A046704 -id:A046704 - cp's OEIS frontend

A075177 Indices of additive primes - primes with prime sum-of-digits, see A046704.

1, 2, 3, 4, 5, 9, 10, 13, 14, 15, 18, 19, 23, 24, 26, 30, 32, 33, 34, 36, 37, 40, 41, 43, 44, 45, 46, 48, 49, 50, 53, 56, 57, 60, 61, 64, 65, 66, 67, 68, 71, 72, 74, 75, 78, 79, 80, 82, 86, 87, 89, 90, 91, 93, 102, 105, 106, 108, 109, 110, 111, 116, 117, 118, 121, 124, 128

Offset: 1

Zak Seidov, Sep 06 2002

There are 107 additive primes among first 200 primes. There are 38455 additive primes among first 100000 primes. Additive primes are in A046704.

			Prime(10) = 29 is an additive prime because 2+9 = 11 is prime.

Zak Seidov, Table of n, a(n) for n=1..4086, a(n)<10^4.

Cf. A046704.

Cf. A007953, A010051.

import Data.List (elemIndices)
a075177 n = a075177_list !! (n-1)
a075177_list = map (+ 1) $
   elemIndices 1 $ map (a010051 . a007953) a000040_list
-- Reinhard Zumkeller, Nov 13 2011

Map[PrimePi[ # ]&, Select[Prime[Range[200]], PrimeQ[Apply[ Plus, IntegerDigits[ # ]]]&]]
Flatten[Position[Prime[Range[150]],?(PrimeQ[Total[IntegerDigits[#]]]&),{1},Heads->False]] (* _Harvey P. Dale, May 29 2013 *)

a(n) = PrimePi(A046704(n)).

A000040(a(n)) = A046704(n).

A156604 a(1)=2; for n > 0, a(n+1) is the smallest prime of A046704 larger than a(n) such that the sequence of digit sums of a(n) is nondecreasing.

Original entry on oeis.org

2, 3, 5, 7, 29, 47, 67, 89, 179, 197, 199, 379, 397, 487, 577, 599, 797, 887, 977, 1499, 1697, 1787, 1877, 1949, 2399, 2579, 2687, 2777, 2939, 2957, 2999, 3989, 4799, 4889, 4999, 6997, 7699, 7789, 7879, 8599, 8689, 8779, 8887, 9679, 9697, 9769, 9787, 9859

Offset: 1

Juri-Stepan Gerasimov, Feb 11 2009

The slowest increasing sequence that is a subsequence of A046704 such that the sequence of A007953(a(n)) is nondecreasing. - R. J. Mathar, Mar 29 2010

			The first several terms after a(1)=2 are
    3 (3 > 2);
    5 (5 > 3);
    7 (7 > 5);
   29 (2 + 9 > 7);
   47 (4 + 7 = 2 + 9);
   67 (6 + 7 > 4 + 9);
   89 (8 + 9 > 6 + 7);
  179 (1 + 7 + 9 = 8 + 9);
  197 (1 + 9 + 7 = 8 + 9).

Cf. A000040, A007953, A067954.

A007953 := proc(n) local d ; add(d,d= convert(n,base,10)) ; end proc:
isA046704 := proc(n) isprime(n) and isprime(A007953(n)) ; end proc:
A156604 := proc(n) option remember ; local psprev,i ; if n = 1 then 2 ; else psprev := A007953(procname(n-1)) ; for i from procname(n-1)+1 do if isA046704(i) then if A007953(i) >= psprev then return i ; end if; end if; end do: end if ; end proc:
seq(A156604(n),n=1..80) ; # R. J. Mathar, Mar 18 2010
From R. J. Mathar, Mar 29 2010: (Start)
A007953 := proc(n) add(d, d= convert(n,base,10)) ; end proc:
isA028834 := proc(n) local d; add(d, d= convert(n,base,10)) ; isprime(%) ; end proc:
isA046704 := proc(n) isprime(n) and isA028834(n) ; end proc:
A156604 := proc(n) option remember; if n = 1 then 2; else for a from procname(n-1)+1 do if isA046704(a) and A007953(a) >= A007953(procname(n-1)) then return a; end if; end do: end if; end proc: seq(A156604(n),n=1..100) ; (End)

Definition, terms and examples corrected by R. J. Mathar, Mar 18 2010

179 and 1877 inserted, and 9 terms after 4889 replaced with the single term 4999, by R. J. Mathar, Mar 29 2010

A146972 Twin primes in A046704.

Original entry on oeis.org

3, 5, 7, 41, 43, 137, 139, 191, 193, 197, 199, 227, 229, 281, 283, 311, 313, 461, 463, 599, 601, 641, 643, 821, 823, 827, 829, 881, 883, 1031, 1033, 1091, 1093, 1277, 1279, 1301, 1303, 1451, 1453, 1721, 1723, 1871, 1873, 2027, 2029, 2081, 2083, 2087, 2089

Offset: 1

Dmitry Kamenetsky, Nov 03 2008

			137 and 139 are twin primes. The sum of their digits are 11 and 13 respectively, which are also twin primes.

A046704

A028834 Numbers whose sum of digits is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 102, 104, 106, 110, 111, 113, 115, 119, 120, 122, 124, 128, 131, 133, 137, 139, 140, 142, 146, 148, 151, 155, 157, 160, 164, 166, 173, 175, 179, 182

Offset: 1

Armand Turpel (armand(AT)vo.lu, armand_t(AT)geocities.com)

			89 included because 8+9 = 17, which is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..25000 (terms 1..1000 from T. D. Noe)
Christian N. K. Anderson, Ulam Spiral for a(n) <= 100000.

Cf. A007953, A028835, A028842, A052018, A052019, A052020, A052021, A052022.
Cf. A010051; A046704 is a subsequence.
Complement of A104211.

a028834 n = a028834_list !! (n-1)
a028834_list = filter ((== 1) . a010051 . a007953) [1..]
-- Reinhard Zumkeller, Nov 13 2011

a:=proc(n) local nn: nn:=convert(n,base,10): if isprime(sum(nn[j],j=1..nops(nn)))=true then n else fi end: seq(a(n),n=1..200); # Emeric Deutsch, Mar 17 2007

Select[Range[200],PrimeQ[Total[IntegerDigits[#]]]&]  (* Harvey P. Dale, Feb 18 2011 *)

is(n)=isprime(sumdigits(n)) \\ Felix Fröhlich, Aug 16 2014

from sympy import isprime
def ok(n): return isprime(sum(map(int, str(n))))
print(list(filter(ok, range(183)))) # Michael S. Branicky, Jun 18 2021

require(gmp); which(sapply(1:1000, function(i) isprime(sum(floor(i/10^(0:(nchar(i)-1)))%%10)))==2) # Christian N. K. Anderson, Apr 22 2024

[x for x in range(200) if (sum(Integer(x).digits(base=10))) in Primes()] # Bruno Berselli, May 05 2014

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A081092 Primes having a prime number of 1's in their binary representation.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443

Offset: 1

Reinhard Zumkeller, Mar 05 2003

Same as primes with prime binary digit sum.
Primes with prime decimal digit sum are A046704.
Sum_{a(n) < x} 1/a(n) is asymptotic to log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 09 2012
A049084(A000120(a(n))) > 0; A081091, A000215 and A081093 are subsequences.

			15th prime = 47 = '101111' with five 1's, therefore 47 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Cf. A000040, A000120, A046704, A081093.

Subsequence of A052294.

a081092 n = a081092_list !! (n-1)
a081092_list = filter ((== 1) . a010051') a052294_list
-- Reinhard Zumkeller, Nov 16 2012

q:= n-> isprime(n) and isprime(add(i,i=Bits[Split](n))):
select(q, [$1..500])[];  # Alois P. Heinz, Sep 28 2023

Clear[BinSumOddQ];BinSumPrimeQ[a_]:=Module[{i,s=0},s=0;For[i=1,i<=Length[IntegerDigits[a,2]],s+=Extract[IntegerDigits[a,2],i];i++ ];PrimeQ[s]]; lst={};Do[p=Prime[n];If[BinSumPrimeQ[p],AppendTo[lst,p]],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime[Range[100]], PrimeQ[Apply[Plus, IntegerDigits[#, 2]]] &] (* Jonathan Sondow, Jun 09 2012 *)

lista(nn) = {forprime(p=2, nn, if (isprime(hammingweight(p)), print1(p, ", ")););} \\ Michel Marcus, Jan 16 2015

from sympy import isprime
def ok(n): return isprime(n.bit_count()) and isprime(n)
print([k for k in range(444) if ok(k)]) # Michael S. Branicky, Dec 27 2023

A070027 Prime numbers whose initial, all intermediate and final iterated sums of digits are primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 83, 101, 113, 131, 137, 151, 173, 191, 223, 227, 241, 263, 281, 311, 313, 317, 331, 353, 401, 421, 443, 461, 599, 601, 641, 797, 821, 887, 911, 977, 1013, 1019, 1031, 1033, 1051, 1091, 1103, 1109, 1123, 1163, 1181, 1213

Offset: 1

Rick L. Shepherd, Apr 14 2002

Subsequence of A046704; actually, exactly those numbers for which the orbit under A007953 is a subset of A046704. - M. F. Hasler, Jun 28 2009
Supersequences: A046704 is primes p with digit sum s(p) also prime; A207294 is primes p with s(p) and s(s(p)) also prime.
Disjoint sequences: A104213 is primes p with s(p) not prime; A207293 is primes p with s(p) also prime, but not s(s(p)); A213354 is primes p with s(p) and s(s(p)) also prime, but not s(s(s(p))); A213355 is smallest prime p with k-fold digit sum s(s(..s(p)..)) also prime for all k < n, but not for k = n. - Jonathan Sondow, Jun 13 2012

			599 is a term because 599, 5+9+9 = 23 and 2+3 = 5 are all prime. 2999 is a term because 2999, 2+9+9+9 = 29, 2+9 = 11 and 1+1 = 2 are all prime. See A062802 and A070026 for related comments.

Alex Costea, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Glyn Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Cf. A070026 (a supersequence), subsequences: A062802, A070028, A070029.

Cf. A007953, A046704, A104213, A207293, A207294, A213354, A213355.

dspQ[n_] := TrueQ[Union[PrimeQ[NestWhileList[Plus@@IntegerDigits[#] &, n, # > 9 &]]] == {True}]; Select[Prime[Range[200]], dspQ] (* Alonso del Arte, Aug 17 2011 *)
isdpQ[n_]:=AllTrue[Rest[NestWhileList[Total[IntegerDigits[#]]&,n,#>9&]],PrimeQ]; Select[Prime[Range[300]],isdpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 12 2017 *)

isA070027(n)={ while(isprime(n), n<9 && return(1); n=vector(#n=eval(Vec(Str(n))),i,1)*n~)} \\ M. F. Hasler, Jun 28 2009

from sympy import isprime
def ok(n): return isprime(n) and (n < 10 or ok(sum(map(int, str(n)))))
print([k for k in range(2, 1214) if ok(k)]) # Michael S. Branicky, May 22 2025

Prime p is a term if and only if p < 10 or A007953(p) is a term. - Michael S. Branicky, May 22 2025

A088136 Primes such that sum of first and last digits is prime.

Original entry on oeis.org

11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 131, 151, 181, 191, 211, 223, 229, 233, 239, 241, 251, 263, 269, 271, 281, 283, 293, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 601, 607, 617, 631, 641, 647, 661, 677, 691

Offset: 1

Zak Seidov, Sep 20 2003

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

Cf. A008040 (primes), A010051 (isprime), A000030 (first digit of n), A010879 (last digit of n).

Cf. A046704, A007953, A088133, A088134, A088135.

Select[Prime[Range[400]],PrimeQ[First[IntegerDigits[#]]+ Last[ IntegerDigits[ #]]]&] (* Harvey P. Dale, Jun 23 2017 *)

select( {is_A088136(p)=isprime(p\10^logint(p,10)+p%10)&&isprime(p)}, primes(99)) \\ M. F. Hasler, Apr 23 2024

from sympy import isprime, primerange
def ok(p): s = str(p); return isprime(int(s[0]) + int(s[-1]))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(691)) # Michael S. Branicky, Nov 23 2021

A062088 Primes with every digit a prime and the sum of the digits a prime.

Original entry on oeis.org

2, 3, 5, 7, 23, 223, 227, 337, 353, 373, 557, 577, 733, 757, 773, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3527, 3727, 5233, 5237, 5273, 5323, 5527, 7237, 7253, 7523, 7723, 7727, 22573, 23327, 25237, 25253, 25523, 27253, 27527, 32233, 32237, 32257

Offset: 1

Amarnath Murthy, Jun 16 2001

			2357 is a prime, each digit is a prime and the sum of digits = 17 is also a prime, so 2357 is a term.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 718 terms from Marius A. Burtea)

Intersection of A019546 and A046704.

prim=primes(1000000);
m=1;
for u=1:100;
    v=prim(u);
    nc=dec2base(v,10)-'0';
    s=sum(nc);
         if and(isprime(nc)==1,isprime(s)==1)
             sol(m)=v;
             m=m+1;
         end
end
sol; % Marius A. Burtea, Dec 08 2018

aQ[p_] := PrimeQ[p] && Module[{d = IntegerDigits[p]}, PrimeQ[Total[d]] && LengthWhile[d, PrimeQ[#] &] == Length[d]]; Select[Range[33000], aQ] (* Amiram Eldar, Dec 08 2018 *)

isok(p) = isprime(p) && isprime(sumdigits(p)) && (#select(x->(! isprime(x)), digits(p)) == 0); \\ Michel Marcus, Dec 08 2018

from sympy import isprime
from itertools import count, islice, product
def agen():
    yield from [2, 3, 5, 7]
    for d in count(2):
        for left in product("2357", repeat=d-1):
            for end in "37":
                ts = "".join(left) + end
                if isprime(sum(map(int, ts))):
                    t = int(ts)
                    if isprime(t): yield t
print(list(islice(agen(), 46))) # Michael S. Branicky, Sep 23 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 21 2001

A054750 Smallest prime number whose digits sum to n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 29, 67, 89, 199, 599, 2999, 4999, 29989, 59999, 79999, 389999, 989999, 6999899, 8989999, 59899999, 89999999, 289999999, 799999999, 3999998999, 19999997999, 79999999999, 399999998999, 599999899999, 999998999999

Offset: 1

G. L. Honaker, Jr., Apr 24 2000

a(n) >= A051885(A000040(n)). Indices n for which the equality holds are listed in A055019.

a(n) >= A046864(n). - Michel Marcus, Nov 01 2015

			a(7)=89 because 8+9=17 and 17 is the 7th prime.

Chris Caldwell and G. L. Honaker, Jr., Prime curio for 8999

Cf. A000040, A067180, A046704, A046864, A068951, A007605.

a[n_]:=Module[{k=2}, While[DigitSum[k]!=Prime[n], k=NextPrime[k]]; k]; Array[a,15] (* Stefano Spezia, Mar 27 2025 *)

a(n) = {my(k=2); my(p=prime(n)); while((sumdigits(k) != prime(n)), k=nextprime(k+1)); k;} \\ Michel Marcus, Nov 01 2015

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 31 2000

Edited and extended by Robert G. Wilson v, Feb 26 2002

A091366 Primes p such that the sum of the cubes of the digits of p is prime.

Original entry on oeis.org

11, 101, 113, 131, 139, 151, 193, 199, 223, 227, 241, 263, 269, 281, 283, 311, 337, 353, 359, 373, 421, 449, 461, 463, 487, 557, 577, 593, 599, 641, 643, 661, 733, 757, 821, 823, 827, 829, 883, 887, 919, 953, 991, 997, 1013, 1031, 1039, 1051, 1093, 1103, 1123

Offset: 1

Chuck Seggelin, Jan 03 2004

Apparently, in most cases the sum of the digits of such primes is also prime, see A091365 for the exceptions.

I conjecture the contrary: the relative density of numbers in this sequence with prime digit sum is 0. - Charles R Greathouse IV, Sep 08 2010

			a(1)=11 because 1^3 + 1^3 = 2 which is prime. a(10)=227 because 2^3 + 2^3 + 7^3 = 359 which is prime.

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

Cf. A046704 (primes whose digits sum to a prime) A052034 (primes whose digits squared sum to a prime) A091365 (primes whose digits cubed sum to a prime but whose digits do not sum to a prime).

Select[Prime[Range[2, 200]], PrimeQ[Total[IntegerDigits[#]^3]]&] (* Vincenzo Librandi, Apr 13 2013 *)

is(n)=my(v);if(!isprime(n),return(0));v=eval(Vec(Str(n)));isprime(sum(i=1,#v,v[i]^3)) \\ Charles R Greathouse IV, Sep 08 2010

a(44) = 997 inserted by Charles R Greathouse IV, Sep 08 2010

cp's OEIS Frontend

A075177 Indices of additive primes - primes with prime sum-of-digits, see A046704.

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A156604 a(1)=2; for n > 0, a(n+1) is the smallest prime of A046704 larger than a(n) such that the sequence of digit sums of a(n) is nondecreasing.

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A146972 Twin primes in A046704.

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A028834 Numbers whose sum of digits is a prime.

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A081092 Primes having a prime number of 1's in their binary representation.

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A070027 Prime numbers whose initial, all intermediate and final iterated sums of digits are primes.

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A088136 Primes such that sum of first and last digits is prime.

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