A106100 -id:A106100 - cp's OEIS frontend

A062339 Primes whose sum of digits is 4.

13, 31, 103, 211, 1021, 1201, 2011, 3001, 10111, 20011, 20101, 21001, 100003, 102001, 1000003, 1011001, 1020001, 1100101, 2100001, 10010101, 10100011, 20001001, 30000001, 101001001, 200001001, 1000000021, 1000001011, 1000010101, 1000020001, 1000200001, 1002000001, 1010000011

Offset: 1

Amarnath Murthy, Jun 21 2001

Is this sequence (and its brothers A062337, A062341 and A062343) infinite?

10^A049054(m)+3 and 3*10^A056807(m)+1 are subsequences. A107715 (primes containing only digits from set {0,1,2,3}) is a supersequence. Terms not containing the digit 3 are either terms of A020449 (primes that contain digits 0 and 1 only) or of A106100 (primes with maximal digit 2) - and thus terms of these sequences' union A036953 (primes containing only digits from set {0,1,2}). - Rick L. Shepherd, May 23 2005

			3001 is a prime with sum of digits = 4, hence belongs to the sequence.

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe)
Amin Witno, Numbers which factor as their digital sum times a prime, International Journal of Open Problems in Computer Science and Mathematics 3:2 (2010), pp. 132-136.

Subsequence of A062338, A107288, and A107715 (primes with digits <= 3).
A159352 is a subsequence.
Cf. A000040 (primes), A052218 (digit sum = 4), A061239 (primes == 4 (mod 9)).
Cf. Primes p with digital sum equal to k: {2, 11 and 101} for k=2; this sequence (k=4), A062341 (k=5), A062337 (k=7), A062343 (k=8), A107579 (k=10), A106754 (k=11), A106755 (k=13), A106756 (k=14), A106757 (k=16), A106758 (k=17), A106759 (k=19), A106760 (k=20), A106761 (k=22), A106762 (k=23), A106763 (k=25), A106764 (k=26), A048517 (k=28), A106766 (k=29), A106767 (k=31), A106768 (k=32), A106769 (k=34), A106770 (k=35), A106771 (k=37), A106772 (k=38), A106773 (k=40), A106774 (k=41), A106775 (k=43), A106776 (k=44), A106777 (k=46), A106778 (k=47), A106779 (k=49), A106780 (k=50), A106781 (k=52), A106782 (k=53), A106783 (k=55), A106784 (k=56), A106785 (k=58), A106786 (k=59), A106787 (k=61), A107617 (k=62), A107618 (k=64), A107619 (k=65), A106807 (k=67), A244918 (k=68), A181321 (k=70).
Cf. A049054 (10^k+3 is prime), A159352 (these primes).
Cf. A056807 (3*10^k+1 is prime), A259866 (these primes).
Cf. A020449 (primes with digits 0 and 1), A036953 (primes with digits <= 2), A106100 (primes with largest digit = 2), A069663, A069664 (smallest resp. largest n-digit prime with minimum digit sum).

[p: p in PrimesUpTo(800000000) | &+Intseq(p) eq 4]; // Vincenzo Librandi, Jul 08 2014

N:= 20: # to get all terms < 10^N
B[1]:= {1}:
B[2]:= {2}:
B[3]:= {3}:
A:= {}:
for d from 2 to N do
   B[4]:= map(t -> 10*t+1,B[3]) union  map(t -> 10*t+3,B[1]);
   B[3]:= map(t -> 10*t, B[3]) union map(t -> 10*t+1,B[2]) union map(t -> 10*t+2,B[1]);
   B[2]:= map(t -> 10*t, B[2]) union map(t -> 10*t+1,B[1]);
   B[1]:= map(t -> 10*t, B[1]);
   A:= A union select(isprime,B[4]);
od:
sort(convert(A,list)); # Robert Israel, Dec 28 2015

Union[FromDigits/@Select[Flatten[Table[Tuples[{0,1,2,3},k],{k,9}],1],PrimeQ[FromDigits[#]]&&Total[#]==4&]] (* Jayanta Basu, May 19 2013 *)
FromDigits/@Select[Tuples[{0,1,2,3},10],Total[#]==4&&PrimeQ[FromDigits[#]]&] (* Harvey P. Dale, Jul 23 2025 *)

for(a=1,20,for(b=0,a,for(c=0,b,if(isprime(k=10^a+10^b+10^c+1),print1(k", "))))) \\ Charles R Greathouse IV, Jul 26 2011

select( {is_A062339(p,s=4)=sumdigits(p)==s&&isprime(p)}, primes([1,10^7])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

A062339_upto_length(L,s=4,a=List(),u=[10^(L-k)|k<-[1..L]])=forvec(d=[[1,L]|i<-[1..s]], isprime(p=vecsum(vecextract(u,d))) && listput(a,p),1); Vecrev(a) \\ M. F. Hasler, Mar 09 2022

Intersection of A052218 (digit sum 4) and A000040 (primes). - M. F. Hasler, Mar 09 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
More terms from Rick L. Shepherd, May 23 2005
More terms from Lekraj Beedassy, Dec 19 2007

A106093 Primes with maximal digit = 9.

Original entry on oeis.org

19, 29, 59, 79, 89, 97, 109, 139, 149, 179, 191, 193, 197, 199, 229, 239, 269, 293, 349, 359, 379, 389, 397, 409, 419, 439, 449, 479, 491, 499, 509, 569, 593, 599, 619, 659, 691, 709, 719, 739, 769, 797, 809, 829, 839, 859, 907, 911, 919, 929, 937, 941, 947

Offset: 1

Zak Seidov and Robert G. Wilson v, May 07 2005

Differs from A062679 in 95th term = 1693; A062679(95) = 1691 = 19*89.

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

Cf. A062679.

Cf. A106100, A106099, A106098, A106097, A106096, A106095, A106094.

[p: p in PrimesUpTo(2000) | 9 in Intseq(p)]; // Vincenzo Librandi, Nov 22 2015

Select[Prime[Range[200]], Max[IntegerDigits[ # ]]==9&]

forprime(p=2, 1e3, if(vecmax(digits(p)) == 9, print1(p, ", "))) \\ Altug Alkan, Nov 22 2015

A106099 Primes with maximal digit = 3.

Original entry on oeis.org

3, 13, 23, 31, 103, 113, 131, 223, 233, 311, 313, 331, 1013, 1031, 1033, 1103, 1123, 1213, 1223, 1231, 1301, 1303, 1321, 2003, 2113, 2131, 2203, 2213, 2311, 2333, 3001, 3011, 3023, 3121, 3203, 3221, 3301, 3313, 3323, 3331

Offset: 1

Zak Seidov, May 07 2005

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

Cf. A106093, A106094, A106095, A106096, A106097, A106098, A106100.

Res:= 3: count:= 1:
A:= {3}: B:= {$1..2}:
for d from 2 while count < 100 do
  A:= {seq(seq(10*a+i,i=0..3),a=A), seq(10*b+3,b=B)}:
  B:= {seq(seq(10*b+i,i=0..2),b=B)}:
  S:= sort(convert(select(isprime,A),list));
  count:= count + nops(S);
  Res:= Res, op(S);
od:
Res; # Robert Israel, Jan 01 2019

Select[Prime[Range[600]], Max[IntegerDigits[ # ]]==3&]

A106097 Primes with maximal digit = 5.

Original entry on oeis.org

5, 53, 151, 251, 353, 503, 521, 523, 541, 1051, 1151, 1153, 1451, 1453, 1511, 1523, 1531, 1543, 1553, 2053, 2153, 2251, 2351, 2503, 2521, 2531, 2543, 2551, 3251, 3253, 3511, 3533, 3541

Offset: 1

Zak Seidov, May 07 2005

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

Cf. A106100, A106099, A106098, A106096, A106095, A106094, A106093.

Select[Prime[Range[500]], Max[IntegerDigits[ # ]]==5&]

A106098 Primes with maximal digit = 4.

Original entry on oeis.org

41, 43, 241, 401, 421, 431, 433, 443, 1423, 1433, 2141, 2143, 2243, 2341, 2411, 2423, 2441, 3041, 3343, 3413, 3433, 4001, 4003, 4013, 4021, 4111, 4133, 4201, 4211, 4231, 4241, 4243

Offset: 1

Zak Seidov, May 07 2005

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

Cf. A106100, A106099, A106097, A106096, A106095, A106094, A106093.

Select[Prime[Range[600]], Max[IntegerDigits[ # ]]==4&]

A221697 Numbers whose largest digit of all divisors is 2.

Original entry on oeis.org

2, 22, 121, 202, 211, 1021, 1201, 2011, 2111, 2221, 2222, 10201, 10211, 12011, 12101, 12211, 12221, 20011, 20021, 20101, 20201, 20222, 21001, 21011, 21101, 21121, 21211, 21221, 22111, 22121, 101021, 101221, 102001, 102101, 102121, 110221, 111121, 111211, 111221, 112111, 112121

Offset: 1

Jaroslav Krizek, Jan 22 2013, corrected Jan 29 2013

Also numbers k such that the largest digit of the concatenation of all the divisors (A037278) of k is 2.
Numbers k such that A209928(k) = 2.
Union of A221698 and A106100.

			10201 is a term because the largest digit of all the divisors of 10201 (1, 101, 10201) is 2.

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

Cf. A037278, A106100, A209928 (largest digit of all divisors of n), A221698.

isA221697 := proc(n)
      local dgs,d;
      dgs := {} ;
    for d in numtheory[divisors](n) do
        dgs := dgs union convert(convert(d,base,10),set) ;
    end do:
    if max(op(dgs)) = 2 then
        true;
    else
        false;
    end if;
end proc:
for n from 2 to 112121 do
    if isA221697(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 30 2013

Select[Range[115000],Max[Flatten[IntegerDigits/@Divisors[#]]]==2&] (* Harvey P. Dale, Dec 15 2014 *)

from sympy import divisors
def ok(n): return '2' == max("".join(map(str, divisors(n))))
print([m for m in range(1, 112122) if ok(m)]) # Michael S. Branicky, Feb 22 2021

from sympy import isprime, divisors
from itertools import count, islice, product
def agen(): # generator of terms
    yield 2
    for d in count(2):
        for f in "12":
            for mid in product("012", repeat=d-2):
                for e in "12": # ending in zero has 5 as divisor
                    s = f+"".join(mid)+e
                    t = int(s)
                    if "2" in s and isprime(t): yield t; continue
                    if "2" == max("".join(map(str, divisors(t)))): yield t
print(list(islice(agen(), 50))) # Michael S. Branicky, Aug 03 2022

A221698 Composite numbers n such that largest digit of all divisors of n is 2.

Original entry on oeis.org

22, 121, 202, 1111, 2222, 10201, 12221, 20222, 22121, 111221, 112211, 202222, 220121, 221111, 222211, 1021211, 1112221, 1122011, 1222201, 2021111, 2022002, 2022121, 2121101, 2122111, 2200202, 2202211, 2211121, 2212111, 2222011, 10212211, 11112211, 11121011

Offset: 1

Jaroslav Krizek, Jan 22 2013

Also composite numbers n such that largest digit of concatenation of all divisors (A037278) of n is 2.
Composite numbers n such that A209928(n) = 2.
Complement of A106100 with respect to A221697.

			Number 10201 is in the sequence because the largest digit of all divisors of 10201 (1, 101, 10201) is 2.

Cf. A209928 (largest digit of all divisors of n), A221697.

t = {}; n = 1; While[Length[t] < 40, n++; m = FromDigits[IntegerDigits[n, 3]]; If[! PrimeQ[m] && Max[Union[Flatten[IntegerDigits[Divisors[m]]]]] <= 2, AppendTo[t, m]]]; t (* T. D. Noe, Jan 30 2013 *)

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A062339 Primes whose sum of digits is 4.

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A106093 Primes with maximal digit = 9.

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A106099 Primes with maximal digit = 3.

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A106097 Primes with maximal digit = 5.

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A106098 Primes with maximal digit = 4.

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A221697 Numbers whose largest digit of all divisors is 2.

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A221698 Composite numbers n such that largest digit of all divisors of n is 2.

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