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

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

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice
• easy

References

• 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).

Links

• 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

Crossrefs

Subsequence of A247104.

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

Programs

Haskell

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

Mathematica

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

PARI

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

Formula

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