A062028 -id:A062028 - cp's OEIS frontend

A176995 Numbers that can be written as (m + sum of digits of m) for some m.

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

Offset: 1

Reinhard Zumkeller, Aug 21 2011

The asymptotic density of this sequence is approximately 0.9022222 (Guaraldo, 1978). - Amiram Eldar, Nov 22 2020

			a(5) = 10 = 5 + (5);
a(87) = 100 = 86 + (8+6);
a(898) = 1000 = 977 + (9+7+7);
a(9017) = 10000 = 9968 + (9+9+6+8).

V. S. Joshi, A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar, Math. Student, Vol. 39 (1971), pp. 327-328. MR0330032 (48 #8371).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Rosalind Guaraldo, On the Density of the Image Sets of Certain Arithmetic Functions - II, The Fibonacci Quarterly, Vol. 16, No. 5 (1978), pp. 481-488.
Andrzej Makowski, On Kaprekar's "junction numbers", Math. Student, Vol. 34, No. 2 (1966), p. 77. MR0223292 (36 #6340); entire issue.
A. Narasinga Rao, On a technique for obtaining numbers with a multiplicity of generators, Math. Student, Vol. 34, No. 2 (1966), pp. 79-84. MR0229573 (37 #5147); entire issue.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.

Complement of A003052, range of A062028.

Cf. A062028, A007953, A055642, A230093.

a176995 n = a176995_list !! (n-1)
a176995_list = filter ((> 0) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

Select[Union[Table[n + Total[IntegerDigits[n]], {n, 77}]], # <= 77 &] (* Jayanta Basu, Jul 27 2013 *)

is_A003052(n)={for(i=1, min(n\2, 9*#digits(n)), sumdigits(n-i)==i && return); n} \\ from A003052
isok(n) = ! is_A003052(n) \\ Michel Marcus, Aug 20 2020

A230093(a(n)) > 0. - Reinhard Zumkeller, Oct 11 2013

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]

A006507 a(n+1) = a(n) + sum of digits of a(n), with a(1)=7.

Original entry on oeis.org

7, 14, 19, 29, 40, 44, 52, 59, 73, 83, 94, 107, 115, 122, 127, 137, 148, 161, 169, 185, 199, 218, 229, 242, 250, 257, 271, 281, 292, 305, 313, 320, 325, 335, 346, 359, 376, 392, 406, 416, 427, 440, 448, 464, 478, 497, 517, 530, 538, 554, 568

Offset: 1

N. J. A. Sloane

a(n) = A004207(n+4) for n > 11. - Reinhard Zumkeller, Oct 14 2013

Editorial Note, Popular Computing (Calabasas, CA), Vol. 4 (No. 37, Apr 1976), p. 12.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 36.
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.
Jeffrey Shallit, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Index entries for Colombian or self numbers and related sequences

Cf. A016052, A007618, A016096.

a006507 n = a006507_list !! (n-1)
a006507_list = iterate a062028 7  -- Reinhard Zumkeller, Oct 14 2013

NestList[#+Total[IntegerDigits[#]]&,7,50] (* Harvey P. Dale, Jan 25 2021 *)

a(n) = A062028(a(n-1)) for n > 1. - Reinhard Zumkeller, Oct 14 2013

More terms from Robert G. Wilson v

A006064 Smallest junction number with n generators.

Original entry on oeis.org

0, 101, 10000000000001, 1000000000000000000000102

Offset: 1

N. J. A. Sloane

Strictly speaking, a junction number is a number n with more than one solution to x+digitsum(x) = n. However, it seems best to start this sequence with n=0, for which there is just one solution, x=0. - N. J. A. Sloane, Oct 31 2013.
a(3) = 10^13 + 1 was found by Narasinga Rao, who reports that Kaprekar verified that it is the smallest term. No details of Kaprekar's proof were given.
a(4) = 10^24 + 102 was conjectured by Narasinga Rao.
a(5) = 10^1111111111124 + 102. - Conjectured by Narasinga Rao, confirmed by Max Alekseyev and N. J. A. Sloane.
a(6) = 10^2222222222224 + 10000000000002. - Max Alekseyev
a(7) = 10^( (10^24 + 10^13 + 115) / 9 ) + 10^13 + 2. - Max Alekseyev
a(8) = 10^( (2*10^24 + 214)/9 ) + 10^24 + 103. - Max Alekseyev

			a(2) = 101 since 101 is the smallest number with two generators: 101 = A062028(91) = A062028(100).
a(4) = 10^24 + 102 = 1000000000000000000000102 has exactly four inverses w.r.t. A062028, namely 999999999999999999999893, 999999999999999999999902, 1000000000000000000000091 and 1000000000000000000000100.

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of expressions for a(n), for n=1..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Terry Trotter, Charlene numbers
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A230093, A230100, A230303, A230857 (highest power of 10).

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

a(n) = the smallest m such that there are exactly n solutions to A062028(x)=m.

Edited, a(5)-a(6) added by Max Alekseyev, Jun 01 2011

a(1) added, a(5) corrected, a(7)-a(8) added by Max Alekseyev, Oct 26 2013

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

A230099 a(n) = n + (product of digits of n).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 70, 78, 86, 94, 102, 110, 118, 126

Offset: 0

N. J. A. Sloane, Oct 12 2013

A230099, A063114, A098736, A230101 are analogs of A092391 and A062028.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A011540, A092391, A062028, A004207, A230093, A003052, A176995.

Cf. also A063114, A098736, A230101, A230103, A230104, A230105, A062028, A230102, A063108.

a230099 n = a007954 n + n  -- Reinhard Zumkeller, Oct 13 2013

with transforms; [seq(n+digprod(n), n=0..200)];

a(n) = if (n, n + vecprod(digits(n)), 0); \\ Michel Marcus, Dec 18 2018

from math import prod
def a(n): return n + prod(map(int, str(n)))
print([a(n) for n in range(78)]) # Michael S. Branicky, Jan 09 2023

a(n) = n iff n contains a digit 0 (A011540). - Bernard Schott, Jul 31 2023

A016096 a(n+1) = a(n) + sum of its digits, with a(1) = 9.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 117, 126, 135, 144, 153, 162, 171, 180, 189, 207, 216, 225, 234, 243, 252, 261, 270, 279, 297, 315, 324, 333, 342, 351, 360, 369, 387, 405, 414, 423, 432, 441, 450, 459, 477, 495, 513, 522, 531, 540

Offset: 1

Robert G. Wilson v

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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Index entries for Colombian or self numbers and related sequences

Cf. A004207, A016052, A007618, A006507.

a016096 n = a016096_list !! (n-1)
a016096_list = iterate a062028 9  -- Reinhard Zumkeller, Oct 14 2013

from itertools import islice
def A016096_gen(): # generator of terms
    a = 9
    while True:
        yield a
        a += sum(int(d) for d in str(a))
A016096_list = list(islice(A016096_gen(),20)) # Chai Wah Wu, Mar 29 2022

a(n) = A062028(a(n-1)) for n > 1. - Reinhard Zumkeller, Oct 14 2013

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

cp's OEIS Frontend

A176995 Numbers that can be written as (m + sum of digits of m) for some m.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A006507 a(n+1) = a(n) + sum of digits of a(n), with a(1)=7.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A006064 Smallest junction number with n generators.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A230099 a(n) = n + (product of digits of n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

PARI

Python

Formula

A016096 a(n+1) = a(n) + sum of its digits, with a(1) = 9.

Views

Author