A243441 -id:A243441 - 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

A243442 Primes p such that, in base 2, p - digitsum(p) is also a prime.

Original entry on oeis.org

5, 23, 71, 83, 101, 113, 197, 281, 317, 353, 359, 373, 401, 467, 599, 619, 683, 739, 751, 773, 977, 1091, 1097, 1103, 1217, 1223, 1229, 1237, 1283, 1303, 1307, 1429, 1433, 1489, 1553, 1559, 1601, 1607, 1613, 1619, 1699, 1873, 1879, 2039, 2347, 2357, 2389

Offset: 1

Anthony Sand, Jun 05 2014

In all bases b, x = n - digitsum(n) is always divisible by b-1, therefore x can be prime only in base 2 and bases b for which b-1 is prime. For example, in base 10, n - digitsum(n) is always divisible by 10 - 1 = 9 -- see A066568 and A068395. In base 8, 9 = 11, therefore 11 - digitsum(11) = 9 - 2 = 7 is divisible by 7.

			5 - digitsum(5,base=2) = 5 - digitsum(101) = 5 - 2 = 3.
23 - digitsum(10111) = 23 - 4 = 19.
71 - digitsum(1000111) = 71 - 4 = 67.
83 - digitsum(1010011) = 83 - 4 = 79.
101 - digitsum(1100101) = 101 - 4 = 97.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A243441.

Select[Prime[Range[400]],PrimeQ[#-Total[IntegerDigits[#,2]]]&] (* Harvey P. Dale, May 15 2019 *)

isok(n) = isprime(n) && isprime(n - hammingweight(n)); \\ Michel Marcus, Jun 05 2014

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","))

A321392 a(n) is the number of bases b > 1 such that prime(n) + digitsum(prime(n), base b) is prime (where prime(n) denotes the n-th prime number).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 4, 3, 5, 6, 7, 7, 7, 7, 10, 11, 10, 12, 11, 11, 12, 11, 13, 16, 14, 13, 10, 14, 13, 21, 19, 19, 17, 20, 21, 24, 26, 25, 25, 25, 23, 26, 26, 24, 26, 29, 33, 27, 30, 31, 28, 32, 33, 32, 34, 34, 34, 32, 31, 34, 37, 37, 41, 36, 38, 41, 44, 45

Offset: 1

Rémy Sigrist, Nov 08 2018

For any prime number p and base b > p, p + digitsum(p, base b) equals twice p and is not prime, hence the sequence is well defined.

For prime(n) + digitsum(prime(n), base b) to be prime, b must be even (see A320866).

			For n = 6, we have prime(6) = 13 and:
  b     13 + sumdigits(13, base b)
  ----  --------------------------
     2  16
     4  17 (prime)
     6  16
     8  19 (prime)
    10  17 (prime)
    12  15
  >=14  26
Hence, a(6) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of (n, b) such that prime(n) + sumdigits(prime(n), base 2*b) is prime and 1 <= n <= 2000 and 1 <= b <= 1000 (where the color is function of floor(prime(n) / (2*b)))

Cf. A000040, A243441, A320866, A321393.

a(n) = my (p=prime(n)); sum(b=1, p\2, isprime(p+sumdigits(p, 2*b)))

a(n) = A321393(A000040(n)).

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","))

A320882 Primes p such that repeated application of A062028 (add sum of digits) yields two other primes in a row: p, A062028(p) and A062028(A062028(p)) are all prime.

Original entry on oeis.org

11, 59, 101, 149, 167, 257, 277, 293, 367, 419, 479, 547, 617, 727, 839, 1409, 1559, 1579, 1847, 2039, 2129, 2617, 2657, 2837, 3449, 3517, 3539, 3607, 3719, 4217, 4637, 4877, 5689, 5779, 5807, 5861, 6037, 6257, 6761, 7027, 7489, 7517, 8039, 8741, 8969, 9371, 9377, 10667, 10847, 10937, 11257, 11279, 11299, 11657

Offset: 1

M. F. Hasler, Nov 06 2018

"Iterates" the idea of A048519 (p and A062028(p) are prime), also considered in A048523, A048524, A048525, A048526, A048527. (This is the union of A048524, A048525, A048526, A048527 etc. A048525(1) = 277 = a(7).)

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

Subsequence of A048519: p and A062028(p) are prime.
Cf. A047791, A048520, A006378, A107740, A243441 (p and p + Hammingweight(p) are prime), A243442 (analog for p - Hammingweight(p)).
Cf. A048523, ..., A048527, A320878, A320879, A320880: primes starting a chain of length 2, ..., 9 under iterations of A062028(n) = n + digit sum of n.

f:= n -> n + convert(convert(n,base,10),`+`):
filter:= proc(n) local x;
if not isprime(n) then return false fi;
x:= f(n);
isprime(x) and isprime(f(x))
end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Dec 17 2020

is_A320882(n,p=n)=isprime(p=A062028(p))&&isprime(A062028(p))&&isprime(n) \\ Putting isprime(n) to the end is more efficient for the frequent case when the terms are already known to be prime.
forprime(p=1,14999,isprime(q=A062028(p))&&isprime(A062028(q))&&print1(p","))

cp's OEIS Frontend

A048519 Prime plus its digit sum equals a prime.

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A243442 Primes p such that, in base 2, p - digitsum(p) is also a prime.

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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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A321392 a(n) is the number of bases b > 1 such that prime(n) + digitsum(prime(n), base b) is prime (where prime(n) denotes the n-th prime number).

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

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