A062339 -id:A062339 - cp's OEIS frontend

A244918 Primes p where the digital sum is equal to 68.

59999999, 69999899, 69999989, 78998999, 88989899, 88999979, 89699999, 89799989, 89989799, 89989979, 89997899, 89997989, 89999699, 89999969, 97889999, 98699999, 98879999, 98899799, 98979989, 98988899, 98989889, 98997989, 98998979, 98999969

Offset: 1

Vincenzo Librandi, Jul 08 2014

			69999899 is a prime with sum of the digits = 68, hence belongs to the sequence.

Vincenzo Librandi and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..45 from Vincenzo Librandi.

Cf. Primes p where the digital sum is equal to k: 2, 11 and 101 for k=2; A062339 (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), this sequence (k=68), A181321 (k=70).

[p: p in PrimesUpTo(100000000) | &+Intseq(p) eq 68];

Select[Prime[Range[10000000]], Total[IntegerDigits[#]]==68 &]

# see code in A107579: the same code can be used to produce this sequence, by giving the initial term p = 6*10**7-1, for digit sum 68. - M. F. Hasler, Mar 16 2022

A001101 Moran numbers: k such that k/(sum of digits of k) is prime.

Original entry on oeis.org

18, 21, 27, 42, 45, 63, 84, 111, 114, 117, 133, 152, 153, 156, 171, 190, 195, 198, 201, 207, 209, 222, 228, 247, 261, 266, 285, 333, 370, 372, 399, 402, 407, 423, 444, 465, 481, 511, 516, 518, 531, 555, 558, 592, 603

Offset: 1

Bill Moran (moran1(AT)llnl.gov)

Witno conjectures that a(n) ~ c*n log(n)^2 for some c. - Charles R Greathouse IV, Jul 26 2011

Bill Moran, Problem 2074: The Moran Numbers, J. Rec. Math., Vol. 25 No. 3, pp. 215, 1993.

Aaron Toponce, Table of n, a(n) for n = 1..10000 (first 1000 terms 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 A005349, Niven (or Harshad) numbers.

Cf. A007953, A010051, A085775, A108780, A130338, A062339.

import Data.List (findIndices)
a001101 n = a001101_list !! (n-1)
a001101_list = map succ $ findIndices p [1..] where
   p n = m == 0 && a010051 n' == 1 where
      (n', m) = divMod n (a007953 n)
-- Reinhard Zumkeller, Jun 16 2011

filter:= proc(n) local q;
  q:= n/convert(convert(n,base,10),`+`);
  q::integer and isprime(q)
end proc:
select(filter, [$1..1000]); # Robert Israel, May 13 2025

Select[Range[700], PrimeQ[ # / Total[IntegerDigits[#]]]&] (* Jean-François Alcover, Nov 30 2011 *)

is(n)=(k->denominator(k)==1&&isprime(k))(n/sumdigits(n)) \\ Charles R Greathouse IV, Jan 10 2014

from sympy import isprime
def ok(n): s = sum(map(int, str(n))); return s and n%s==0 and isprime(n//s)
print([k for k in range(604) if ok(k)]) # Michael S. Branicky, Mar 28 2022

Name corrected by Charles R Greathouse IV, Jan 10 2014

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

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

A056807 Numbers k such that 3*10^k + 1 is prime.

Original entry on oeis.org

1, 3, 7, 10, 28, 36, 67, 81, 147, 483, 643, 1020, 1900, 2620, 10453, 27720, 52824, 105589, 111988, 618853, 665829

Offset: 1

Robert G. Wilson v, Aug 22 2000

			k = 3 gives (3*10^3+1) = 3000+1 = 3001, which is prime.

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Makoto Kamada, Prime numbers of the form 300...001.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A056797, A062339, A101823, A199683, A259866.

Do[ If[ PrimeQ[ 3*10^k + 1], Print[ k ]], {k, 0, 20000}]

is(k)=isprime(3*10^k+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A101823(n) + 1.

a(13)-a(14) from Julien Peter Benney (jpbenney(AT)ftml.net), Nov 23 2004
a(15) from Hugo Pfoertner, Jan 18 2005
a(16)-a(17) from Robert G. Wilson v, Jan 18 2005
a(18) from Roman Makarchuk, Dec 05 2008 confirmed as next term by Ray Chandler, Mar 02 2012
a(19) from Alexander Gramolin, Feb 24 2012 confirmed as next term by Ray Chandler, Mar 02 2012
a(20)-a(21) from Kamada data by Robert Price, Jan 26 2015

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 *)

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 *)

A157711 Primes made up of 0's and four 1's only.

Original entry on oeis.org

10111, 1011001, 1100101, 10010101, 10100011, 101001001, 1000001011, 1000010101, 1010000011, 1100010001, 10000001101, 10001000011, 10001001001, 10001100001, 10100000011, 10100001001, 11000000101, 11001000001

Offset: 1

Lekraj Beedassy, Mar 04 2009

Intersection of A062339 and A020449. Subsequence of A235154. - Felix Fröhlich, Nov 19 2014

Primes that are the sum of four distinct powers of ten (A038446). - Jeppe Stig Nielsen, May 18 2023

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Hans Havermann, The one-millionth term.
Hans Havermann, The two-millionth term.

Cf. A020449, A038446, A062339, A235154, A383675 (number of n-digit terms).

for d from 4 to 18 do for c from 0 to 2^d-1 do bdgs := convert(c,base,2) ; if add(i,i=bdgs) = 3 then p := 10^d+add(op(i,bdgs)*10^(i-1),i=1..nops(bdgs)) ; if isprime(p) then printf("%d,",p) ; fi; fi; od: od: # R. J. Mathar, Mar 06 2009

Flatten[Select[FromDigits/@Permutations[Join[{1,1,1,1},PadRight[{},7,0]]],PrimeQ]] // Union (* Harvey P. Dale, May 09 2019 *)

for(n=0, 10, forprime(p=10^n, (10^(n+1)-1)/9, if(vecmax(digits(p))==1, if(sumdigits(p)==4, print1(p, ", "))))) \\ Felix Fröhlich, Nov 19 2014

my(M=20);for(i=3, M, for(j=2,i-1, for(k=1, j-1, my(p=10^i+10^j+10^k+1); isprime(p)&&print1(p,", ")))) \\ Jeppe Stig Nielsen, May 18 2023

Extended by numerous authors, Mar 06 2009

cp's OEIS Frontend

A244918 Primes p where the digital sum is equal to 68.

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A001101 Moran numbers: k such that k/(sum of digits of k) is prime.

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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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A106754 Primes p with digital sum equal to 11.

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