A048519 -id:A048519 - cp's OEIS frontend

A356570 a(n) is the first prime that starts a sequence of exactly n consecutive primes that are in A048519.

19, 11, 97, 72461, 346373, 2587093, 1534359019, 1010782220887

Offset: 1

J. M. Bergot and Robert Israel, Aug 12 2022

a(n) = prime(k) for the least possible k such that prime(i) for k <= i < k+n has the property that it plus its digit sum is prime, but prime(k-1) and prime(k+n) do not have the property.

			a(3) = 97 because 97, 101, 103 are 3 consecutive primes with 97+9+7 = 113, 101+1+0+1 = 103, 103+1+0+3=107, all prime, but the prime before 97 is 89 and the prime after 103 is 107, and 89+8+9 = 106 and 107+1+0+7 = 115 are not prime; 97 is the least prime for which this works.

Cf. A048519.

N:= 6: # for a(1)..a(N)
V:= Vector(N): count:= 0:
p:= 2: s:= 0:
while count < N do
  p:= nextprime(p)
  if isprime(p+convert(convert(p,base,10),`+`)) then
  if s = 0 then q:= p fi;
  s:= s+1
else
  if s >= 1 and s <= N and V[s] = 0 then
    V[s]:= q; count:= count+1
  fi;
  s:= 0
fi
od:
convert(V,list);

seq[len_, pmax_] := Module[{s = Table[0, {len}], v = {}, p = 2, c = 0, pfirst = 2, i}, While[c < len && p < pmax, If[PrimeQ[p + Plus @@ IntegerDigits[p]], AppendTo[v, p]; If[pfirst == 0, pfirst = p], i = Length[v]; v = {}; If[0 < i <= len && s[[i]] == 0, s[[i]] = pfirst]; pfirst = 0]; p = NextPrime[p]]; s]; seq[6, 10^7] (* Amiram Eldar, Aug 14 2022 *)

a(8) from Amiram Eldar, Aug 15 2022

A048520 Primes expressible as the sum of a prime plus its digit sum.

Original entry on oeis.org

13, 17, 29, 47, 61, 73, 79, 83, 103, 107, 113, 137, 163, 173, 181, 191, 241, 271, 281, 293, 307, 317, 383, 397, 409, 433, 439, 443, 499, 521, 523, 563, 577, 607, 631, 641, 709, 743, 757, 809, 821, 859, 877, 881, 967, 1019, 1063, 1069, 1103, 1163, 1181

Offset: 1

Patrick De Geest, May 15 1999

			a(15) = 181 which is 167 + (1+6+7).

M. F. Hasler, Table of n, a(n) for n = 1..10000, Nov 08 2018

Cf. A007953, A047791, A048519.

Sort[Select[Table[p=Prime[n];p+Total[IntegerDigits[p]],{n,195}],PrimeQ]] (* Jayanta Basu, May 03 2013 *)
Select[#+Total[IntegerDigits[#]]&/@Prime[Range[200]],PrimeQ]//Sort (* Harvey P. Dale, Sep 02 2023 *)

is_A048520(n)=#select(p->p+sumdigits(p)==n,primes([n-9*#digits(n),n-2]))&&isprime(n) \\ M. F. Hasler, Nov 08 2018

Offset corrected to 1 by M. F. Hasler, Nov 08 2018

A090009 Begins the earliest length-n chain of primes such that any term in the chain equals the previous term increased by the sum of its digits.

Original entry on oeis.org

2, 11, 11, 277, 37783, 516493, 286330897, 286330897, 56676324799

Offset: 1

Joseph L. Pe, Jan 27 2004

From the second term on, subsequence of A[2] := A048519. Due to the "exclusive" definition of this sequence, A048523(1) > a(2), but for k >= 3, a(k) = A[k](1) for A[3..9] = A048524 .. A048527, A320878 .. A320880. - M. F. Hasler, Nov 09 2018

			11 begins the earliest chain 11, 13, 17 of three primes such that any term in the chain equals the previous term increased by the sum of its digits, viz., 13 = 11 + 2, 17 = 13 + 4. Hence a(3) = 11.

Carlos Rivera, Puzzle 163. P+SOD(P)

Cf. A047791, A048519, A062028 (n + digit sum of n).

Cf. A048523 .. A048527, A320878 .. A320880.

A090009(n,P=2)=forprime(p=P,,P=p;for(i=2,n,isprime(P=A062028(P))||next(2));return(p))
P=0; A090009_vec=vector(6,n,P=A090009(n,P)) \\ Takes long for n > 6. - M. F. Hasler, Nov 09 2018

a(7)-a(8) from Donovan Johnson, Jan 08 2013

a(9) from Giovanni Resta, Jan 14 2013

A243441 Primes p such that p + A000120(p) is also a prime, where A000120 = sum of digits in base 2 = Hamming weight.

Original entry on oeis.org

2, 3, 5, 17, 43, 163, 277, 311, 347, 373, 461, 479, 571, 643, 673, 821, 853, 857, 881, 977, 983, 1013, 1093, 1103, 1117, 1181, 1223, 1297, 1427, 1433, 1439, 1481, 1523, 1607, 1613, 1621, 1823, 1861, 1871, 1873, 2003, 2083, 2281, 2333, 2393, 2417, 2467, 2549

Offset: 1

Anthony Sand, Jun 05 2014

			2 + digitsum(2,base=2) = 2 + digitsum(10) = 2 + 1 = 3, which is prime.
3 + digitsum(11) = 3 + 2 = 5.
5 + digitsum(101) = 5 + 2 = 7.
17 + digitsum(10001) = 17 + 2 = 19.
43 + digitsum(101011) = 43 + 4 = 47.

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

Cf. A000120, A092391 (n + A000120(n)), A048519 (analog for base 10).

Cf. A243442 (analog for p - A000120(p)).

Select[Prime@ Range@ 400, PrimeQ[# + Total@ IntegerDigits[#, 2]] &] (* Michael De Vlieger, Nov 06 2018 *)

lista(lim) = forprime(p=2,lim, if (isprime(p+hammingweight(p)), print1(p, ", "))); \\ Michel Marcus, Jun 10 2014

Name edited by M. F. Hasler, Nov 07 2018

A048523 Primes for which only one iteration of 'Prime plus its digit sum equals a prime' is possible.

Original entry on oeis.org

13, 19, 37, 53, 71, 73, 97, 103, 127, 163, 181, 233, 271, 307, 383, 389, 431, 433, 499, 509, 563, 587, 631, 701, 743, 787, 811, 857, 859, 947, 1009, 1049, 1061, 1087, 1153, 1171, 1223, 1283, 1423, 1483, 1489, 1553, 1597, 1601, 1607, 1733, 1801, 1861, 1867

Offset: 1

Patrick De Geest, May 15 1999

Sequence A048519 lists the primes for which at least (rather than exactly) one iteration of A062028 is "possible". See A048524 .. A048527 and A320878 .. A320880 for further subsequences, and A090009 for the list of their initial terms, starting chains of length >= 3 .. 9. - M. F. Hasler, Nov 09 2018

			prime 1999 -> 1999 + (1+9+9+9) = prime 2027 -> next iteration yields composite 2038.

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

Cf. A047791, A048519.

ppd1Q[n_]:=PrimeQ[Rest[NestList[#+Total[IntegerDigits[#]]&,n,2]]] == {True,False}; Select[Prime[Range[300]],ppd1Q] (* Harvey P. Dale, Nov 10 2011 *)

A048527 Primes for which only five iterations of 'Prime plus its digit sum equals a prime' are possible.

Original entry on oeis.org

516493, 1056493, 1427383, 1885943, 3166183, 3805183, 4241593, 6621283, 7646953, 12912283, 17987839, 32106493, 107152093, 120224773, 131144473, 133210873, 139388891, 142782877, 150326173, 155382923, 177865819, 184081943, 227795839, 242376877, 264174877

Offset: 1

Patrick De Geest, May 15 1999

			516493 -> 516521 -> 516541 -> 516563 -> 516589 -> 516623 -> next iteration yields a composite.

Lars Blomberg, Table of n, a(n) for n = 1..3000

Cf. A047791, A048519, A062028 (n + digit sum of n).

Cf. A048523, A048524, A048525, A048526.

forprime(p=2,,isprime(pp=A062028(p))&& !for(i=2,5,isprime(pp=A062028(pp))||next(2))&& !isprime(A062028(pp))&& print1(p",")) \\ M. F. Hasler, Nov 08 2018

Offset changed to 1 and a(15)-a(24) from Lars Blomberg, Dec 04 2013

A107288 Primes whose digit sum is a square.

Original entry on oeis.org

13, 31, 79, 97, 103, 211, 277, 349, 367, 439, 457, 547, 619, 673, 691, 709, 727, 853, 907, 997, 1021, 1069, 1087, 1201, 1249, 1429, 1447, 1483, 1609, 1627, 1663, 1699, 1753, 1789, 1861, 1879, 1933, 1951, 1987, 2011, 2239, 2293, 2347, 2383, 2437, 2473, 2617, 2671

Offset: 1

Zak Seidov, May 20 2005

Primes in A028839. [K. D. Bajpai, Jul 08 2014]

From Altug Alkan and Waldemar Puszkarz, Apr 10 2016: All terms are congruent to 1 mod 6. Proof: For n > 2, prime(n) is 1 or 5 mod 6. If p is 5 mod 6, then it is of the form 3*k-1. For numbers of this form, the sum of digits is also of this form, as can be seen through the kind of reasoning used in proving that numbers divisible by 3 have the sum of digits divisible by 3. However, 3*k-1 can never be a square, meaning n^2+1 is never divisible by 3: any n is equal to one of 0, 1, 2 mod 3, thus by the rules of modular arithmetic, n^2+1 is 1 or 2 mod 3, never 0. Hence p must be congruent to 1 mod 6.

			79 is in the sequence because it is prime. Also, (7 + 9) = 16 = 4^2.
997 is in the sequence because it is prime. Also, (9 + 9 + 7) = 25 = 5^2.

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

Cf. A000040, A007953, A028839, A048519.

Cf. A244863 (Semiprimes whose digit sum is square).

with(numtheory): A107288:= proc() local a; a:=add(i,i = convert((n),base,10))(n); if isprime(n) and root(a,2)=floor(root(a,2)) then RETURN (n); fi; end: seq(A107288 (), n=1..5000); # K. D. Bajpai, Jul 08 2014

bb = {}; Do[If[IntegerQ[Sqrt[Apply[Plus, IntegerDigits[p = Prime[n]]]]], bb = Append[bb, p]], {n, 500}]; bb

lista(nn) = {forprime(p=2, nn, if (issquare(sumdigits(p)), print1(p, ", ")););} \\ Michel Marcus, Apr 09 2016

Terms a(47) and a(48) added by K. D. Bajpai, Jul 08 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.

A320880 Primes such that iteration of A062028 (n + its digit sum) yields 8 primes in a row.

Original entry on oeis.org

56676324799, 373169411809, 2121959132809, 10180781225809, 14328311692789, 17429111275789, 32594135422789, 34327062247789, 39262151325799, 57404320087789, 60760513291789, 63116080460809, 66105224572789, 92054642332789, 98606700040789

Offset: 1

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

Term a(1) is immediate to find from the nearly equal terms A320879(7..8); terms a(2..9) were found by G. Resta as answer to Rivera's Puzzle 163, cf. link.

Carlos Rivera, Puzzle 163. P+SOD(P)

Cf. A047791, A048519, A062028 (n + digit sum of n).
Cf. A048523 .. A048527.
Subsequence of A320879 which is subsequence of A320878.
a(1) = A090009(9) = start of first chain of 9 primes under iteration of A062028.

is_A320880(n)=isprime(n=A062028(n))&& is_A320879(n)

A320880 = { n in A320879 | A062028(n) in A320879 }.

a(10)-a(15) from Lars Blomberg, Feb 10 2019

A048524 Primes for which only two iterations of 'Prime plus its digit sum equals a prime' are possible.

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, May 15 1999

			prime 727 -> 727 + (7+2+7) = prime 743 -> 743 + (7+4+3) = prime 757 -> next iteration yields composite 776.

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

Cf. A047791, A048519.

Select[Prime[Range[1400]],Boole[PrimeQ[Rest[NestList[ #+Total[ IntegerDigits[ #]]&,#,3]]]] == {1,1,0}&] (* Harvey P. Dale, Oct 31 2018 *)

Changed offset by Lars Blomberg, Dec 05 2013

cp's OEIS Frontend

A356570 a(n) is the first prime that starts a sequence of exactly n consecutive primes that are in A048519.

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A048520 Primes expressible as the sum of a prime plus its digit sum.

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A090009 Begins the earliest length-n chain of primes such that any term in the chain equals the previous term increased by the sum of its digits.

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A243441 Primes p such that p + A000120(p) is also a prime, where A000120 = sum of digits in base 2 = Hamming weight.

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A048523 Primes for which only one iteration of 'Prime plus its digit sum equals a prime' is possible.

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A048527 Primes for which only five iterations of 'Prime plus its digit sum equals a prime' are possible.

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A107288 Primes whose digit sum is a square.

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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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