A048519 -id:A048519 - cp's OEIS frontend

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

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","))

A320879 Primes such that iteration of A062028 (n + its digit sum) yields 7 primes in a row.

Original entry on oeis.org

286330897, 10858338851, 12869802851, 15845166851, 29837412851, 45480846799, 56676324799, 56676324863, 68105187851, 73915118861, 114737845853, 129282912851, 154648223809, 155738371853, 207036953861, 271077075851, 358515148853, 373169411809, 373169411861, 395705343799

Offset: 1

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

The first 15 terms are immediately calculated from A320878(1..200) using the formula.

Lars Blomberg, Table of n, a(n) for n = 1..445 (Terms < 10^14)
Carlos Rivera, Puzzle 163. P+SOD(P)

Cf. A047791, A048519, A062028 (n + digit sum of n).
Cf. A048523 .. A048527.
a(1) = A090009(8) = start of first chain of 8 primes under iteration of A062028.
Subsequence of A320878; A320880 is a subsequence.

is_A320879(n)=isprime(n=A062028(n))&& is_A320878(n) \\ If possible, use this to select terms from a sufficiently large precomputed array A320878:
A320879 = select( is_A320879, A320878) \\ or, in that case, the more efficient:
A320879 = select( p->setsearch(A320878, A062028(p)), A320878) \\ or: vecextract(A320878, select(i->A062028(A320878[i])==A320878[i+1], [1..#A320878-1]))

A320879 = { n in A320878 | A062028(n) in A320878 } = { n = A320878(k) | A062028(n) = A320878(k+1) }.

a(16)-a(20) from Lars Blomberg, Feb 10 2019

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","))

A048525 Primes for which only three iterations of 'Prime plus its digit sum equals a prime' are possible.

Original entry on oeis.org

277, 1559, 5779, 7489, 11279, 15091, 22093, 37811, 43579, 46279, 48541, 49957, 53479, 54751, 60589, 68473, 72883, 74821, 83621, 85621, 90793, 91921, 93901, 97501, 107981, 110899, 111799, 120193, 153379, 157739, 170299, 180731, 184441

Offset: 1

Patrick De Geest, May 15 1999

			prime 7489 -> 7489 + (7+4+8+9) = prime 7517 -> 7517 + (7+5+1+7) = prime 7537 -> 7537 + (7+5+3+7) = prime 7559 -> next iteration yields composite 7585.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A047791, A048519.

tiQ[p_]:=Boole[PrimeQ[NestList[#+Total[IntegerDigits[#]]&,p,4]]]=={1,1,1,1,0}; Select[Prime[Range[20000]],tiQ] (* Harvey P. Dale, Dec 02 2024 *)

A048526 Primes for which only four iterations of 'Prime plus its digit sum equals a prime' are possible.

Original entry on oeis.org

37783, 85601, 259631, 268721, 350941, 371939, 378901, 516521, 665111, 733331, 883331, 967781, 1047929, 1056521, 1081721, 1258811, 1427411, 1480573, 1515929, 1584901, 1614929, 1842131, 1875311, 1885981, 2027801, 2044873, 2450531

Offset: 1

Patrick De Geest, May 15 1999

			prime 37783 -> 37783 + (3+7+7+8+3) = prime 37811 -> 37811 + (3+7+8+1+1) = prime 37831 -> 37831 + (3+7+8+3+1) = prime 37853 -> 37853 + (3+7+8+5+3) = prime 37879 -> next iteration yields composite 37913.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A047791, A048519.

Changed offset by Lars Blomberg, Dec 05 2013

A244863 Semiprimes whose digit sum is a perfect square.

Original entry on oeis.org

4, 9, 10, 22, 121, 169, 178, 187, 202, 259, 295, 301, 358, 394, 466, 493, 529, 538, 565, 583, 655, 718, 745, 763, 781, 799, 817, 835, 862, 871, 889, 898, 934, 943, 961, 979, 1003, 1111, 1159, 1177, 1186, 1195, 1267, 1285, 1294, 1339, 1357, 1366, 1393, 1438, 1465

Offset: 1

K. D. Bajpai, Jul 07 2014

Subsequence of A028839.

			178 is in the sequence because 178 = 2*89 (semiprime) and 1+7+8 = 16 (square).
187 is in the sequence because 187 = 11*17 (semiprime) and 1+8+7 = 16 (square).

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A007953, A028839, A048519.

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

select(n -> numtheory:-bigomega(n)=2 and issqr(convert(convert(n,base,10),`+`)),
[$1..3000]); # Robert Israel, Jul 09 2014

Select[Range[3000], PrimeOmega[#] == 2 && IntegerQ[Sqrt[Apply [Plus, IntegerDigits[#]]]] &]

A244733 Semiprimes sp such that sp plus its digit sum is a perfect square.

Original entry on oeis.org

38, 86, 161, 614, 662, 998, 1145, 1355, 1829, 2189, 2483, 4607, 5027, 5315, 6377, 7199, 8258, 11435, 13214, 15611, 17933, 19574, 20153, 21305, 21878, 24014, 26867, 30599, 32738, 34199, 36077, 38387, 38777, 40778, 42422, 46211, 51509, 52874, 56618, 58541, 59987

Offset: 1

K. D. Bajpai, Jul 12 2014

			86 is in the sequence because 86 = 2* 43, which is semiprime. Also, 86 + (8 + 6) = 100 = 10^2.
614 is in the sequence because 614 = 2* 307, which is semiprime. Also, 614 + (6 + 1 + 4) = 625 = 25^2.

K. D. Bajpai, Table of n, a(n) for n = 1..1150

Cf. A007953, A028839, A048519, A107288, A244863.

Select[Range[50000], PrimeOmega[#] == 2 && IntegerQ[Sqrt[# + Apply[Plus, IntegerDigits[#]]]] &]

A245064 Primes p such that p minus its digit sum is a perfect cube.

Original entry on oeis.org

2, 3, 5, 7, 31, 37, 223, 227, 229, 743, 1741, 1747, 3391, 5851, 5857, 9281, 9283, 13841, 19709, 27011, 27017, 35963, 35969, 46681, 46687, 59341, 74101, 91141, 110603, 110609, 132679, 373273, 474581, 474583, 729023, 804383, 1061227, 1259743, 1259749, 1481573, 2000393

Offset: 1

K. D. Bajpai, Jul 11 2014

			37 is in the sequence because it is prime. Also, 37 - (3 + 7 ) = 27 = 3^3: a perfect cube.
743 is in the sequence because it is prime. Also, 743 - (7 + 4 + 3) = 729 = 9^3: a perfect cube.

K. D. Bajpai and Jens Kruse Andersen, Table of n, a(n) for n = 1..10000 (first 274 terms from K. D. Bajpai)

Cf. A000578, A048519, A107288.

dmax:= 9; # to get all entries < 10^dmax
cmax:= floor(10^(dmax/3));
count:= 0;
for m from 0 to cmax do
   for p from m^3 to m^3 + 9*dmax do
      if p - convert(convert(p,base,10),`+`) = m^3 and isprime(p) then
         count:= count+1;
         A[count]:= p;
      fi
   od
od;
{seq(A[i],i=1..count)}; # Robert Israel, Jul 15 2014

Select[Prime[Range[200000]], IntegerQ[CubeRoot[# - Apply[Plus, IntegerDigits[#]]]] &]

digsum(n) = my(d=eval(Vec(Str(n)))); sum(i=1, #d, d[i])
s=[]; forprime(p=2, 2002000, if(ispower(p-digsum(p), 3), s=concat(s, p))); s \\ Colin Barker, Jul 15 2014

cp's OEIS Frontend

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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A320879 Primes such that iteration of A062028 (n + its digit sum) yields 7 primes in a row.

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

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Maple

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

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Maple

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A048525 Primes for which only three iterations of 'Prime plus its digit sum equals a prime' are possible.

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Mathematica

A048526 Primes for which only four iterations of 'Prime plus its digit sum equals a prime' are possible.

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A244863 Semiprimes whose digit sum is a perfect square.

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