A028834 -id:A028834 - cp's OEIS frontend

A092595 Numbers k such that the sum of decimal digits of k and k+1 are both prime numbers, i.e., both k and k+1 are in A028834.

2, 11, 20, 29, 49, 101, 110, 119, 139, 199, 200, 209, 229, 289, 319, 379, 409, 469, 559, 649, 739, 829, 919, 1001, 1010, 1019, 1039, 1099, 1100, 1109, 1129, 1189, 1219, 1279, 1309, 1369, 1459, 1549, 1639, 1729, 1819, 1909, 2000, 2009, 2029, 2089, 2119, 2179

Offset: 1

Labos Elemer, Mar 17 2004

Interestingly, the following numbers n are such that the sum of the decimal digits of n, n+1, and n+2 are all prime numbers: 199, 1099, 10099, 19999, 100099, 109999, . . . -Harvey P. Dale, Jan 28 2025

			For k=4429, digitsum(k) = 4 + 4 + 2 + 9 = 19, digitsum(k+1) = 4 + 4 + 3 + 0 = 11.

J.W.L. (Jan) Eerland, Table of n, a(n) for n = 1..10000.

Cf. A028834.

t=Table[0, {256}]; j=1; Do[s=Apply[Plus, IntegerDigits[n]]; s1=Apply[Plus, IntegerDigits[n+1]]; If[PrimeQ[s]&&PrimeQ[s1], Print[n]; t[[j]]=n; j=j+1], {n, 1, 10000}]; t
DeleteCases[ParallelTable[If[PrimeQ[Total[IntegerDigits[n]]]&&PrimeQ[Total[IntegerDigits[n+1]]],n,a],{n,2,952999}],a] (* J.W.L. (Jan) Eerland, Dec 20 2021 *)
SequencePosition[Table[If[PrimeQ[Total[IntegerDigits[n]]],1,0],{n,2500}],{1,1}][[;;,1]] (* Harvey P. Dale, Jan 28 2025 *)

isok(n) = isprime(sumdigits(n)) && isprime(sumdigits(n+1)); \\ Michel Marcus, Jul 29 2017

A005349 Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Both spellings, "Harshad" or "harshad", are in use. It is a Sanskrit word, and in Sanskrit there is no distinction between upper- and lower-case letters. - N. J. A. Sloane, Jan 04 2022
z-Niven numbers are numbers n which are divisible by (A*s(n) + B) where A, B are integers and s(n) is sum of digits of n. Niven numbers have A = 1, B = 0. - Ctibor O. Zizka, Feb 23 2008
A070635(a(n)) = 0. A038186 is a subsequence. - Reinhard Zumkeller, Mar 10 2008
A049445 is a subsequence of this sequence. - Ctibor O. Zizka, Sep 06 2010
Complement of A065877; A188641(a(n)) = 1; A070635(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2011
A001101, the Moran numbers, are a subsequence. - Reinhard Zumkeller, Jun 16 2011
A140866 gives the number of terms <= 10^k. - Robert G. Wilson v, Oct 16 2012
The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1984). - Amiram Eldar, Jul 10 2020
From Amiram Eldar, Oct 02 2023: (Start)
Named "Harshad numbers" by the Indian recreational mathematician Dattatreya Ramchandra Kaprekar (1905-1986) in 1955. The meaning of the word is "giving joy" in Sanskrit.
Named "Niven numbers" by Kennedy et al. (1980) after the Canadian-American mathematician Ivan Morton Niven (1915-1999). During a lecture given at the 5th Annual Miami University Conference on Number Theory in 1977, Niven mentioned a question of finding a number that equals twice the sum of its digits, which appeared in the children's pages of a newspaper. (End)

			195 is a term of the sequence because it is divisible by 15 (= 1 + 9 + 5).

Paul Dahlenberg and T. Edgar, Consecutive factorial base Niven numbers, Fib. Q., 56:2 (2018), 163-166.
D. R. Kaprekar, Multidigital Numbers, Scripta Math., Vol. 21 (1955), p. 27.
Robert E. Kennedy and Curtis N. Cooper, On the natural density of the Niven numbers, Abstract 816-11-219, Abstracts Amer. Math. Soc., 6 (1985), 17.
Robert E. Kennedy, Terry A. Goodman, and Clarence H. Best, Mathematical Discovery and Niven Numbers, The MATYC Journal, Vol. 14, No. 1 (1980), pp. 21-25.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 381.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 171.

N. J. A. Sloane, Table of n, a(n) for n = 1..11872 (all a(n) <= 100000)
Bob Albrecht, Don Albers, and Jim Conlan, Problem #22 Two-digit Niven numbers, Programming Problems, Recreational Computing Magazine, Vol. 9, No. 1, Issue 46 (1980), p. 59.
Curtis N. Cooper and Robert E. Kennedy, On an asymptotic formula for the Niven numbers, International Journal of Mathematics and Mathematical Sciences, Vol. 8, No. 3 (1985), pp. 537-543.
Curtis N. Cooper and Robert E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly 96 (1989), no. 2, 118-124.
Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.
Jean-Marie De Koninck and Nicolas Doyon, Large and Small Gaps Between Consecutive Niven Numbers, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.
Nicolas Doyon, Les fascinants nombres de Niven, Thèse de la Faculté des Sciences et de Génie de l'Université Laval, Québec, Novembre 2006 (in French).
Ömer Eğecioğlu and Bünyamin Şahin, On twin EP numbers, Transact. Comb. (2025) Vol. 14, Iss. 4, Art. No. 4, 261-270. See p. 262.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Robert E. Kennedy, Digital sums, Niven numbers, and natural density, Crux Mathematicorum, Vol. 8 (1982), pp. 131-135.
Robert E. Kennedy and Curtis N. Cooper, On the natural density of the Niven numbers, The College Mathematics Journal, Vol. 15, No. 4 (Sep., 1984), pp. 309-312.
Project Euler, Harshad Numbers: Problem 387.
Terry Trotter, Niven Numbers for Fun and Profit. [archived page]
Gérard Villemin, Nombres de Harshad (French).
Elaine E. Visitacion, Renalyn T. Boado, Mary Ann V. Doria, and Eduard M. Albay, On Harshad Number, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 134-138. [archived]
Eric Weisstein's World of Mathematics, Digit and Harshad Numbers.
Wikipedia, Harshad number.

Cf. A001101, A007602, A007953, A028834, A038186, A049445, A052018, A052019, A052020, A052021, A052022, A065877, A070635, A113315, A188641.
Cf. A001102 (a subsequence).
Cf. A118363 (for factorial-base analog).
Cf. A330927, A154701, A141769, A330928, A330929, A330930 (start of runs of 2, 3, ..., 7 consecutive Niven numbers).

Filtered([1..230],n-> n mod List(List([1..n],ListOfDigits),Sum)[n]=0); # Muniru A Asiru

a005349 n = a005349_list !! (n-1)
a005349_list = filter ((== 0) . a070635) [1..]
-- Reinhard Zumkeller, Aug 17 2011, Apr 07 2011

[n: n in [1..250] | n mod &+Intseq(n) eq 0];  // Bruno Berselli, May 28 2011

[n: n in [1..250] | IsIntegral(n/&+Intseq(n))];  // Bruno Berselli, Feb 09 2016

s:=proc(n) local N:N:=convert(n,base,10):sum(N[j],j=1..nops(N)) end:p:=proc(n) if floor(n/s(n))=n/s(n) then n else fi end: seq(p(n),n=1..210); # Emeric Deutsch

harshadQ[n_] := Mod[n, Plus @@ IntegerDigits@ n] == 0; Select[ Range[1000], harshadQ] (* Alonso del Arte, Aug 04 2004 and modified by Robert G. Wilson v, Oct 16 2012 *)
Select[Range[300],Divisible[#,Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Sep 07 2015 *)

is(n)=n%sumdigits(n)==0 \\ Charles R Greathouse IV, Oct 16 2012

A005349 = [n for n in range(1,10**6) if not n % sum([int(d) for d in str(n)])] # Chai Wah Wu, Aug 22 2014

[n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)] # Freddy Barrera, Jul 27 2018

A046704 Additive primes: sum of digits is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131, 137, 139, 151, 157, 173, 179, 191, 193, 197, 199, 223, 227, 229, 241, 263, 269, 281, 283, 311, 313, 317, 331, 337, 353, 359, 373, 379, 397, 401, 409, 421, 443, 449, 461, 463, 467, 487, 557, 571, 577, 593

Offset: 1

Felice Russo

Sum_{a(n) < x} 1/a(n) is asymptotic to (3/2)*log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 07 2012

Harman 2012 also shows, under a conjecture about primes in short intervals, that there are 3/2 * x/(log x log log x) terms up to x. - Charles R Greathouse IV, Nov 17 2014

			The digit sums of 11 and 13 are 1+1=2 and 1+3=4. Since 2 is prime and 4 is not, 11 is a member and 13 is not. - _Jonathan Sondow_, Jun 07 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Glyn Harman, Counting primes whose sum of digits is prime, J. Integer Seq., 15 (2012), Article 12.2.2.
Glyn Harman, Primes whose sum of digits is prime and metric number theory, Bull. Lond. Math. Soc. 44:5 (2012), pp. 1042-1049.

Indices of additive primes are in A075177.
Cf. A046703, A119450 = Primes with odd digit sum, A081092 = Primes with prime binary digit sum, A104213 = Primes with nonprime digit sum.
Cf. A007953, A010051; intersection of A028834 and A000040.

a046704 n = a046704_list !! (n-1)
a046704_list = filter ((== 1) . a010051 . a007953) a000040_list
-- Reinhard Zumkeller, Nov 13 2011

[ p: p in PrimesUpTo(600) | IsPrime(&+Intseq(p)) ];  // Bruno Berselli, Jul 08 2011

select(n -> isprime(n) and isprime(convert(convert(n,base,10),`+`)), [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Nov 17 2014

Select[Prime[Range[100000]], PrimeQ[Apply[Plus, IntegerDigits[ # ]]]&]

isA046704(n)={local(s,m);s=0;m=n;while(m>0,s=s+m%10;m=floor(m/10));isprime(n) & isprime(s)} \\ Michael B. Porter, Oct 18 2009

is(n)=isprime(n) && isprime(sumdigits(n)) \\ Charles R Greathouse IV, Dec 26 2013

A052018 Numbers k with the property that the sum of the digits of k is a substring of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 200, 300, 400, 500, 600, 700, 800, 900, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1063

Offset: 1

Patrick De Geest, Nov 15 1999

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

Cf. A052019, A052020, A052021, A052022, A007953, A005349, A028834.
Cf. A175688. - Reinhard Zumkeller, Aug 15 2010
Cf. A055642, A004426.
Cf. A119246, A138166.

import Data.List (isInfixOf)
a052018 n = a052018_list !! (n-1)
a052018_list = filter f [0..] where
   f x = show (a007953 x) `isInfixOf` show x
-- Reinhard Zumkeller, Jun 18 2013

sdssQ[n_]:=Module[{idn=IntegerDigits[n],s,len},s=Total[idn];len= IntegerLength[ s]; MemberQ[Partition[idn,len,1],IntegerDigits[s]]]; Join[{0},Select[Range[1100],sdssQ]] (* Harvey P. Dale, Jan 02 2013 *)

loop = (str(n) for n in range(399))
print([int(n) for n in loop if str(sum(int(k) for k in n)) in n]) # Jonathan Frech, Jun 05 2017

A028842 Numbers whose product of digits is prime.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 21, 31, 51, 71, 112, 113, 115, 117, 121, 131, 151, 171, 211, 311, 511, 711, 1112, 1113, 1115, 1117, 1121, 1131, 1151, 1171, 1211, 1311, 1511, 1711, 2111, 3111, 5111, 7111, 11112, 11113, 11115, 11117, 11121, 11131, 11151

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..130
Index entries for 10-automatic sequences

Cf. A007954, A028843, A028834, A046703 (subsequence of primes).

Select[Range[11160], PrimeQ[Times@@IntegerDigits[#]] &] (* Jayanta Basu, Jun 02 2013 *)

isok(n) = isprime(vecprod(digits(n))); \\ Michel Marcus, Apr 17 2020

is(n)=my(d=digits(n),p); for(i=1,#d,if(d[i]==1,next); if(isprime(d[i]) && !p, p=1, return(0))); p \\ Charles R Greathouse IV, Apr 18 2020

[x for x in range(10^5) if (prod(Integer(x).digits(base=10))) in Primes()] # Bruno Berselli, May 05 2014

(1 to 10000).filter(n => List(2, 3, 5, 7).contains(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)) // _Alonso del Arte, Apr 14 2020

More terms from Erich Friedman.

Name edited by Jianing Song, Jul 07 2025

A089392 Magnanimous primes: primes with the property that inserting a "+" in any place between two digits yields a sum which is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 227, 229, 281, 401, 443, 449, 467, 601, 607, 647, 661, 683, 809, 821, 863, 881, 2221, 2267, 2281, 2447, 4001, 4027, 4229, 4463, 4643, 6007, 6067, 6803, 8009, 8221, 8821, 20261, 24407, 26881, 28429, 40427

Offset: 1

Amarnath Murthy, Nov 10 2003

Original definition: Let the digits of n be abcd. Then bcd+a, cd+ab, d+abc, abcd, etc. must all be primes. If n is a k-digit number then it must produce k such primes.
Partition the digits of n into two groups by placing a '+' sign anywhere inside; the result of the expression is prime in every case. Conjecture: sequence is infinite. 11 is the largest term with all odd digits. 2 is the only member with all even digits. Observation: all two-digit primes with the most significant digit even are members.
In contradiction to the above conjecture, it is rather expected that this sequence is finite, cf. the link to C. Rivera's "Puzzle 401", and G. Resta's web page. Concerning the statement about 2 and 11, one can say that all terms except 2, 11 and 101 consist of even digits followed by a final odd digit. - M. F. Hasler, Dec 25 2014
Primes among the magnanimous numbers A252996. - M. F. Hasler, Dec 25 2014

			2267 is a member which gives primes 2+267 = 269, 22+67 = 89, 226+7 = 233 and 2267 itself.

Zak Seidov, Table of n, a(n) for n = 1..84
Eric Angelini et al., Insert "+" and always get a prime, Dec
2014
Giovanni Resta, magnanimous numbers, 2013.
Carlos Rivera, Puzzle 401. Magnanimous primes, 2007, The Prime Puzzles & Problems Connection.

Cf. A089393, A089394, A089695, A028834, A182175, A088134, A221699, A227823, A252996.

with(combinat): ds:=proc(s) local j: RETURN(add(s[j]*10^(j-1),j=1..nops(s))):end: for d from 1 to 6 do sch:=[seq([1,op(i),d+1],i=[[],seq([j],j=2..d)])]: for n from 10^(d-1) to 10^d-1 do sn:=convert(n,base,10): fl:=0: for s in sch do m:=add(j,j=[seq(ds(sn[s[i]..s[i+1]-1]),i=1..nops(s)-1)]): if not isprime(m) then fl:=1: break fi od: if fl=0 then printf("%d, ",n) fi od od: # C. Ronaldo

mpQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];And@@PrimeQ[ Table[ FromDigits[Take[idn,i]]+FromDigits[Take[idn,-(len-i)]],{i,len}]]]; Select[Range[41000],mpQ] (* Harvey P. Dale, Nov 06 2013 *)

is_A089392(n)={!for(i=1,#Str(n),ispseudoprime([1,1]*(divrem(n,10^i)))||return)} \\ M. F. Hasler, Dec 25 2014

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    return all(isprime(int(s[:i])+int(s[i:])) for i in range(1, len(s)))
print([k for k in range(357) if ok(k)]) # Michael S. Branicky, Oct 14 2024

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 25 2004
Comments edited by Zak Seidov, Jan 29 2013
Edited by M. F. Hasler, Dec 25 2014

A052019 Sum of digits of prime p is substring of p.

Original entry on oeis.org

2, 3, 5, 7, 109, 139, 149, 179, 199, 911, 919, 1009, 1063, 1109, 1163, 1181, 1327, 1381, 1409, 1427, 1481, 1609, 1627, 1663, 1709, 1811, 2099, 2137, 2399, 2699, 2711, 2713, 2719, 2999, 3613, 3617, 4513, 4517, 4519, 5413, 5417, 5419, 6113, 6133, 6143

Offset: 1

Patrick De Geest, Nov 15 1999

Cf. A052018, A052020, A052021, A052022, A007953, A005349, A028834 & A158473.

fQ[n_] := Block[ {id = IntegerDigits@ n}, pid = Plus @@ id; MemberQ[ Partition[id, IntegerLength@ pid, 1], IntegerDigits@ pid]]; Select[ Prime@ Range@ 802, fQ] (* Robert G. Wilson v, Aug 16 2011 *)
Select[Prime[Range[1000]],SequenceCount[IntegerDigits[#], IntegerDigits[ Total[ IntegerDigits[ #]]]]> 0&] (* The program uses the SequenceCount function from Mathematica version 10 *)  (* Harvey P. Dale, Sep 30 2015 *)

A104211 Integers n such that the sum of the digits of n is not prime.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 13, 15, 17, 18, 19, 22, 24, 26, 27, 28, 31, 33, 35, 36, 37, 39, 40, 42, 44, 45, 46, 48, 51, 53, 54, 55, 57, 59, 60, 62, 63, 64, 66, 68, 69, 71, 72, 73, 75, 77, 78, 79, 80, 81, 82, 84, 86, 87, 88, 90, 91, 93, 95, 96, 97, 99, 100, 103, 105, 107, 108, 109, 112

Offset: 1

Cino Hilliard, Mar 13 2005

			Sum of digits of 15=6 not prime.

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

Complement of A028834.

Cf. A000040, A007953.

Select[Range[250], !PrimeQ[Total[IntegerDigits[ # ]]]&] (* Vincenzo Librandi, Apr 17 2013 *)

lista(n) = { my(x,y); for(x=1,n, y=sumdigits(x); if(!isprime(y), print1(x,", ") ) ); }

A052020 Sum of digits of k is a prime proper factor of k.

Original entry on oeis.org

12, 20, 21, 30, 50, 70, 102, 110, 111, 120, 133, 140, 200, 201, 209, 210, 230, 247, 300, 308, 320, 322, 364, 407, 410, 476, 481, 500, 506, 511, 605, 629, 700, 704, 715, 782, 803, 832, 874, 902, 935, 1002, 1010, 1011, 1015, 1020, 1040, 1066, 1088, 1100, 1101

Offset: 1

Patrick De Geest, Nov 15 1999

For each prime p there are infinitely many terms with sum of digits p. - Robert Israel, Feb 26 2017

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

Cf. A052018, A052019, A052021, A052022, A007953, A005349, A028834.

filter:= proc(n) local s;
  s:= convert(convert(n,base,10),`+`);
  isprime(s) and (n mod s = 0)
end proc:
select(filter, [$10..10^4]); # Robert Israel, Feb 26 2017

Select[Range[0, 2000], With[{s = DigitSum[#]}, s < # && Divisible[#, s] && PrimeQ[s]] &] (* Paolo Xausa, May 18 2024 *)

from sympy import isprime
def ok(n):
    sd = sum(map(int, str(n)))
    return 1 < sd < n and n%sd == 0 and isprime(sd)
print([k for k in range(1102) if ok(k)]) # Michael S. Branicky, Dec 20 2021

A052021 Sum of digits of n is the largest prime factor of n.

Original entry on oeis.org

2, 3, 5, 7, 12, 50, 70, 308, 320, 364, 476, 500, 605, 700, 704, 715, 832, 935, 1088, 1183, 1547, 1729, 2401, 2584, 2618, 2704, 2926, 3080, 3200, 3536, 3640, 3952, 4225, 4760, 4784, 4913, 5000, 5491, 5525, 5819, 5831, 6050, 6175, 6517, 6647, 7000, 7040, 7150

Offset: 1

Patrick De Geest, Nov 15 1999

			13685 has sum of digits '23' and 13685 = 5*7*17*'23'.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A052018, A052019, A052020, A052022, A007953, A005349, A028834.

a052021 n = a052021_list !! (n-1)
a052021_list = tail $ filter (\x -> a007953 x == a006530 x) [1..]
-- Reinhard Zumkeller, Nov 06 2011

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc:
A006530 := proc(n) numtheory[factorset](n) ; max(op(%)) ; end proc:
for n from 1 to 8000 do if A007953(n) = A006530(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, May 30 2010

Select[Range[2,8000],FactorInteger[#][[-1,1]]==Total[IntegerDigits[#]]&] (* Harvey P. Dale, Oct 17 2012 *)

{n: A007953(n) = A006530(n)}. - R. J. Mathar, May 30 2010

Single-digit primes added by R. J. Mathar, May 30 2010

Offset corrected by Reinhard Zumkeller, Nov 05 2011

cp's OEIS Frontend

A092595 Numbers k such that the sum of decimal digits of k and k+1 are both prime numbers, i.e., both k and k+1 are in A028834.

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A005349 Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.

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A046704 Additive primes: sum of digits is a prime.

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A052018 Numbers k with the property that the sum of the digits of k is a substring of k.

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A028842 Numbers whose product of digits is prime.

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A089392 Magnanimous primes: primes with the property that inserting a "+" in any place between two digits yields a sum which is prime.

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