A244918 -id:A244918 - 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

A062341 Primes whose sum of digits is 5.

Original entry on oeis.org

5, 23, 41, 113, 131, 311, 401, 1013, 1031, 1103, 1301, 2003, 2111, 3011, 4001, 10103, 10211, 10301, 11003, 12011, 12101, 13001, 20021, 20201, 21011, 21101, 30011, 100103, 101021, 101111, 102101, 103001, 120011, 121001, 200003, 200201, 201011, 201101, 202001

Offset: 1

Amarnath Murthy, Jun 21 2001

			1301 belongs to the sequence since it is a prime with sum of digits = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..14312 (first 100 terms from Harvey P. Dale)

Cf. A000040 (primes), A007953 (sum of digits), A052219 (digit sum = 5).
Cf. A062339 (same for digit sum s = 4), A062337 (s = 7), and others listed in A244918 (s = 68).
Subsequence of A062340 (primes with sum of digits divisible by 5).

[p: p in PrimesUpTo(250000) | &+Intseq(p) eq 5]; // Vincenzo Librandi, Jul 08 2014

T:= n-> `if`(n=1, 5, sort(select(isprime, [seq(seq(seq(
    10^(n-1)+1+10^i+10^j+10^k, k=1..j), j=1..i), i=1..n-1),
    seq(10^(n-1)+3+10^i, i=1..n-1)]))[]):
seq(T(n), n=1..8);  # Alois P. Heinz, Dec 28 2015

Select[Prime[Range[20000]],Total[IntegerDigits[#]]==5&] (* Harvey P. Dale, Nov 24 2013 *)

\\ From M. F. Hasler, Mar 09 2022: (Start)
select( {is_A062341(p,s=5)=sumdigits(p)==s&&isprime(p)}, primes([1,10^6])) \\ 2nd optional parameter for similar sequences with other digit sums.
A062341_upto_length(L,s=5,a=List(),u=[10^k|k<-[0..L-1]])={forvec(d=[[1,L]|i<-[1..s]], isprime(p=vecsum(vecextract(u,d))) && listput(a,p),1); Set(a)} \\ (End)

from sympy import primerange as primes
def ok(p): return sum(map(int, str(p))) == 5
print(list(filter(ok, primes(1, 202002)))) # Michael S. Branicky, May 23 2021

Intersection of A000040 (primes) with A052219 (digit sum 5). - M. F. Hasler, Mar 09 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001

A062343 Primes whose sum of digits is 8.

Original entry on oeis.org

17, 53, 71, 107, 233, 251, 431, 503, 521, 701, 1061, 1151, 1223, 1511, 1601, 2141, 2213, 2411, 3023, 3041, 3203, 3221, 4013, 4211, 5003, 5021, 6011, 6101, 7001, 10007, 10061, 10133, 10151, 10223, 10313, 10331, 10601, 11213, 11321, 11411

Offset: 1

Amarnath Murthy, Jun 21 2001

			1151 belongs to the sequence since it is a prime with sum of digits = 8.

Alois P. Heinz, Table of n, a(n) for n = 1..42745 (first 500 terms from Vincenzo Librandi)

Cf. A000040 (primes), A007953 (sum of digits), A052222 (digit sum = 8).
Cf. A062339 (same for digit sum s = 4), A062341 (s = 5), A062337 (s = 7), A107579 (s = 10), and others listed in A244918 (s = 68).
Subsequence of A062342 (primes with digit sum divisible by 8).

[p: p in PrimesUpTo(20000) | &+Intseq(p) eq 8]; // Vincenzo Librandi, Jul 08 2014

A062343 := proc(n)
    option remember ;
    local p ;
    if n = 1 then
        17;
    else
        p := nextprime(procname(n-1)) ;
        while true do
            if digsum(p) = 8 then # digsum in oeis.org/transforms.txt
                return p;
            else
                p := nextprime(p) ;
            end if;
        end do:
    end if;
end proc:
seq(A062343(n),n=1..80) ; # R. J. Mathar, May 22 2025

Select[Prime[Range[500000]], Total[IntegerDigits[#]]==8 &] (* Vincenzo Librandi, Jul 08 2014 *)

select( {is_A062343(p, s=8)=sumdigits(p)==s&&isprime(p)}, primes([1, 12345])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

{A062343_upto_length(L, s=8, 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 A000040 (primes) and A052222 (digit sum 8). - M. F. Hasler, Mar 09 2022

More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001

A107579 Primes with digit sum 10.

Original entry on oeis.org

19, 37, 73, 109, 127, 163, 181, 271, 307, 433, 523, 541, 613, 631, 811, 1009, 1063, 1117, 1153, 1171, 1423, 1531, 1621, 1801, 2017, 2053, 2143, 2161, 2251, 2341, 2503, 2521, 3061, 3313, 3331, 3511, 4051, 4231, 5023, 5113, 6121, 6211, 6301, 8011, 8101

Offset: 1

Zak Seidov, May 16 2005

Subset of A061237 and A117674.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1001..3000 from Vincenzo Librandi and Zak Seidov, terms 1..1000 from Vincenzo Librandi)

Cf. A000040 (primes), A007953 (sum of digits), A052224 (digit sum = 10).
Cf. A061237 (sum of digits == 1 (mod 9)).
Subsequence of A062340 (primes with digit sum divisible by 5).
Cf. A062339 (same for digit sum s = 4), A062341 (s = 5), A062343 (s = 8),  A106754 (s = 11), and others listed in A244918 (s = 68).

[p: p in PrimesUpTo(10000) | &+Intseq(p) eq 10]; // Vincenzo Librandi, Jul 08 2014

a:=proc(n) local nn: nn:=convert(n,base,10): if isprime(n)=true and add(nn[j], j=1..nops(nn))=10 then n else end if end proc: seq(a(n),n=1..10^4); # Emeric Deutsch, Mar 06 2008

Select[Prime[Range[100000]], Total[IntegerDigits[#]]==10 &] (* Vincenzo Librandi, Jul 08 2014 *)

forprime(p=19,8101,if(10==sumdigits(p),print(p","))) \\ Zak Seidov, Oct 08 2016

(A107579_nxt(p)=until(isprime(p=A228915(p)),); p); A107579_first(N=100)=vector(N, i, p=if(i>1, A107579_nxt(p), 19)) \\ M. F. Hasler, Mar 15 2022

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def agen(b=10, sod=10): # generator for any base, sum-of-digits
    if 0 <= sod < b:
        yield sod
    nzdigs = [i for i in range(1, b) if i <= sod]
    nzmultiset = []
    for d in range(1, b):
        nzmultiset += [d]*(sod//d)
    for d in count(2):
        fullmultiset = [0]*(d-1-(sod-1)//(b-1)) + nzmultiset
        for firstdig in nzdigs:
            target_sum, restmultiset = sod - int(firstdig), fullmultiset[:]
            restmultiset.remove(firstdig)
            for p in multiset_permutations(restmultiset, d-1):
                if sum(p) == target_sum:
                    t = int("".join(map(str, [firstdig]+p)), b)
                    if isprime(t):
                        yield t
                    if p[0] == target_sum:
                        break
print(list(islice(agen(), 45))) # Michael S. Branicky, Mar 10 2022

from sympy import isprime
def A107579(p=19):
    "Return a generator of the sequence of all primes >= p with the same digit sum as p."
    while True:
        if isprime(p): yield p
        p = A228915(p) # skip to next larger integer with the same digit sum
a=A107579(); [next(a) for  in range(50)] # _M. F. Hasler, Mar 16 2022

Intersection of A000040 (primes) and A052224 (digit sum = 10). - M. F. Hasler, Mar 09 2022

Edited by N. J. A. Sloane, Feb 20 2009 at the suggestion of Pacha Nambi

A062337 Primes whose sum of digits is 7.

Original entry on oeis.org

7, 43, 61, 151, 223, 241, 313, 331, 421, 601, 1033, 1051, 1123, 1213, 1231, 1303, 1321, 2113, 2131, 2203, 2221, 2311, 3121, 3301, 4003, 4021, 4111, 4201, 5011, 5101, 10141, 10303, 10321, 10501, 11113, 11131, 11311, 12211, 12301, 13003, 14011, 20023, 20113

Offset: 1

Amarnath Murthy, Jun 21 2001

There are O((log n)^6) members of this sequence below n.

			601 is a prime with sum of the digits = 7, hence belongs to the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A062336. See also A000579, A118703 (no digit 0)

Cf. similar sequences listed in A244918.

[p: p in PrimesUpTo(250000) | &+Intseq(p) eq 7]; // Vincenzo Librandi, Jul 08 2014

A062337 := proc(n)
    option remember ;
    local p ;
    if n = 1 then
        7;
    else
        p := nextprime(procname(n-1)) ;
        while true do
            if digsum(p) = 7 then # digsum in oeis.org/transforms.txt
                return p;
            else
                p := nextprime(p) ;
            end if;
        end do:
    end if;
end proc:
seq(A062337(n),n=1..80) ; # R. J. Mathar, May 22 2025

Select[Prime[Range[3000]], Plus @@ IntegerDigits[ # ] == 7 &] (* Zak Seidov, Feb 17 2005 *)

A062337(lim)={my(pow=ceil(log(floor(lim)-.5)/log(10)),n);print("Checking for members of A062337 up to 10^"pow);for(a=0,pow-1,for(b=0,a,for(c=0,b,for(d=0,c,for(e=0,d,for(f=0,e,n=10^a+10^b+10^c+10^d+10^e+10^f+1;if(isprime(n),print1(n","))))))))};

select( {is_A062337(p, s=7)=sumdigits(p)==s&&isprime(p)}, primes([1, 14321])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

{A062337_upto_length(L, s=7, 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 A000040 (primes) and A052221 (digit sum 7). - M. F. Hasler, Mar 09 2022

More terms from Larry Reeves (larryr(AT)acm.org), Jul 06 2001

Comments and program from Charles R Greathouse IV, Sep 11 2009

A106754 Primes p with digital sum equal to 11.

Original entry on oeis.org

29, 47, 83, 137, 173, 191, 227, 263, 281, 317, 353, 443, 461, 641, 821, 911, 1019, 1091, 1109, 1163, 1181, 1217, 1307, 1361, 1433, 1451, 1523, 1613, 1721, 1811, 1901, 2027, 2063, 2081, 2153, 2207, 2243, 2333, 2351, 2423, 2441, 2531, 2621, 2711, 2801, 3251

Offset: 1

Zak Seidov, May 16 2005

Alois P. Heinz, Table of n, a(n) for n = 1..41771 (first 2000 terms from Vincenzo Librandi)

Cf. A000040 (primes), A007953 (sum of digits), A166311 (digit sum = 11).
Cf. A062339 (same for digit sum s = 4), ..., A107579 (s = 10),  A106755 (s = 13), and others listed in A244918 (s = 68).
Subsequence of A119891 (prime trios: chain of prime sums of digits; also has as subsequence A106762 (s = 23), A106774 (s = 41), etc).

[p: p in PrimesUpTo(10000) | &+Intseq(p) eq 11]; // Vincenzo Librandi, Jul 08 2014

Select[Prime[Range[100000]], Total[IntegerDigits[#]]==11 &] (* Vincenzo Librandi, Jul 08 2014 *)

select( {is_A106754(n)=sumdigits(n)==11&&isprime(n)}, primes([1, 3333])) \\ M. F. Hasler, Mar 09 2022

Intersection of A000040 (primes) and A166311 (digit sum = 11), also equals { p in A000040 | A007953(p) = 11 }. - M. F. Hasler, Mar 09 2022

A106787 Primes with digit sum = 61.

Original entry on oeis.org

8989999, 8999899, 9989899, 9999889, 16999999, 17989999, 17999899, 18889999, 18899899, 18899989, 18979999, 18989899, 18998899, 18999889, 18999979, 18999997, 19889899, 19899889, 19979899, 19987999, 19988989, 19989997, 19999789, 19999897, 26998999, 27799999

Offset: 1

Zak Seidov, May 16 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..3960

Cf, similar sequences listed in A244918.

[p: p in PrimesUpTo(30000000) | &+Intseq(p) eq 61]; // Vincenzo Librandi, Jul 09 2014

Select[Prime[Range[3000000]], Total[IntegerDigits[#]]==61 &] (* Vincenzo Librandi, Jul 09 2014 *)

isok(n) = isprime(n) && (sumdigits(n) == 61); \\ Michel Marcus, Oct 09 2013

More terms from Joshua Zucker, May 17 2006

More terms from Michel Marcus, Oct 05 2013

A107618 Primes with digit sum = 64.

Original entry on oeis.org

19999999, 29999899, 29999989, 39979999, 39999979, 47999899, 48899899, 48989989, 48997999, 48999799, 48999889, 49989799, 49999699, 49999897, 56999989, 58799899, 58898989, 58988899, 58997899, 59698999, 59788999

Offset: 1

Zak Seidov, May 18 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..700

Cf. A062339, A062341, A062343, A106754 - A106787, A107617, A107619.

Cf. similar sequences listed in A244918.

[p: p in PrimesUpTo(69000000) | &+Intseq(p) eq 64]; // Vincenzo Librandi, Jul 09 2014

Select[Prime[Range[3600000]],Total[IntegerDigits[#]]==64&] (* Harvey P. Dale, Jan 19 2012 *)

A106760 Primes with digit sum = 20.

Original entry on oeis.org

389, 479, 569, 587, 659, 677, 839, 857, 929, 947, 983, 1289, 1487, 1559, 1667, 1847, 1973, 2099, 2297, 2459, 2477, 2549, 2657, 2693, 2729, 2819, 2837, 2909, 2927, 2963, 3089, 3359, 3449, 3467, 3539, 3557, 3593, 3719, 3863, 3881, 3917, 4079, 4259, 4349

Offset: 1

Zak Seidov, May 16 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..9000

Cf. similar sequences listed in A244918.

[p: p in PrimesUpTo(5000) | &+Intseq(p) eq 20]; // Vincenzo Librandi, Jul 08 2014

Select[Prime[Range[2000]], Total[IntegerDigits[#]]==20 &] (* Vincenzo Librandi, Jul 08 2014 *)

A107617 Primes with digit sum = 62.

Original entry on oeis.org

9899999, 18899999, 18999989, 19899989, 19998899, 19998989, 27989999, 27999899, 28998989, 28999979, 29789999, 29798999, 29969999, 29979899, 29988899, 29988989, 29989889, 29997899, 29998799, 29998889, 29999699, 36998999

Offset: 1

Zak Seidov, May 18 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A062339, A062341, A062343, A106754 - A106787, A107618, A107619.

Cf. similar sequences listed in A244918.

[p: p in PrimesUpTo(38000000) | &+Intseq(p) eq 62]; // Vincenzo Librandi, Jul 09 2014

Select[Prime[Range[600000]], Total[IntegerDigits[#]]==62 &] (* Vincenzo Librandi, Jul 09 2014 *)

cp's OEIS Frontend

A062339 Primes whose sum of digits is 4.

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A062341 Primes whose sum of digits is 5.

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A062343 Primes whose sum of digits is 8.

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A107579 Primes with digit sum 10.

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A062337 Primes whose sum of digits is 7.

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