(isprime(p*10+k), t++)); if(t==3, print1(p, ", ")))

A240689 The number of values of the digit k for which prime(n)*10+k is prime.

Original entry on oeis.org

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

Offset: 1

Colin Barker, Apr 10 2014

			a(16) = 0 because prime(16) = 53, and 531, 533, 537 and 539 are not prime.
a(5) = 1 because prime(5) = 11, and 113 is prime, but 111, 117 and 119 are not prime.
a(1) = 2 because prime(1) = 2, and 23 and 29 are prime, but 21 and 27 are not prime.
a(4) = 3 because prime(4) = 7, and 71, 73 and 79 are prime, but 77 is not prime.
a(8) = 4 because prime(8) = 19, and 191, 193, 197 and 199 are all prime.

Colin Barker, Table of n, a(n) for n = 1..10000

Cf. A067267, A119289, A240678, A240679, A240680.

Count[#,?PrimeQ]&/@Table[10*Prime[n]+k,{n,90},{k,{1,3,7,9}}] (* _Harvey P. Dale, Mar 25 2018 *)

forprime(p=2, 10000, t=0; forstep(k=1, 9, 2, if(isprime(p*10+k), t++)); print1(t, ", "))

A155762 Prime numbers p such that prepending any single decimal digit to p does not produce a prime.

Original entry on oeis.org

2, 5, 149, 401, 509, 773, 809, 1021, 1103, 1289, 1301, 1451, 1697, 1709, 1747, 1877, 1889, 2087, 2389, 2521, 2663, 3373, 3511, 3631, 3733, 3779, 3821, 3919, 3947, 3989, 4003, 4073, 4241, 4289, 4339, 4637, 4643, 4801, 4931, 5039, 5113, 5387, 5417, 5477

Offset: 1

Dmitry Kamenetsky, Jan 26 2009

149 is in the sequence, because the following numbers are all composite: 1149, 2149, 3149, 4149, 5149, 6149, 7149, 8149 and 9149.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A119289.

Select[Prime@Range@1000, NoneTrue[#+10^IntegerLength@#*Range@9, PrimeQ]&] (* Hans Rudolf Widmer, May 28 2022 *)

from sympy import isprime, primerange
def ok(p): return not any(isprime(int(d+str(p))) for d in "123456789")
print(list(filter(isprime, primerange(2, 5500)))) # Michael S. Branicky, May 28 2022

A239747 Super-prime leaders: right-truncatable primes p with property that appending any single decimal digit to p does not produce a prime.

Original entry on oeis.org

53, 317, 599, 797, 2393, 3793, 3797, 7331, 23333, 23339, 31193, 31379, 37397, 73331, 373393, 593993, 719333, 739397, 739399, 2399333, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133

Offset: 1

Arkadiusz Wesolowski, Mar 26 2014

The name "super-prime leaders" is not due to the author.

			2393 belongs to this sequence because 2393, 239, 23 and 2 are all prime; 10*2393 + k, for k = 0 to 9, are all composite.

Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, p. 292.

Chris Caldwell, The Prime Glossary, Right-truncatable prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 53
Kvant magazine, The simplest prime numbers, (in Russian) No 11, 1979. (beware of typo)
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Subsequence of A024770 and of A119289.

f=1; for(n=2, 73939133, v=n; t=1; while(isprime(n), if(!Mod(f, n^2)==0, t=t*n); c=n; n=(c-lift(Mod(c, 10)))/10); if(n==0, f=f*t); n=v); s=Set(factor(f)[, 1]); for(k=1, #s, p=s[k]; if(!Mod(f, p^2)==0, print1(p, ", ")));

A024770 INTERSECT A119289.

A240843 Primes p with property that appending or prepending any single decimal digit to p does not produce a prime.

Original entry on oeis.org

773, 1103, 1301, 3947, 3989, 4241, 4637, 4931, 5039, 5387, 5417, 6803, 6917, 6971, 7229, 7451, 7703, 7753, 10211, 10303, 10337, 10607, 10657, 10723, 10859, 11117, 11399, 11423, 11489, 11717, 11813, 11971, 11987, 12119, 12329, 12541, 12653, 12659, 12907, 12983

Offset: 1

Arkadiusz Wesolowski, Apr 13 2014

			1103 belongs to this sequence because 10*1103 + k and k*10^4 + 1103, for k = 1 to 9, are all composite.

Subsequence of A119289 and of A155762.

fQ[n_] := Block[{e = Floor[ Log10@ n] + 1, r = Range@ 9}, Union@ Flatten[ PrimeQ[{10 n + r, r*10^e + n}]] == {False}]; Select[ Prime@ Range@ 1550, fQ] (* Robert G. Wilson v, Apr 15 2014 *)

for(n=2, 12983, v=n; if(isprime(n), s=#Str(v); t=0; for(k=1, 9, if(isprime(10*v+k)||isprime(k*10^s+v), break, t++)); if(t==9, print1(v, ", "))); n=v);

A119289 INTERSECT A155762.

A346979 Count of the prime decimal descendants of n.

Original entry on oeis.org

83, 63, 23, 22, 23, 11, 29, 23, 3, 4, 54, 1, 9, 14, 6, 7, 3, 4, 7, 40, 0, 4, 19, 15, 8, 7, 10, 14, 5, 6, 2, 7, 0, 16, 9, 11, 12, 13, 4, 1, 34, 1, 8, 14, 5, 1, 13, 5, 5, 16, 6, 0, 9, 0, 24, 4, 6, 19, 2, 9, 25, 16, 0, 7, 4, 4, 3, 11, 2, 7, 7, 4, 1, 15, 2, 8, 8

Offset: 0

Ya-Ping Lu, Aug 09 2021

The number of direct decimal descendants (i.e., decimal children) of n is A038800(n). The number of prime decimal descendants of the n-th prime is A214342(p_n). a(n) is the number of prime decimal descendants of n, which include the prime decimal children of n, the prime decimal children of the prime decimal children of n, and so on.
a(0) = Sum_{m=1..4} (A214342(m) + 1); a(1) = Sum_{m=5..8} (A214342(m) + 1).
a(A032352(m)) = 0; a(A119289(m)) = 0.
A214342 is a subset, as A214342(m) = a(prime(m)).
Conjecture 1: a(n) <= 83. Conjecture 2: lim_{n->oo} (n0/n) = 1, where n0 is the number of zero terms, a(k) = 0, for k <= n.

			a(4) = 23. The 23 prime decimal descendants of 4 are shown in the tree below.
       _____ 4__________________________
      /      |                          \
     41   ___43______________            47
    /    /   |               \             \
  419  431  433               439          479
            / \              /   \        /   \
        4337  4339         4391  4397   4793  4799
             /  |  \        |     |     /  \
        43391 43397 43399 43913 43973 47933 47939
                            |
                         439133
                            |
                        4391339

Cf. A030665, A032352, A038800, A119289, A122072, A214342, A218255, A256481.

Table[Length@Rest@Flatten[FixedPointList[(b=#;Select[Flatten[(a=#;FromDigits/@(Join[IntegerDigits@a,{#}]&/@If[b=={0},Range@9,{1,3,7,9}]))&/@b],PrimeQ])&,{n}]],{n,0,76}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

from sympy import isprime
def p_count(k):
    global ct; d = [2, 3, 5, 7] if k == 0 else [1, 3, 7, 9]
    for i in range(4):
        m = 10*k + d[i]
        if isprime(m): ct += 1; p_count(m)
    return ct
for n in range(100):
    ct = 0; print(p_count(n))

cp's OEIS Frontend

A232125 Smallest prime such that the n numbers obtained by removing 1 digit on the right are also prime, while no digit can be added on the right to get another prime.

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A240678 Primes p such that p*10+k is prime for exactly one value of the digit k.

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A240679 Primes p such that p*10+k is prime for exactly two values of the digit k.

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A240680 Primes p such that p*10+k is prime for exactly three values of the digit k.

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A240689 The number of values of the digit k for which prime(n)*10+k is prime.

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A155762 Prime numbers p such that prepending any single decimal digit to p does not produce a prime.

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A239747 Super-prime leaders: right-truncatable primes p with property that appending any single decimal digit to p does not produce a prime.

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A240843 Primes p with property that appending or prepending any single decimal digit to p does not produce a prime.

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