A062028 -id:A062028 - cp's OEIS frontend

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

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

A181664 Let f(m) = number of steps needed to reach a Harshad number when the map k->A062028(l) is iterated starting at m; a(n) = smallest m such that f(m) = n.

Original entry on oeis.org

15, 19, 14, 28, 23, 16, 22, 65, 55, 142, 134, 130, 119, 109, 95, 79, 71, 58, 47, 37, 32, 25, 17, 13, 11, 44, 256, 245, 235, 815, 1313, 1489, 1469, 1510, 1493, 1480, 1829, 1828, 1814, 1789, 1772, 3115, 4295, 4276, 4262, 4246, 4229, 4216, 4196, 4177, 4163, 4147, 4183, 4166, 4153, 4142, 4132, 4118, 4111, 4094, 4081, 8914, 8885, 8857, 8834, 8809, 8783, 8761, 8741, 8722, 8699, 8674, 8648, 8626, 8597, 8569, 8546, 8530, 8513, 8491, 8471, 8452, 8429, 8413, 8387, 8365, 8345, 8326, 8312, 8287, 8270, 8248, 8228, 8209, 8186, 8170, 8153, 8140, 31085, 31072

Offset: 1

N. J. A. Sloane, Nov 18 2010

Terms were computed by D. S. McNeil, Claudio Meller and Hans Havermann.

Eric Angelini, Posting to Sequence Fans Mailing List, Sep 20 2010

Eric Angelini, Chasing base-10 Harshad numbers
E. Angelini, Chasing base-10 Harshad numbers [Cached copy, with permission]
Hans Havermann, Steps to reach a Harshad

Cf. A062028, A005349, A181614.

A292202 The n-th iteration of A062028 starting with n.

Original entry on oeis.org

2, 8, 15, 28, 25, 33, 59, 77, 90, 79, 109, 111, 134, 137, 129, 148, 184, 189, 218, 197, 204, 226, 250, 231, 284, 284, 297, 313, 325, 309, 341, 398, 354, 418, 418, 414, 488, 440, 438, 478, 529, 465, 545, 554, 531, 628, 658, 561, 620, 677, 624, 697, 697, 657, 785, 818, 735, 865, 835, 762, 851, 851

Offset: 1

Peter Weiss, Sep 11 2017

a(n) == n*2^n (mod 9). This has a period of 18. - Robert Israel, Sep 11 2017

			n=5: 5+5=10, 10+1+0=11, 11+1+1=13, 13+1+3=17, 17+1+7=25. After 5 iterations you get 25, so a(5)=25.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Plot of a(n)/n for n = 1 .. 4000

Cf. A007618, A006507, A062028, A016096.

A062028:= proc(t) option remember; t + convert(convert(t,base,10),`+`):end proc:
seq((A062028@@n)(n), n=1..100); # Robert Israel, Sep 11 2017

Table[Nest[# + Total@ IntegerDigits@ # &, n, n], {n, 62}] (* Michael De Vlieger, Sep 11 2017 *)

a(n) = my(x=n); for (k=1, n, x += sumdigits(x)); x; \\ Michel Marcus, Sep 12 2017

A181614 a(n) = number of steps needed to reach a Harshad number when the map k->A062028(l) is iterated starting at n, or -1 if a Harshad number is never reached.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 24, 3, 1, 6, 23, 0, 2, 0, 0, 7, 5, 0, 22, 6, 0, 4, 1, 0, 3, 21, 5, 5, 2, 0, 20, 3, 4, 0, 4, 0, 1, 26, 0, 3, 19, 0, 2, 0, 3, 25, 26, 0, 9, 2, 2, 18, 24, 0, 25, 1, 0, 23, 8, 4, 1, 24, 1, 0, 17, 0, 23, 22, 2, 7, 24, 3, 16, 0, 0, 23, 22, 0

Offset: 1

N. J. A. Sloane, Nov 18 2010

Eric Angelini, Posting to Sequence Fans Mailing List, Sep 20 2010

Hans Havermann, Steps to reach a Harshad

Cf. A005349, A062028, A181664.

a(22) onward from John W. Layman, Nov 19 2010

A292512 Sequence A: Start with n, add the sum of digits of n (A062028) and repeat. Sequence B: Start with n, add the sum of base-100 digits of n and repeat. a(n) is the smallest common number > n.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 221, 341, 24, 109, 218, 30, 1171, 173, 36, 406, 80, 84, 88, 851, 96, 163, 104, 54, 218, 346, 120, 628, 1171, 231, 173, 181, 72, 197, 406, 213, 538, 260, 237, 1003, 1705, 90, 184, 719, 1041, 1015, 365, 111, 320, 127, 117, 418, 488, 114, 1487, 137, 120, 122, 199, 126, 1171, 298, 231, 134, 677

Offset: 1

Peter Weiss, Sep 18 2017

If you start with n=1 and take a third sequence C (n + sum of base-1000 digits of n), the first common numbers of the three sequences are 2, 4, 8, 16 and 1027975.
The common numbers for the first ten primes are:
                    2 -> 4, 8, 16, 1027975, ...
                    3 -> 24, 96, 60342, ...
                    5 -> 10, 469534, ...
                    7 -> 14, 131558, ...
                   11 -> 923428, ...
                   13 -> 668495, ...
                   17 -> 81820, ...
                   19 -> 2061797, ...
                   23 -> 2227118, ...
                   29 -> 12278, ...

			n=10: Sequence A: 10, 11, 13, 17, 25, 32, 37, 47, 58, 71, 79, 95, 109, 119, 130, 134, 142, 149, 163, 173, 184, 197, 214, 221, ...
Sequence B: 10, 20, 40, 80, 160, 221, ...
-> 221 is the first common number > 10, so a(n)=221.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A286660, A062028, A007618, A006507, A016096.

With[{m = 10^3}, Table[With[{A = Rest@ NestList[# + Total@ IntegerDigits@ # &, n, m]}, NestWhile[# + Total@ IntegerDigits[#, 100] &, n, FreeQ[A, #] &, 1, m]], {n, 68}]] (* Michael De Vlieger, Sep 23 2017 *)

a(n) = my (A=n + sumdigits(n), B=n + sumdigits(n,100)); while (1, if (A==B, return (A), ARémy Sigrist, Sep 23 2017

A320870 Irregular table: row n >= 0 lists numbers m >= 0 such that n = A062028(m) := m + sum of digits of m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 11, 7, 12, 8, 13, 9, 14, 15, 20, 16, 21, 17, 22, 18, 23, 19, 24, 25, 30, 26, 31, 27, 32, 28, 33, 29, 34, 35, 40, 36, 41, 37, 42, 38, 43, 39, 44, 45, 50, 46, 51, 47, 52, 48, 53, 49, 54, 55, 60, 56, 61, 57, 62, 58, 63, 59, 64, 65, 70, 66, 71, 67, 72, 68, 73, 69, 74, 75, 80, 76, 81, 77, 82, 78, 83, 79, 84, 85, 90

Offset: 0

M. F. Hasler, Nov 09 2018

Row lengths are given by A230093.

			The first nonempty rows are:
    n  | list of m
    0  | 0        // since 0 = 0 + 0
    2  | 1        // since 2 = 1 + 1
    4  | 2        // etc.
    6  | 3        // Below 10 every odd row is empty, but thereafter,
    8  | 4        // only rows 20, 31, 42, ..., 108 (steps of 11),
   10  | 5        // 110, 121, 132, ..., 198, etc. are empty.
   11  | 10       // Since 11 = 10 + (1 + 0)
   12  | 6
   13  | 11       // The first prime that yields a prime: 11 + (1 + 1) = 13.
     (...)
  100  | 86       // The first row of length 2 is 101:
  101  | 91, 100  // 101 = 91 + (9 + 1) = 100 + (1 + 0 + 0)
  102  | 87
     (...)

Robert Israel, Table of n, a(n) for n = 0..9971 (rows 0 to 10000, flattened)

Cf. A007953 (sum of digits of n), A062028 (n + digit sum of n).
Cf. A230093 (number of m such that m + (sum of digits of m) is n).
Cf. A006064 (least m with row length n),
Cf. A003052 (Self or Colombian numbers: rows of length 0), A006378 (Colombian primes).
Cf. A320881 (indices of rows containing a prime), A048520 (primes among these).

N:= 100: # for rows 0 to N, flattened
for i from 0 to N do V[i]:= NULL od:
for i from 0 to N-1 do
  v:= convert(convert(i,base,10),`+`);
  if v <= N then V[v]:= V[v],i fi
od:
seq(V[i],i=1..N); # Robert Israel, Jul 21 2025

A320870_row(n)=if(n,select(m->m+sumdigits(m)==n,[max(n-9*logint(n,10)+8,n\/2)..n-1]),[0])

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","))

A292566 Primes that can be reached with their prime-index, if you start with the prime-index and use iterations of A062028.

Original entry on oeis.org

2, 11, 107, 191, 587, 719, 1061, 1171, 1181, 1259, 1327, 1487, 1597, 1619, 1933, 1949, 2011, 2141, 2269, 2477, 2803, 2999, 3041, 3049, 3079, 3169, 3229, 3259, 3617, 3733, 4493, 4799, 5009, 5023, 5171, 5261, 5581, 5657, 6131, 6211, 6301, 6311, 6421, 6451, 6529

Offset: 1

Peter Weiss, Sep 19 2017

If p is in the sequence, its index A000720(p) is not divisible by 3. - Robert Israel, Sep 19 2017

			The prime-index of 11 is 5: 5+5=10, 10+1+0=11 -> after two iterations you reach 11, so 11 is in the sequence.

Cf. A000720, A062028, A007618, A004207.

f:= proc(n) local t, p;
  p:= ithprime(n);
  t:= n;
  do
    t:= t + convert(convert(t,base,10),`+`);
    if t > p then return NULL
    elif t = p then return p
    fi
  od;
end proc:
map(f, [$1..1000]); # Robert Israel, Sep 19 2017

ok[p_] := Block[{n = PrimePi@ p}, While [n < p, n += Total@ IntegerDigits@ n]; n == p]; Select[Prime@ Range@ 600, ok] (* Giovanni Resta, Sep 19 2017 *)

is(n) = my(x=primepi(n)); while(1, x=x+sumdigits(x); if(x==n, return(1), if(x > n, return(0))))
forprime(p=1, 7000, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Sep 19 2017

More terms from Felix Fröhlich, Sep 19 2017

cp's OEIS Frontend

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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A181664 Let f(m) = number of steps needed to reach a Harshad number when the map k->A062028(l) is iterated starting at m; a(n) = smallest m such that f(m) = n.

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A292202 The n-th iteration of A062028 starting with n.

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A181614 a(n) = number of steps needed to reach a Harshad number when the map k->A062028(l) is iterated starting at n, or -1 if a Harshad number is never reached.

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A292512 Sequence A: Start with n, add the sum of digits of n (A062028) and repeat. Sequence B: Start with n, add the sum of base-100 digits of n and repeat. a(n) is the smallest common number > n.

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A320870 Irregular table: row n >= 0 lists numbers m >= 0 such that n = A062028(m) := m + sum of digits of m.

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