A047791 -id:A047791 - cp's OEIS frontend

A062028 a(n) = n + sum of the digits of n.

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

Offset: 0

Amarnath Murthy, Jun 02 2001

a(n) = A248110(n,A007953(n)). - Reinhard Zumkeller, Oct 01 2014

			a(34) = 34 + 3 + 4 = 41, a(40) = 40 + 4 = 44.

Harry J. Smith and N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1000 terms were computed by Harry J. Smith)
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A006378, A048521, A107740, A066568, A064806, A176995 (range), A003052, A230093, A248110.
Indices of: A047791 (primes), A107743 (composites), A066564 (squares), A084661 (cubes).
Iterations: A004207 (start=1), A016052 (start=3), A007618 (start=5), A006507 (start=7), A016096 (start=9).

a062028 n = a007953 n + n  -- Reinhard Zumkeller, Oct 11 2013

with(numtheory): for n from 1 to 100 do a := convert(n,base,10):
c := add(a[i],i=1..nops(a)): printf(`%d,`,n+c); od:
A062028 := n -> n+add(i,i=convert(n,base,10)) # M. F. Hasler, Nov 08 2018

Table[n + Total[IntegerDigits[n]], {n, 0, 100}]

A062028(n)=n+sumdigits(n) \\ M. F. Hasler, Jul 19 2015

def a(n): return n + sum(map(int, str(n)))
print([a(n) for n in range(71)]) # Michael S. Branicky, Jan 09 2023

a(n) = n + A007953(n).

a(n) = A160939(n+1) - 1. - Filip Zaludek, Oct 26 2016

More terms from Vladeta Jovovic, Jun 05 2001

A048519 Prime plus its digit sum equals a prime.

Original entry on oeis.org

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

A006378 Prime self (or Colombian) numbers: primes not expressible as the sum of an integer and its digit sum.

Original entry on oeis.org

3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, 2099, 2213, 2347, 2437, 2459, 2503, 2549, 2593, 2617, 2683, 2729, 2819, 2953, 3023, 3067

Offset: 1

N. J. A. Sloane

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.
D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.
Jeffrey Shallit, personal communication c. 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
B. Recamán, Problem E2408, Amer. Math. Monthly, 81 (1974), p. 407.
T. Trotter, Charlene Numbers (Copy as of Jan. 2018 on web.archive.org; original page no longer available.) Originally published Nov. 2010.
Index entries for Colombian or self numbers and related sequences

Subsequence of A247104.

Cf. A003052, A007953, A047791, A048521, A062028, A000040, A107740, A049084.

a006378 n = a006378_list !! (n-1)
a006378_list = map a000040 $ filter ((== 0) . a107740) [1..]
-- Reinhard Zumkeller, Sep 27 2014

With[{nn=3200},Complement[Prime[Range[PrimePi[nn]]],Table[n+Total[ IntegerDigits[n]],{n,nn}]]] (* Harvey P. Dale, Dec 30 2011 *)

select( is_A006378(n)=is_A003052(n)&&isprime(n), primes([1,3000])) \\ M. F. Hasler, Nov 08 2018

A107740(A049084(a(n))) = 0. [Corrected by Reinhard Zumkeller, Sep 27 2014]

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

A107740 Number of numbers m such that prime(n) = m + (digit sum of m).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1

Offset: 1

Reinhard Zumkeller, May 23 2005

a(A049084(A006378(n))) = 0; a(A049084(A048521(n))) > 0. [Corrected by Reinhard Zumkeller, Sep 27 2014]
a(n) <= 2 for n <= 10^5. Conjecture: sequence is bounded.
I would rather conjecture the opposite. Of course a(n) >= m implies n >= A006064(m), having more than A230857(m) digits, i.e., 14, 25 and 1111111111125 digits of n, for a(n) = 3, 4, 5. - M. F. Hasler, Nov 09 2018

			A000040(26) = 101 = 91 + (9 + 1) = 100 + (1 + 0 + 0): a(26) = # {91, 100} = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007953, A000040, A062028, A047791, A107741.

Cf. A006378, A048521, A049084, A055642, A062028.

a107740 n = length [() | let p = a000040 n,
                         m <- [max 0 (p - 9 * a055642 p) .. p - 1],
                         a062028 m == p]
-- Reinhard Zumkeller, Sep 27 2014

Table[p=Prime[n];c=0;i=1;While[iJayanta Basu, May 03 2013 *)

apply( A107740(n)=A230093(prime(n)), [1..150]) \\ M. F. Hasler, Nov 08 2018

a(n) = A230093(prime(n)), i.e.: A107740 = A230093 o A000040. - 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

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

A048521 Primes expressible as the sum of an integer plus its digit sum.

Original entry on oeis.org

2, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 313

Offset: 1

Patrick De Geest, May 15 1999

			a(24) = prime 113 which is 106 + (1+0+6) (or 97 + (9+7)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A047791, A048520.
Cf. A006378, A062028, A107741.
Cf. A000040, A107740, A049084.

a048521 n = a048521_list !! (n-1)
a048521_list = map a000040 $ filter ((> 0) . a107740) [1..]
-- Reinhard Zumkeller, Sep 27 2014

t={};Do[p=Prime[n];c=0;i=1;While[iJayanta Basu, May 03 2013 *)
Union[Select[Table[n+Total[IntegerDigits[n]],{n,400}],PrimeQ]] (* Harvey P. Dale, Jul 14 2014 *)

A107740(A049084(a(n))) > 0.

Formula and also offset corrected by Reinhard Zumkeller, Sep 27 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.

cp's OEIS Frontend

A062028 a(n) = n + sum of the digits of n.

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A048519 Prime plus its digit sum equals a prime.

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A006378 Prime self (or Colombian) numbers: primes not expressible as the sum of an integer and its digit sum.

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

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A107740 Number of numbers m such that prime(n) = m + (digit sum of m).

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