A048520 -id:A048520 - cp's OEIS frontend

A048519 Prime plus its digit sum equals a prime.

11, 13, 19, 37, 53, 59, 71, 73, 97, 101, 103, 127, 149, 163, 167, 181, 233, 257, 271, 277, 293, 307, 367, 383, 389, 419, 431, 433, 479, 499, 509, 547, 563, 587, 617, 631, 701, 727, 743, 787, 811, 839, 857, 859, 947, 1009, 1049, 1061, 1087, 1153, 1171

Offset: 1

Patrick De Geest, May 15 1999

For any prime p, p +- digitsum(p, base b) can't be prime unless the base b is even, since in an odd base, an odd number always has an odd digit sum (powers of b are congruent to b (mod 2)), so p +- digitsum(p, base b) is even for odd b. This sequence is for b = 10 (where "-" is also excluded, see comment in A243442), see A243441 for b = 2. - M. F. Hasler, Nov 06 2018

See subsequence A048523 for primes which only once give another prime under iteration of A062028, and A048524 .. A048527, A320878 .. A320880 for primes starting longer chains. See A090009 for their initial terms, starting the earliest chain of given length. - M. F. Hasler, Nov 09 2018

			a(9) = prime 97 because 97 + sum-of-digits(97) = 97 + 16 = 113 also a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007953 (digit sum), A062028 (n + digit sum of n), A047791 (A062028(n) is prime), A048520.
Cf. A006378, A107740.
Cf. A000040, A010051, A065073.
Cf. A048523 .. A048527, A320878, A320879, A320880, A090009.

a048519 n = a048519_list !! (n-1)
a048519_list = map a000040 $ filter ((== 1) . a010051' . a065073) [1..]
-- Reinhard Zumkeller, Sep 27 2014

[p: p in PrimesUpTo(1200) | IsPrime(q) where q is p+&+Intseq(p)]; // Vincenzo Librandi, Jan 30 2018

select(n -> isprime(n) and isprime(n + convert(convert(n,base,10),`+`)), [$1..10^4]); # Robert Israel, Aug 10 2014

Select[Prime[Range[500]],PrimeQ[#+Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Oct 03 2011 *)

select( is(p)=isprime(p+sumdigits(p))&&isprime(p), primes([0,2000])) \\ M. F. Hasler, Aug 08 2014, edited Nov 09 2018

Primes in A047791, i.e., intersection of A047791 and A000040. - M. F. Hasler, Nov 08 2018

A048521 Primes expressible as the sum of an integer plus its digit sum.

Original entry on oeis.org

2, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 313

Offset: 1

Patrick De Geest, May 15 1999

			a(24) = prime 113 which is 106 + (1+0+6) (or 97 + (9+7)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A047791, A048520.
Cf. A006378, A062028, A107741.
Cf. A000040, A107740, A049084.

a048521 n = a048521_list !! (n-1)
a048521_list = map a000040 $ filter ((> 0) . a107740) [1..]
-- Reinhard Zumkeller, Sep 27 2014

t={};Do[p=Prime[n];c=0;i=1;While[iJayanta Basu, May 03 2013 *)
Union[Select[Table[n+Total[IntegerDigits[n]],{n,400}],PrimeQ]] (* Harvey P. Dale, Jul 14 2014 *)

A107740(A049084(a(n))) > 0.

Formula and also offset corrected by Reinhard Zumkeller, Sep 27 2014

A320878 Primes such that iteration of A062028 (n + its digit sum) yields 6 primes in a row.

Original entry on oeis.org

286330897, 286330943, 388098901, 955201943, 1776186851, 1854778853, 2559495863, 2647782901, 3517793911, 3628857863, 3866728909, 3974453911, 4167637819, 4269837799, 5083007887, 5362197829, 5642510933, 6034811933, 8180784851, 8214319903

Offset: 1

Zak Seidov and M. F. Hasler, Nov 08 2018

In contrast to A048523, ..., A048527, this definition uses "at least" for the number of successive primes. This allows easier computation of subsequences of terms which yield even more primes in a row.

One can nonetheless compute the terms of this sequence by considering possible pre-images under A062028 of terms of A048527. This gives the terms which yield exactly 6 primes in a row (i.e., A320878 \ A320879), and one has to take the union with further iterates of this procedure (which successively yields A320879 \ A320880, etc).

Lars Blomberg, Table of n, a(n) for n = 1..7626 (Terms < 10^14. The first 200 from M. F. Hasler)
Carlos Rivera, Puzzle 163. P+SOD(P)

Cf. A062028 (n + digit sum of n), A047791 (A062028(n) is prime), A048519 (primes among these).
Cf. A048523 .. A048527, A320879.
a(1) = A090009(7) = start of first chain of 7 primes under iteration of A062028.
Cf. A320881, A048520.
Cf. A230093 (number of m s.th. m + (sum of digits of m) = n) and references there.

is_A320878(n,p=n)={for(i=1,6, isprime(p=A062028(p))||return);isprime(n)}
forprime(p=286e6,,is_A320878(p)&& print1(p","))
/* much faster, using the precomputed array A048527, as follows: */
PP(n)=select(p->p+sumdigits(p)==n,primes([n-9*#digits(n),n-2])) \\ Returns list of prime predecessors for A062028. (PP(n) nonempty <=> n in A320881.)
A320878=[]; my(S=A048527); while(#S=Set(concat(apply(PP,S))), A320878=setunion(A320878,S)) \\ Yields 211 terms from A048527[1..3000]

Numbers n in A048519 for which A062028(n) is in A048527, form the subset A320878 \ A320879.

A065073 a(n) = prime(n) + (sum of digits of prime(n)).

Original entry on oeis.org

4, 6, 10, 14, 13, 17, 25, 29, 28, 40, 35, 47, 46, 50, 58, 61, 73, 68, 80, 79, 83, 95, 94, 106, 113, 103, 107, 115, 119, 118, 137, 136, 148, 152, 163, 158, 170, 173, 181, 184, 196, 191, 202, 206, 214, 218, 215, 230, 238, 242, 241, 253, 248, 259, 271, 274, 286

Offset: 1

Bodo Zinser, Nov 09 2001

			a(5) = 13 because p(5) = 11 and 11 + (1 + 1) = 13.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A062028, A007953, A000040.

Cf. A047791, A048519, A048520, A006378, A107740.

a065073 = a062028 . a000040  -- Reinhard Zumkeller, Sep 27 2014

[NthPrime(n) + &+Intseq(NthPrime(n), 10): n in [1..80]]; // Vincenzo Librandi, Nov 07 2018

Table[ Prime[n] + Apply[ Plus, IntegerDigits[ Prime[n]]], {n, 1, 75} ]

forprime(p=2,300,print1(p+sumdigits(p),",")) \\ Edited by M. F. Hasler, Nov 06 2018

A065073(n)=sumdigits(n=prime(n))+n \\ M. F. Hasler, Nov 06 2018

a(n) = A062028(A000040(n)). - M. F. Hasler, Nov 06 2018

More terms from Larry Reeves (larryr(AT)acm.org) and Robert G. Wilson v, Nov 13 2001

A320866 Primes such that p + digitsum(p, base 4) is again a prime.

Original entry on oeis.org

5, 7, 13, 17, 19, 37, 59, 67, 97, 127, 173, 193, 223, 233, 277, 359, 379, 439, 499, 563, 569, 599, 607, 631, 653, 691, 733, 769, 811, 821, 829, 877, 919, 929, 937, 967, 1009, 1019, 1087, 1093, 1163, 1193, 1213, 1223, 1229, 1297, 1319, 1373, 1399, 1423, 1481, 1483, 1559, 1571, 1597, 1613, 1619, 1627, 1657, 1699, 1733, 1777

Offset: 1

M. F. Hasler, Nov 06 2018

Such primes exist only for even bases b. See A243441, A320867, A320868 and A048519 for the analog in base 2, 6, 8 and 10, respectively. Also, as in base 10, there are no such primes (except 5 and 7) when + is changed to -, see comment in A243442.

			5 = 4 + 1 = 11[4] (in base 4), and 5 + 1 + 1 = 7 is again prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A047791, A048519 (base 10 analog), A048520, A006378, A107740, A243441 (base 2 analog: p + Hammingweight(p) is prime), A243442 (analog for p - Hammingweight(p)), A320867 (analog for base 6), A320868 (analog for base 8).

Select[Prime[Range[300]],PrimeQ[#+Total[IntegerDigits[#,4]]]&] (* Harvey P. Dale, Feb 06 2020 *)

forprime(p=1,1999,isprime(p+sumdigits(p,4))&&print1(p","))

A320867 Primes such that p + digitsum(p, base 6) is again a prime.

Original entry on oeis.org

11, 19, 23, 31, 41, 53, 61, 79, 109, 137, 151, 167, 179, 229, 233, 263, 271, 331, 347, 359, 419, 439, 467, 541, 557, 587, 599, 607, 653, 719, 797, 809, 839, 863, 997, 1019, 1049, 1097, 1109, 1237, 1283, 1291, 1301, 1321, 1373, 1427, 1439, 1487, 1523, 1549, 1607, 1621, 1697, 1709, 1733, 1741, 1867

Offset: 1

M. F. Hasler, Nov 06 2018

Such primes exist only for an even base b. See A048519, A243441, A320866 and A320868 for the analog in base 10, 2, 4 and 8, respectively. Also, as in base 10, there are no such primes (except 7 and 11) when + is changed to -, see comment in A243442.

			11 = 6 + 5 = 15[6] (in base 6), and 11 + 1 + 5 = 17 is again prime.

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

Cf. A047791, A048519 (base 10 analog), A048520, A006378, A107740, A243441 (base 2 analog: p + Hammingweight(p) is prime), A243442 (analog for p - Hammingweight(p)), A320866 (analog for base 4), A320868 (analog for base 8).

filter:= n -> isprime(n) and isprime(n+convert(convert(n,base,6),`+`)):
select(filter, [seq(i,i=3..2000,2)]); # Robert Israel, Mar 22 2020

forprime(p=1,1999,isprime(p+sumdigits(p,6))&&print1(p","))

A320868 Primes such that p + digitsum(p, base 8) is again a prime.

Original entry on oeis.org

13, 29, 31, 41, 47, 61, 67, 71, 83, 97, 157, 193, 229, 241, 271, 283, 373, 397, 409, 431, 449, 467, 503, 587, 601, 607, 761, 787, 929, 971, 991, 1039, 1087, 1091, 1163, 1181, 1213, 1217, 1237, 1249, 1289, 1291, 1307, 1423, 1453, 1471, 1511, 1543, 1553, 1559, 1627, 1657, 1741, 1811, 1847, 1867, 1973, 1999

Offset: 1

M. F. Hasler, Nov 06 2018

Such primes exist only for an even base b. See A048519, A243441, A320866 and A320867 for the analog in base 10, 2, 4 and 6, respectively. Also, as in base 10, there are no such primes (except 11 and 13) when + is changed to -, see comment in A243442.

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

Cf. A047791, A048519 (base 10 analog), A048520, A006378, A107740, A243441 (base 2 analog: p + Hammingweight(p) is prime), A243442 (analog for p - Hammingweight(p)), A320866 (analog for base 4), A320867 (analog for base 6).

digsum:= proc(n,b) convert(convert(n,base,b),`+`) end proc:
select(p -> isprime(p) and isprime(p+digsum(p,8)), [seq(i,i=3..10000,2)]); # Robert Israel, Nov 07 2018

forprime(p=1,1999,isprime(p+sumdigits(p,8))&&print1(p","))

A320881 Numbers equal to a prime plus its digit sum.

Original entry on oeis.org

4, 6, 10, 13, 14, 17, 25, 28, 29, 35, 40, 46, 47, 50, 58, 61, 68, 73, 79, 80, 83, 94, 95, 103, 106, 107, 113, 115, 118, 119, 136, 137, 148, 152, 158, 163, 170, 173, 181, 184, 191, 196, 202, 206, 214, 215, 218, 230, 238, 241, 242, 248, 253, 259, 271, 274, 281, 286, 292, 293, 296, 307, 316

Offset: 1

M. F. Hasler, Nov 08 2018

Sequence A048520 lists the primes in this sequence.

			a(1) = 4 = 2 + 2 = (the smallest prime, 2 = prime(1)) + (digit sum of 2).
Similarly, a({2, 3, 5}) = 2*prime({2, 3, 4}), since the digit sum of single-digit primes is the prime itself.
a(4) = 13 = 11 + (1 + 1) = A048520(1), the first prime in this sequence.
a(6) = 17 = 13 + (1 + 3) = A048520(2), the second prime in this sequence.

Cf. A062028 (n + its digit sum), A047791 (A062028(n) is prime), A048519 (primes in A047791).

is_A320881(n)=select(p->p+sumdigits(p)==n, primes([n-9*#digits(n), n-2])) \\ Returns the list of all "solutions"; this has the boolean value of true iff the list is nonempty. - M. F. Hasler, Nov 08 2018

A320869 Primes such that p + digitsum(p, base 16) is again a prime.

Original entry on oeis.org

17, 19, 23, 29, 31, 53, 59, 89, 127, 149, 151, 157, 179, 181, 211, 223, 241, 251, 263, 269, 331, 359, 367, 397, 419, 431, 449, 457, 461, 463, 487, 541, 563, 571, 593, 599, 601, 631, 659, 661, 701, 733, 761, 769, 809, 811, 839, 907, 911, 941, 971, 997, 1049, 1087, 1109, 1171, 1201, 1237, 1283, 1289, 1291

Offset: 1

M. F. Hasler, Nov 06 2018

Such primes exist only for an even base b. See A048519, A243441, A320866, A320867 and A320868 for the analog in base 10, 2, 4, 6 and 8, respectively. Also, as in base 10, there are no such primes when + is changed to -, see comment in A243442.

			17 = 16 + 1 = 11[16] (in base 16), and 17 + 1 + 1 = 19 is again prime.

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

Cf. A047791, A048519 (base 10 analog), A048520, A006378, A107740, A243441 (base 2 analog: p + Hammingweight(p) is prime), A243442 (analog for p - Hammingweight(p)), A320866 (analog for base 4), A320867 (analog for base 6), A320868 (analog for base 8).

digsum:= (n,b) -> convert(convert(n,base,b),`+`):
select(p -> isprime(p) and isprime(p+digsum(p,16)), [2,seq(i,i=3..1000,2)]); # Robert Israel, Nov 07 2018

forprime(p=1,1999,isprime(p+sumdigits(p,16))&&print1(p","))

A320870 Irregular table: row n >= 0 lists numbers m >= 0 such that n = A062028(m) := m + sum of digits of m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 11, 7, 12, 8, 13, 9, 14, 15, 20, 16, 21, 17, 22, 18, 23, 19, 24, 25, 30, 26, 31, 27, 32, 28, 33, 29, 34, 35, 40, 36, 41, 37, 42, 38, 43, 39, 44, 45, 50, 46, 51, 47, 52, 48, 53, 49, 54, 55, 60, 56, 61, 57, 62, 58, 63, 59, 64, 65, 70, 66, 71, 67, 72, 68, 73, 69, 74, 75, 80, 76, 81, 77, 82, 78, 83, 79, 84, 85, 90

Offset: 0

M. F. Hasler, Nov 09 2018

Row lengths are given by A230093.

			The first nonempty rows are:
    n  | list of m
    0  | 0        // since 0 = 0 + 0
    2  | 1        // since 2 = 1 + 1
    4  | 2        // etc.
    6  | 3        // Below 10 every odd row is empty, but thereafter,
    8  | 4        // only rows 20, 31, 42, ..., 108 (steps of 11),
   10  | 5        // 110, 121, 132, ..., 198, etc. are empty.
   11  | 10       // Since 11 = 10 + (1 + 0)
   12  | 6
   13  | 11       // The first prime that yields a prime: 11 + (1 + 1) = 13.
     (...)
  100  | 86       // The first row of length 2 is 101:
  101  | 91, 100  // 101 = 91 + (9 + 1) = 100 + (1 + 0 + 0)
  102  | 87
     (...)

Robert Israel, Table of n, a(n) for n = 0..9971 (rows 0 to 10000, flattened)

Cf. A007953 (sum of digits of n), A062028 (n + digit sum of n).
Cf. A230093 (number of m such that m + (sum of digits of m) is n).
Cf. A006064 (least m with row length n),
Cf. A003052 (Self or Colombian numbers: rows of length 0), A006378 (Colombian primes).
Cf. A320881 (indices of rows containing a prime), A048520 (primes among these).

N:= 100: # for rows 0 to N, flattened
for i from 0 to N do V[i]:= NULL od:
for i from 0 to N-1 do
  v:= convert(convert(i,base,10),`+`);
  if v <= N then V[v]:= V[v],i fi
od:
seq(V[i],i=1..N); # Robert Israel, Jul 21 2025

A320870_row(n)=if(n,select(m->m+sumdigits(m)==n,[max(n-9*logint(n,10)+8,n\/2)..n-1]),[0])

cp's OEIS Frontend

A048519 Prime plus its digit sum equals a prime.

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A048521 Primes expressible as the sum of an integer plus its digit sum.

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A320878 Primes such that iteration of A062028 (n + its digit sum) yields 6 primes in a row.

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A065073 a(n) = prime(n) + (sum of digits of prime(n)).

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A320866 Primes such that p + digitsum(p, base 4) is again a prime.

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A320867 Primes such that p + digitsum(p, base 6) is again a prime.

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A320868 Primes such that p + digitsum(p, base 8) is again a prime.

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