A048519 -id:A048519 - cp's OEIS frontend

A320881 Numbers equal to a prime plus its digit sum.

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

A092529 Primes p such that both the digit sum of p plus p and the digit product of p plus p are also primes.

Original entry on oeis.org

163, 233, 293, 431, 499, 563, 617, 743, 1423, 1483, 1489, 1867, 2273, 2543, 2633, 3449, 4211, 4217, 4273, 4547, 4729, 5861, 6121, 6529, 6637, 6653, 6761, 6857, 6949, 7681, 8273, 8431, 8837, 8839, 9649, 9689

Offset: 1

Ray G. Opao, Apr 08 2004

Intersection of A048519 and A092518.

Zeros are not permitted in p; thus, for example, 101 is not included. - Harvey P. Dale, May 25 2013

			a(2) = 233: 233+(2+3+3) = 233+8 = 241, which is prime. 233+(2*3*3) = 233+18 = 251, which is prime.

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

Cf. A048519, A092518.

filter:= proc(p) local L;
  if not isprime(p) then return false fi;
  L:= convert(p,base,10);
  if member(0,L) then return false fi;
  isprime(p + convert(L,`+`)) and isprime(p + convert(L,`*`))
end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Feb 20 2024

pppQ[n_]:=Module[{idn=IntegerDigits[n]},!MemberQ[idn,0]&&And@@PrimeQ[ {n+ Total[idn], n+Times@@idn}]]; Select[Prime[Range[1200]],pppQ] (* Harvey P. Dale, May 25 2013 *)

More terms from Robert G. Wilson v, Apr 10 2004

A128717 Primes that yield another prime if one adds either the sum of its digits or the product of its digits.

Original entry on oeis.org

101, 103, 163, 233, 293, 307, 431, 499, 509, 563, 617, 701, 743, 1009, 1049, 1061, 1087, 1409, 1423, 1483, 1489, 1601, 1607, 1801, 1867, 2017, 2039, 2053, 2273, 2543, 2633, 2903, 3041, 3067, 3089, 3449, 3607, 4013, 4057, 4079, 4211, 4217, 4273, 4507

Offset: 1

J. M. Bergot, Jun 27 2007

			163 + (1+6+3) = 173, 163 + 1*6*3 = 181; 173 and 181 are prime numbers.

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

Cf. A048519.

filter:= proc(n) local S;
  S:= convert(n,base,10);
  isprime(n+convert(S,`+`)) and isprime(n+convert(S,`*`))
end proc:
A:= NULL:
p:= 2:
while p < 10000 do
  if filter(p) then A:= A, p fi;
  p:= nextprime(p)
od:
A; # Robert Israel, Jul 12 2017

Select[Prime[Range[500]], PrimeQ[ # + Plus @@ IntegerDigits[ # ]] && PrimeQ[ # + Times @@ IntegerDigits[ # ]] &]

isok(n) = isprime(n) && (isprime(n+sumdigits(n)) && ((d=digits(n)) && isprime(n+prod(k=1, #d, d[k])))); \\ Michel Marcus, Jul 12 2017

Edited, corrected and extended by Stefan Steinerberger, Jul 14 2007

A230087 Primes such that prime plus its digit sum is a perfect square.

Original entry on oeis.org

2, 17, 179, 347, 467, 521, 1433, 1583, 2111, 3347, 10601, 12527, 25889, 28541, 32027, 33113, 39569, 39971, 41201, 43661, 45767, 55667, 58061, 59513, 61001, 62969, 63977, 67061, 70199, 77261, 92387, 92993, 100469, 109541, 120401, 122477, 130307, 156011, 163193

Offset: 1

K. D. Bajpai, Oct 08 2013

Number of primes obtained from the sequence ‘prime plus its digit sum is perfect square’ is 150 for n = 1 to 3*10^5, while the sequence for ‘perfect cube’ yields only 11 primes for the same range of n. Hence, sequence for ‘square’ is framed.

Subsequence of primes of A066564. - Michel Marcus, Jun 02 2015

			a(2) = 17 is prime. Digit sum of 17 = 8, 17 + 8 = 25 = 5^2.
a(5) = 467 is prime. Digit sum of 467 = 17, 467 + 17 = 484 = 22^2.

K. D. Bajpai and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 2..150 from Bajpai)

Cf. A048519.

Cf. A107288 (Primes whose digit sum is square).

[p: p in PrimesUpTo(6*10^5) | IsSquare(p+(&+Intseq(p)))]; // Vincenzo Librandi, Jun 02 2015

KD:= proc() local a,b,c,d; a:= ithprime(n);b:=add( i,i = convert((a), base, 10))(a); c:=a+b; d:=evalf(sqrt(c)); if d=floor(d) then return (a) :fi;end:seq(KD(),n=1..50000);

for(n=2,1e4,forprime(p=n^2-9*#digits(n^2),n^2, if(p+sumdigits(p) == n^2, print1(p", ")))) \\ Charles R Greathouse IV, Oct 08 2013

a(1) from Charles R Greathouse IV, Oct 08 2013

A247896 Primes that produce a different prime when one of its digits is added to it.

Original entry on oeis.org

29, 43, 61, 67, 89, 167, 227, 239, 263, 269, 281, 349, 367, 389, 439, 457, 461, 463, 487, 499, 521, 563, 601, 607, 613, 641, 643, 647, 653, 677, 683, 821, 827, 983, 1063, 1229, 1277, 1283, 1289, 1361, 1367, 1423, 1427, 1429, 1447, 1481, 1483, 1489, 1549, 1601

Offset: 1

Paolo P. Lava, Sep 26 2014

From an idea of Eric Angelini (see seqfan link).

Digit 0 is not considered because the new primes must be different from the starting numbers. Therefore, 101 is not part of the sequence, because the only prime that results from adding one of its digits is 101 + 0 = 101, which is the same number, while 601 is acceptable because 601 + 6 = 607, a prime.

			The number 29 is prime, and 29 + 2 = 31 is also prime.
The same with 487, which produces 487 + 4 = 491, a prime.

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Eric Angelini, Primes adding one of their digit to themselves (+chains)

Cf. A047791, A048519.

Cf. A000040, A010051.

a247896 n = a247896_list !! (n-1)
a247896_list = filter f a000040_list where
   f p = any ((== 1) . a010051') $
             map (+ p) $ filter (> 0) $ map (read . return) $ show p
-- Reinhard Zumkeller, Sep 27 2014

P:=proc(q) local a,b,k,n,ok;
for n from 1 to q do a:=ithprime(n); ok:=0;
for k from 1 to ilog10(a)+1 do
b:=trunc((a mod 10^k)/10^(k-1)); if b>0 then
if isprime(a+b) then ok:=1; break; fi; fi; od;
if ok=1 then print(a); fi; od; end: P(10^6);

/* Description: Generates a vector containing this kind of terms between m^u1 and m^u2 for this definition applied by adding base B digits to the original number in decimal. Here (u1,m,B)=(1,3,10) by default. */
LstThem(u2,u1=1,m=3,B=10)={
  my(L:list=List(),y);
  forprime(x=m^u1,m^u2,
    y=vecsort(digits(x,B),,8);
    if(sum(j=1,#y,y[j]&&isprime(x+y[j])),
      listput(L,x)));
  vector(#L,i,L[i])} \\ R. J. Cano, Sep 27 2014

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","))

A225519 Primes of the form p + sum of squares of digits of p, where p is prime.

Original entry on oeis.org

13, 23, 41, 67, 101, 103, 113, 131, 157, 181, 191, 227, 281, 379, 421, 457, 461, 467, 547, 659, 677, 677, 751, 809, 811, 829, 839, 877, 1039, 1039, 1091, 1093, 1109, 1187, 1201, 1223, 1319, 1361, 1439, 1453, 1531, 1567, 1571, 1613, 1663, 1693, 1753, 1789

Offset: 1

Jayanta Basu, May 09 2013

Primes generated by A076162.

			41 is a member since 41=31+(3^2+1^2).

Cf. A048519, A048520, A076162.

Sort[Select[Table[p=Prime[n]; p+Total[IntegerDigits[p]^2],{n,262}],PrimeQ]]

cp's OEIS Frontend

A320881 Numbers equal to a prime plus its digit sum.

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A092529 Primes p such that both the digit sum of p plus p and the digit product of p plus p are also primes.

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A128717 Primes that yield another prime if one adds either the sum of its digits or the product of its digits.

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A230087 Primes such that prime plus its digit sum is a perfect square.

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Magma

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A247896 Primes that produce a different prime when one of its digits is added to it.

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Haskell

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

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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.

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