A048521 -id:A048521 - 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

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]

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

A107741 Smallest number m such that prime(n) = m + (digit sum of m), a(n)=0 if no such m exists.

Original entry on oeis.org

1, 0, 0, 0, 10, 11, 13, 14, 16, 19, 0, 32, 34, 35, 37, 0, 52, 53, 56, 58, 59, 71, 73, 76, 0, 91, 92, 94, 95, 97, 122, 124, 127, 128, 142, 143, 146, 149, 160, 163, 166, 167, 181, 182, 184, 185, 0, 215, 217, 218, 0, 232, 233, 238, 250, 253, 256, 257, 0, 271

Offset: 1

Reinhard Zumkeller, May 23 2005

If a(n)>0 then: A000040(n)=A062028(a(n)) and A107740(n)>0.

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

Cf. A047791, A048521.

Cf. A000040, A055642, A062028.

a107741 n = if null ms then 0 else head ms  where
   ms = [m | let p = a000040 n,
             m <- [max 0 (p - 9 * a055642 p) .. p - 1], a062028 m == p]
-- Reinhard Zumkeller, Sep 27 2014

Data error corrected by Reinhard Zumkeller, Sep 27 2014

A048528 Primes expressible in two ways as the sum of an integer and its digit sum.

Original entry on oeis.org

101, 103, 107, 109, 113, 307, 311, 313, 317, 509, 521, 709, 719, 911, 919, 1009, 1013, 1213, 1217, 1409, 1607, 1609, 1613, 1619, 1621, 1811, 1823, 2017, 2027, 2111, 2113, 2309, 2311, 2521, 2711, 2713, 2719, 2917, 2927, 3011, 3209, 3217, 3221, 3407, 3413, 3613, 3617, 3623

Offset: 1

Patrick De Geest, May 15 1999

			313 = 296 + (2+9+6) and 313 = 305 + (3+0+5).

Delbert L. Johnson, Table of n, a(n) for n = 1..20000

Cf. A047791, A048521.

Transpose[Select[Tally[Select[#+Total[IntegerDigits[#]]&/@ Range[ 5000], PrimeQ]],#[[2]]==2&]][[1]] (* Harvey P. Dale, May 09 2013 *)

from collections import Counter
from sympy import isprime
def a_list(upto):
    return [i for i, j in Counter([i+sum(map(int, str(i))) for i in range(upto)]).items() if j>1 and isprime(i)]
print(a_list(4000)) # Nicholas Stefan Georgescu, Mar 02 2023

Corrected and extended by Harvey P. Dale, May 09 2013

A224966 Numbers n such that n^2+sum-of-digits(n^2) is prime.

Original entry on oeis.org

1, 4, 10, 16, 31, 32, 40, 41, 43, 62, 71, 76, 94, 95, 97, 98, 121, 142, 158, 163, 164, 166, 179, 188, 208, 211, 214, 227, 229, 259, 260, 265, 284, 301, 313, 317, 320, 328, 331, 340, 352, 355, 356, 365, 380, 382, 386, 392, 397, 401, 418, 424, 425, 431, 436, 439

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 21 2013

This is the sequence of indices of prime numbers in A171613.

The Ulam spiral for this sequence is a near-perfect line y=-x (see links).

			a(12)=76 because 76^2=5776, and 5776+(5+7+7+6)=5801, which is prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, The Ulam Spiral for the prime numbers derived from this sequence, i.e. a(n)^2+sum of digits(a(n)^2)

Cf. A048521.
Cf. numbers of the form n^2+sum-of-digits(n^2) A171613, and subsets A171614, A171615.
Cf. A062028.

library(gmp); digsum<-function(x) sum(as.numeric(unlist(strsplit(as.character(x),split=""))))
ans=as.bigz(rep(0,100)); n=1; i=as.bigz(1)
while(n<=100) {
if(isprime((w=i^2+digsum(i^2)))) ans[(n=n+1)-1]=i
i=i+1
}; ans

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

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

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A107741 Smallest number m such that prime(n) = m + (digit sum of m), a(n)=0 if no such m exists.

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A048528 Primes expressible in two ways as the sum of an integer and its digit sum.

Views

Author

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Programs

Mathematica

Python

Extensions

A224966 Numbers n such that n^2+sum-of-digits(n^2) is prime.

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