A090009 -id:A090009 - 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

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 *)

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

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

A235680 The smallest first term of a sequence of n primes such that, after the first, each is equal to the previous prime plus the sum of all of its digits, plus the product of all of its nonzero digits.

Original entry on oeis.org

2, 191, 163, 151, 127, 1644997, 36778597, 935715673, 50682890749, 16390560362269, 63334172492839

Offset: 1

Carlos Rivera, Jan 13 2014

The following term produced in each sequence after the n-th is necessarily a composite integer.

			Example for n=8: a(8)=935715673 because after it the seven primes are 936311069, 936337351, 936490481, 936677149, 938391809, 938811763 and 939029537, with 936311069 = 935715673 + 9*3*5*7*1*5*6*7*3+(9+3+5+7+1+5+6+7+3) and so on...

C. Rivera, Puzzle 727. Extending the OEIS sequence A235680

Cf. A233692, A233783, A090009.

a(9) from Giovanni Resta, Jan 13 2014

a(10)-a(11) from Giovanni Resta, Feb 22 2014

A225544 a(n) begins the earliest chain of exactly n distinct primes such that any term in the chain equals the previous term increased by the product of its digits.

Original entry on oeis.org

2, 29, 23, 347, 293, 239, 57487, 486193, 1725121513, 1221261395831, 28549657193411

Offset: 1

Giovanni Resta, May 10 2013

A chain ends either at a composite number, or at a prime which contains a zero, since the subsequent primes in the chain are identical.

			23 starts the earliest chain of length 3, since 23+2*3 = 29, 29+2*9 = 47 and 47+4*7 = 75, where the first 3 terms are distinct and prime, so a(3) = 23. The last distinct term in the chain starting at 1725121513 is the prime 1725980623 which contains a zero and thus generates itself.

Cf. A090009, A145280.

seq = 0*Range[8]; p = 2; While[p < 500000, v = Length@ NestWhileList[# + Times @@ IntegerDigits@# &, p, PrimeQ@#2 && #1 != #2 &, 2] - 1; If[ seq[[v]] == 0, seq[[v]] = p]; p = NextPrime@p]; seq

A376693 a(n) is the first k such that if x(1) = k and x(i+1) = A062028(x(i)), x(1) to x(n) are all semiprimes but x(n+1) is not.

Original entry on oeis.org

1, 4, 15, 22, 39, 33, 291, 23174, 90137, 119135, 1641362, 1641337, 7113362, 471779113

Offset: 0

Robert Israel, Oct 01 2024

			a(4) = 39 because 39 = 3 * 13 is a semiprime, A062028(39) = 39 + 3 + 9 = 51 = 3 * 17 is a semiprime, A062028(51) = 51 + 5 + 1 = 57 = 3 * 19 is a semiprime, A062028(57) = 57 + 5 + 7 = 69 = 3 * 23 is a semiprime, but A062028(69) = 69 + 6 + 9 = 84 = 2^2 * 3 * 7 is not a semiprime.

Cf. A001358, A062028, A090009, A108638.

f:= proc(n) local x,i;
x:= n;
for i from 0 do
  if numtheory:-bigomega(x) <> 2 then return i fi;
  x:= x + convert(convert(x,base,10),`+`);
od
end proc:
V:= Array(0..12): count:= 0:
for i from 1 while count < 13 do
  v:= f(i);
  if v <= 12 and V[v] = 0 then V[v]:= i; count:= count+1 fi
od:
convert(V,list);

a(n) = if(n==0, return(1)); for(k=1, oo, if(bigomega(k) == 2, my(c=1, t=k+sumdigits(k)); while(bigomega(t) == 2, c += 1; t += sumdigits(t)); if(c == n, return(k)))); \\ Daniel Suteu, Nov 03 2024

a(13) from Daniel Suteu, Nov 03 2024

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A048519 Prime plus its digit sum equals a prime.

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

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

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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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A235680 The smallest first term of a sequence of n primes such that, after the first, each is equal to the previous prime plus the sum of all of its digits, plus the product of all of its nonzero digits.

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A225544 a(n) begins the earliest chain of exactly n distinct primes such that any term in the chain equals the previous term increased by the product of its digits.

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A376693 a(n) is the first k such that if x(1) = k and x(i+1) = A062028(x(i)), x(1) to x(n) are all semiprimes but x(n+1) is not.

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