A208272 -id:A208272 - cp's OEIS frontend

A045708 Primes with first digit 2.

2, 23, 29, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221

Offset: 1

Felice Russo

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

Cf. A000040.
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509
Cf. A000030, subsequence of A208272.
Column k=2 of A262369.

a045708 n = a045708_list !! (n-1)
a045708_list = filter ((== 2) . a000030) a000040_list
-- Reinhard Zumkeller, Mar 16 2012

[p: p in PrimesUpTo(2300) | Intseq(p)[#Intseq(p)] eq 2]; // Vincenzo Librandi, Aug 08 2014

Select[Table[Prime[n], {n, 3000}], First[IntegerDigits[#]]==2 &] (* Vincenzo Librandi, Aug 08 2014 *)

from sympy import isprime
def agen(limit=float('inf')):
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(1, 10**digits, 2):
      test = adder + i
      if test > limit: return
      if isprime(test): yield test
    digits, adder = digits+1, adder*10
agento = lambda lim: agen(limit=lim)
print(list(agento(2222))) # Michael S. Branicky, Feb 23 2021

from sympy import primepi
def A045708(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<1)-1,x))-primepi(min(3*m-1,x))+sum(primepi(((m:=10**i)<<1)-1)-primepi(3*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

See A045707 for comments on density of these sequences.

More terms from Erich Friedman.

Offset fixed by Reinhard Zumkeller, Mar 15 2012

A257667 Primes containing a digit 5.

Original entry on oeis.org

5, 53, 59, 151, 157, 251, 257, 353, 359, 457, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 653, 659, 751, 757, 853, 857, 859, 953, 1051, 1151, 1153, 1259, 1451, 1453, 1459, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579

Offset: 1

Vincenzo Librandi, May 03 2015

Subsequence of primes of A011535. - Michel Marcus, May 03 2015

Primes in A062671. - Bruno Berselli, May 03 2015

Shujing Lyu, Table of n, a(n) for n = 1..100000

Cf. prime numbers containing the string k: A208270 (k=1), A208272 (k=2), A212525 (k=3), this sequence (k=5), A257668 (k=7), A166571 (k=10), A166572 (k=11), A243529 (k=12), A166573 (k=13), A243530 (k=14), A243531 (k=15), A243532 (k=16), A166579 (k=17), A243527 (k=111), A166580 (k=222), A166581 (k=333), A166582 (k=444).

Cf. A011535, A062671, A243531 (subsequence).

[p: p in PrimesUpTo(1600) | 5 in Intseq(p)];

Select[Prime[Range[250]], ! StringFreeQ[ToString[#], "5"] &]

forprime(p=1, 1600, if(vecsearch(vecsort(digits(p)), 5), print1(p, ", "))) \\ Derek Orr, May 05 2015; corrected by Michel Marcus, Oct 30 2023

[p for p in primes(1600) if 5 in p.digits(base=10)] # Bruno Berselli, May 03 2015

a(n) ~ n log n. - Charles R Greathouse IV, Nov 01 2022

A243529 Prime numbers containing the string 12.

Original entry on oeis.org

127, 1123, 1129, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 2129, 3121, 4127, 4129, 6121, 7121, 7127, 7129, 8123, 9127, 11213, 11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287

Offset: 1

Vincenzo Librandi, Jun 06 2014

Subsequence of A208272. [Bruno Berselli, May 06 2015]

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

Cf. similar sequences listed in A257667.

Select[Prime[Range[2000]], ! StringFreeQ[ToString[#], "12" ] &]

contains(n,k)=my(N=digits(n),K=digits(k)); for(i=0,#N-#K, for(j=1,#K,if(N[i+j]!=K[j],next(2))); return(1)); 0
is(n)=isprime(n) && contains(n,12) \\ Charles R Greathouse IV, Jun 20 2014

a(n) ~ n log n. - Charles R Greathouse IV, Jun 20 2014

A284290 Primes containing a digit 4.

Original entry on oeis.org

41, 43, 47, 149, 241, 347, 349, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 541, 547, 641, 643, 647, 743, 941, 947, 1049, 1249, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489

Offset: 1

Jaroslav Krizek, Mar 24 2017

Subsequence of A011534 and A062669.

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

Cf. Primes containing a digit k for k = 0 - 9: A056709 (k = 0), A208270 (k = 1), A208272 (k = 2), A212525 (k = 3), A284290 (k = 4), A257667 (k = 5), A284291 (k = 6), A257668 (k = 7), A284292 (k = 8), A106093 (k = 9).

[p: p in PrimesUpTo(10000) | 4 in Intseq(p)]

Select[Range[1500], PrimeQ[#] && MemberQ[IntegerDigits[#], 4] &] (* Amiram Eldar, Nov 09 2019 *)

A284291 Primes containing a digit 6.

Original entry on oeis.org

61, 67, 163, 167, 263, 269, 367, 461, 463, 467, 563, 569, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 761, 769, 863, 967, 1061, 1063, 1069, 1163, 1361, 1367, 1567, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663

Offset: 1

Jaroslav Krizek, Mar 24 2017

Subsequence of A011536 and A062673.

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

Primes containing a digit k for k = 0 - 9: A056709 (k = 0), A208270 (k = 1), A208272 (k = 2), A212525 (k = 3), A284290 (k = 4), A257667 (k = 5), A284291 (k = 6), A257668 (k = 7), A284292 (k = 8), A106093 (k = 9).

[p: p in PrimesUpTo(10000) | 6 in Intseq(p)];

Select[Range[2000], PrimeQ[#] && MemberQ[IntegerDigits[#], 6] &] (* Amiram Eldar, Nov 09 2019 *)

A284292 Primes containing a digit 8.

Original entry on oeis.org

83, 89, 181, 281, 283, 383, 389, 487, 587, 683, 787, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 983, 1087, 1181, 1187, 1283, 1289, 1381, 1481, 1483, 1487, 1489, 1583, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867

Offset: 1

Jaroslav Krizek, Mar 25 2017

Subsequence of A011538 and A062677.

Differs from A062677 which contains also the composites 6889 = 83^2, 7387 = 83*89, 23489=83*283, 25187=89*283, 31789 = 83*383 etc. - R. J. Mathar, Mar 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. Primes containing a digit k for k = 0 - 9: A056709 (k = 0), A208270 (k = 1), A208272 (k = 2), A212525 (k = 3), A284290 (k = 4), A257667 (k = 5), A284291 (k = 6), A257668 (k = 7), this sequence (k = 8), A106093 (k = 9).

[p: p in PrimesUpTo(10000) | 8 in Intseq(p)];

isA284292 := proc(n)
    if isprime(n) then
        convert(convert(n,base,10),set) ;
        if 8 in % then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 1 to 2000 do
    if isA284292(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 27 2017

Select[Prime@ Range@ 500, MemberQ[ IntegerDigits@ #, 8] &] (* Giovanni Resta, Mar 25 2017 *)

from sympy import primerange
print([n for n in primerange(2, 2000) if '8' in str(n)]) # Indranil Ghosh, Mar 25 2017

A036433 Number of divisors is a digit in the base 10 representation of n.

Original entry on oeis.org

1, 2, 14, 23, 29, 34, 46, 63, 68, 74, 76, 78, 88, 94, 116, 127, 128, 134, 138, 141, 142, 143, 145, 146, 164, 182, 184, 186, 189, 194, 196, 211, 214, 223, 227, 229, 233, 236, 238, 239, 241, 247, 248, 249, 251, 254, 257, 258, 261, 263, 268, 269, 271, 274, 277

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR concept formation program.

			14 has 4 divisors and 4 is a digit in the base 10 representation of 14.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A045708, A208272.

Cf. A000005.

a036433 n = a036433_list !! (n-1)
a036433_list = filter f [1..] where
   f x = d < 10 && ("0123456789" !! d) `elem` show x where d = a000005 x
-- Reinhard Zumkeller, Mar 15 2012

Select[Range[300],MemberQ[IntegerDigits[#],DivisorSigma[0,#]]&] (* Harvey P. Dale, Sep 02 2013 *)

from sympy import divisor_count
A036433_list = []
for i in range(1,10**5):
    d = divisor_count(i)
    if d < 10 and str(d) in str(i):
        A036433_list.append(i) # Chai Wah Wu, Jan 07 2015

A208273 Composite numbers containing a digit 2.

Original entry on oeis.org

12, 20, 21, 22, 24, 25, 26, 27, 28, 32, 42, 52, 62, 72, 82, 92, 102, 112, 120, 121, 122, 123, 124, 125, 126, 128, 129, 132, 142, 152, 162, 172, 182, 192, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221

Offset: 1

Jaroslav Krizek, Mar 04 2012

Subsequence of A011532. Complement of A208272 with respect to A011532.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A208272 (primes containing a digit 2), A011532 (numbers containing a digit 2).

filter:= proc(n) not isprime(n) and member(2,convert(n,base,10)) end proc:
select(filter, [$4..300]); # Robert Israel, Oct 09 2024

Select[Range[300], ! PrimeQ[#] && MemberQ[IntegerDigits[#], 2] &] (* T. D. Noe, Mar 06 2012 *)
Select[Range[300],CompositeQ[#]&&DigitCount[#,10,2]>0&] (* Harvey P. Dale, Sep 08 2024 *)

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A045708 Primes with first digit 2.

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A257667 Primes containing a digit 5.

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A243529 Prime numbers containing the string 12.

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A284290 Primes containing a digit 4.

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A284291 Primes containing a digit 6.

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A284292 Primes containing a digit 8.

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A036433 Number of divisors is a digit in the base 10 representation of n.

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