A004207 -id:A004207 - cp's OEIS frontend

A230287 First differences of A016052/3 (= A230286).

1, 2, 1, 2, 1, 2, 1, 2, 4, 2, 4, 5, 4, 5, 1, 2, 1, 2, 4, 2, 4, 5, 4, 5, 1, 2, 1, 2, 4, 2, 4, 5, 4, 5, 4, 2, 4, 2, 4, 5, 4, 5, 4, 5, 4, 2, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 7, 5, 4, 5, 4, 5, 7, 8, 4, 5, 4, 5, 7, 8, 4, 5, 7, 5, 7, 5, 4, 5, 7, 8, 7, 2, 1, 2, 4, 2, 4, 5, 4, 5, 1, 2, 1, 2, 4, 2, 4, 5, 4, 5, 4

Offset: 1

Reinhard Zumkeller and N. J. A. Sloane, Oct 16 2013

This sequence captures the essence of A016052.

Essentially the same as A084228/3.

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

Cf. A004207, A016052, A230286, A084228, A065075.

a230287 n = a230287_list !! (n-1)
a230287_list = zipWith (-) (tail a230286_list) a230286_list

A230297 a(n) = A010062(n) written in binary: a(n+1) = a(n) + hammingweight(a(n)) in binary.

Original entry on oeis.org

1, 10, 11, 101, 111, 1010, 1100, 1110, 10001, 10011, 10110, 11001, 11100, 11111, 100100, 100110, 101001, 101100, 101111, 110100, 110111, 111100, 1000000, 1000001, 1000011, 1000110, 1001001, 1001100, 1001111, 1010100, 1010111, 1011100, 1100000, 1100010, 1100101, 1101001, 1101101, 1110010, 1110110, 1111011, 10000001, 10000011

Offset: 0

N. J. A. Sloane, Oct 17 2013

Is there any way to tell by looking at a binary number whether or not it is a term of this sequence?

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences.

Cf. A010062.
Essentially the same as A157845.
Cf. A004207 (base-10 analog); A007088 (n in binary), A010062 (this written in base 10), A000120 (Hammingweight), A092391 (A000120(n) + n), A028897 (convert binary to decimal).

s[0] = 1; s[n_] := s[n] = s[n-1] + DigitCount[s[n-1], 2, 1]; Table[FromDigits[IntegerDigits[s[n], 2]], {n, 0, 50}] (* Amiram Eldar, Jul 28 2023 *)

(A230297(n)=A007088(A010062(n))); A230297_vec(N)={vector(N,i, if(i>1, A007088(N+=hammingweight(N)), N=1))} \\ M. F. Hasler, Nov 18 2019

a(n) = A157845(n+1) = A007088(A010062(n)) = A007088(A092391(A028897(a(n-1)))). - M. F. Hasler, Nov 18 2019

A036228 a(1) = 31; a(n+1) = a(n) + sum of decimal digits of a(n).

Original entry on oeis.org

31, 35, 43, 50, 55, 65, 76, 89, 106, 113, 118, 128, 139, 152, 160, 167, 181, 191, 202, 206, 214, 221, 226, 236, 247, 260, 268, 284, 298, 317, 328, 341, 349, 365, 379, 398, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620, 628, 644, 658

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

Elements >= 214 can be found in A007618

D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
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, A016052, A249043.

NestList[#+Total[IntegerDigits[#]]&,31,60] (* Harvey P. Dale, Jan 30 2020 *)

Edited by Charles R Greathouse IV, Aug 02 2010

A157845 a(0) = 1, a(n) = sum of binary digits of all prior terms, expressed in binary.

Original entry on oeis.org

1, 1, 10, 11, 101, 111, 1010, 1100, 1110, 10001, 10011, 10110, 11001, 11100, 11111, 100100, 100110, 101001, 101100, 101111, 110100, 110111, 111100, 1000000, 1000001, 1000011, 1000110, 1001001, 1001100, 1001111, 1010100, 1010111, 1011100, 1100000, 1100010

Offset: 0

Oliver K. Seet, Mar 07 2009

Equals A230297 = A010062 converted from decimal to binary, prefixed by another initial 1. - M. F. Hasler, Nov 18 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A004207 (base-10 analog); A007088 (n in binary), A010062 (this written in base 10), A000120 (Hammingweight), A092391 (A000120(n) + n), A028897 (convert binary to decimal).

b:= proc(n) option remember; `if`(n<2, 1, b(n-1)+
      add(i, i=convert(a(n-1), base, 10)))
    end:
a:= n-> convert(b(n), binary):
seq(a(n), n=0..44);  # Alois P. Heinz, Nov 18 2019

s[0] = s[1] = 1; s[n_] := s[n] = s[n-1] + DigitCount[s[n-1], 2, 1]; Table[FromDigits[IntegerDigits[s[n], 2]], {n, 0, 50}] (* Amiram Eldar, Jul 28 2023 *)

lista(nn) = {my(s = 1); my(t = 1); print1(t, ", "); for (i=1, nn, sb = binary(s); t = subst(Pol(sb), x, 10); print1(t, ", "); s += hammingweight(sb););}

apply( A157845(n)=fromdigits(binary(A010062(n-!!n))), [0..40]) \\ M. F. Hasler, Nov 18 2019

a(n) = A230297(n-1) = A007088(A010062(n-1)) = A007088(A092391(A028897(a(n-1)))) for n > 0. - M. F. Hasler, Nov 18 2019

a(11) corrected and extended by R. J. Mathar, Mar 12 2009

More terms from Michel Marcus, Apr 19 2014

A112435 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 8, 16, 23, 28, 38, 45, 42, 41, 41, 36, 34, 32, 31, 30, 28, 29, 33, 34, 37, 44, 42, 37, 41, 39, 41, 38, 43, 40, 39, 39, 46, 45, 47, 54, 51, 45, 44, 43, 39, 42, 42, 39, 43, 43, 38, 43, 44, 40, 37, 40, 33, 32, 29, 36, 35, 39, 45, 49, 51, 48, 52, 47

Offset: 1

Alexandre Wajnberg, Dec 11 2005

Variation on Angelini's A112395. The sequence cycles at a(16)=38 and the loop has 312 terms. It is exactly the same loop as in A112433 "Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 2" and the first term of this 4-loop is the first term of that 2-loop . Computed by Gilles Sadowski.

			a(16)=38 because 4 + 4 + 8 + 1+6 +2+3 + 2+8 = 38

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A112395, A004207, A065075, A001370.

nxt[{t_,a_}]:={Take[Join[t,IntegerDigits[Total[t]]],-10],Total[t]}; Join[ {0,0,0,0,0,0,0,0,0},NestList[nxt,{{0,0,0,0,0,0,0,0,0,4},4},80][[All,2]]] (* Harvey P. Dale, Jun 22 2019 *)

A229527 Start with 1, skip (sum of digits of n) numbers, accept next number.

Original entry on oeis.org

1, 3, 7, 15, 22, 27, 37, 48, 61, 69, 85, 99, 118, 129, 142, 150, 157, 171, 181, 192, 205, 213, 220, 225, 235, 246, 259, 276, 292, 306, 316, 327, 340, 348, 364, 378, 397, 417, 430, 438, 454, 468, 487, 507, 520, 528, 544, 558, 577, 597

Offset: 1

Dave Durgin, Sep 25 2013

			a(1)=1, a(2)=1+1+1=3, a(3)=3+3+1=7, a(4)=7+7+1=15, a(5)=15+1+5+1=22, a(6)=22+2+2+1=27, ...

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A007612, A004207, A160939.

a[n_] := a[n] = a[n - 1] + 1 + Plus @@ IntegerDigits@a[n - 1]; a[1] = 1; Array[a, 50] (* Robert G. Wilson v, Aug 01 2018 *)

from itertools import islice
def A229527_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a += sum(map(int,str(a)))+1
A229527_list = list(islice(A229527_gen(),40)) # Chai Wah Wu, Aug 09 2025

a(n+1) = a(n) + (sum of digits of a(n)) + 1.

A230299 Define a sequence b_s by b_s(1)=s, b_s(k+1)=b_s(k)+(sum of digits of b_s(k)); a(n) is the number of steps needed for b_n to reach a term in one of b_0, b_1, b_3 or b_9, or a(n) = -1 if b_n never joins one of these four sequences.

Original entry on oeis.org

0, 0, 0, 0, 0, 52, 0, 11, 0, 0, 51, 50, 0, 49, 10, 0, 0, 48, 0, 9, 50, 0, 49, 0, 0, 47, 48, 0, 0, 8, 0, 49, 46, 0, 47, 48, 0, 45, 0, 0, 7, 46, 7, 47, 6, 0, 45, 44, 6, 0, 46, 0, 5, 5, 0, 45, 44, 0, 43, 4, 5, 4, 0, 0, 4, 44, 4, 43, 3, 0, 0, 42, 0, 3, 3, 4, 43, 0

Offset: 0

N. J. A. Sloane and Reinhard Zumkeller, Oct 21 2013

We conjecture that a(n) is never -1.

Cf. A230107, A062028, A004207, A016052, A007618, A006507, A016096.

read transforms; # to get digsum
M:=2000;
# f(s) returns the sequence k->k+digsum(k) starting at s
f:=proc(s) global M; option remember; local n,k,s1;
s1:=[s]; k:=s;
for n from 1 to M do  k:=k+digsum(k);
s1:=[op(s1),k]; od: end;
# g(s) returns (x,p), where x = first number in common between
# f(s) and one of f(1), f(3), f(9) and p is the position where it occurred.
# If f(s) and all of f(1), f(3), f(9) are disjoint for M terms, returns (-1,-1)
S1:=convert(f(1),set):
S3:=convert(f(3),set):
S9:=convert(f(9),set):
g:=proc(s) global f,S1,S3,S9; local t1,p,T0,T1,T2;
T0:=f(s):
T1:=convert(T0,set);
if ((s mod 9) = 3) or ((s mod 9) = 6) then   T2:= T1 intersect S3;   t1:=min(T2);   if (t1 = infinity) then RETURN(-1,-1); else     member(t1,T0,'p'); RETURN(t1,p-1); fi;
elif ((s mod 9) = 0) then   T2:= T1 intersect S9;   t1:=min(T2);   if (t1 = infinity) then RETURN(-1,-1); else     member(t1,T0,'p'); RETURN(t1,p-1); fi;
else   T2:= T1 intersect S1;   t1:=min(T2);   if (t1 = infinity) then RETURN(-1,-1); else     member(t1,T0,'p'); RETURN(t1,p-1); fi;
fi;
end;
[seq(g(n)[2],n=1..45)];

Terms a(46) and beyond from Lars Blomberg, Jan 10 2018

A243259 Numbers n such that n appears in the sequence x(i) = x(i-1) +/- digitsum(x(i-1)), where even digitsums are added, odd digitsums are subtracted and x(0) = n.

Original entry on oeis.org

99, 108, 117, 198, 207, 216, 297, 306, 315, 396, 405, 414, 495, 504, 513, 594, 603, 612, 693, 702, 711, 792, 801, 810, 972, 990, 999, 1008, 1098, 1107, 1116, 1197, 1206, 1215, 1296, 1305, 1314, 1395, 1404, 1413, 1494, 1503, 1512, 1593, 1602, 1611, 1692, 1701

Offset: 1

Anthony Sand, Jun 02 2014

The sequence begins with x(0) = n and continues by adding or subtracting the digitsum. When the digitsum(x(i-1)) is even, x(i) = x(i-1) + digitsum(x(i-1)), otherwise x(i) = x(i-1) - digitsum(x(i-1)).

			digitsum(99) = 18, 18 is even, so 99 + 18 = 117. digitsum(117) = 9, 9 is odd, so 177 - 9 = 108. 108 - 9 = 99, hence 99 belongs to sequence.
108 - 9 = 99, 99 + 18 = 117, 117 - 9 = 108, hence 108 is in the sequence.
117 - 9 = 108. 108 - 9 = 99. 99 + 18 = 117.
198 + 18 = 216. 216 - 9 = 207. 207 - 9 = 198.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A004207, A243260.

x(i) = x(i-1) + digitsum(x(i-1)) * (1 - (digitsum(x(i-1)) mod 2) * 2).

A243260 Numbers n such that n appears in the sequence x(i) = x(i-1) +/- digitsum(x(i-1)), where even digit sums are subtracted, odd digit sums are added and x(0) = n.

Original entry on oeis.org

81, 90, 99, 171, 180, 189, 261, 270, 279, 351, 360, 369, 441, 450, 459, 531, 540, 549, 621, 630, 639, 711, 720, 729, 801, 810, 819, 1071, 1080, 1089, 1161, 1170, 1179, 1251, 1260, 1269, 1341, 1350, 1359, 1431, 1440, 1449, 1521, 1530, 1539, 1611, 1620, 1629

Offset: 1

Anthony Sand, Jun 02 2014

The auxiliary sequence begins with x(0) = n and continues by adding or subtracting the digit sum. When the digitsum(x(i-1)) is even, x(i) = x(i-1) - digitsum(x(i-1)), otherwise x(i) = x(i-1) + digitsum(x(i-1)).

			digitsum(81) = 9, 9 is odd, so 81 + 9 = 90. 90 + 9 = 99. digitsum(99) = 18, 18 is even, so 99 - 18 = 81, so 81 is in the list.
90 + 9 = 99. 99 - 18 = 81. 81 + 9 = 90.
99 - 18 = 81. 81 + 9 = 90. 90 + 9 = 99.
171 + 9 = 180. 180 + 9 = 189. 189 - 18 = 171.
180 + 9 = 189. 189 - 18 = 171. 171 + 9 = 180.
Starting with n=81, we have 81+9(odd)=90, 90+9(odd)=99, 99-18(even)=81 for the auxiliary x(i) sequence; so 81 is in the main sequence; starting with n=90 or 99 will lead to the same cycle loop, so 90, 99 are also in this sequence.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A243259, A004207.

x(i) = x(i-1) + digitsum(x(i-1)) * -(1 - (digitsum(x(i-1)) mod 2) * 2).

A249043 a(1) = 42; a(n+1) = a(n) + sum of decimal digits of a(n).

Original entry on oeis.org

42, 48, 60, 66, 78, 93, 105, 111, 114, 120, 123, 129, 141, 147, 159, 174, 186, 201, 204, 210, 213, 219, 231, 237, 249, 264, 276, 291, 303, 309, 321, 327, 339, 354, 366, 381, 393, 408, 420, 426, 438, 453, 465, 480, 492, 507, 519, 534, 546, 561, 573, 588, 609, 624, 636

Offset: 1

N. J. A. Sloane, Oct 30 2014

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, A016052, A036228.

Cf. A007953, A062028.

a249043 n = a249043_list !! (n-1)
a249043_list = iterate a062028 42
-- Reinhard Zumkeller, Oct 31 2014

a(n+1) = A062028(a(n)). - Reinhard Zumkeller, Oct 31 2014

cp's OEIS Frontend

A230287 First differences of A016052/3 (= A230286).

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A230297 a(n) = A010062(n) written in binary: a(n+1) = a(n) + hammingweight(a(n)) in binary.

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A036228 a(1) = 31; a(n+1) = a(n) + sum of decimal digits of a(n).

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A157845 a(0) = 1, a(n) = sum of binary digits of all prior terms, expressed in binary.

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Maple

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A112435 Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 4.

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Mathematica

A229527 Start with 1, skip (sum of digits of n) numbers, accept next number.

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A230299 Define a sequence b_s by b_s(1)=s, b_s(k+1)=b_s(k)+(sum of digits of b_s(k)); a(n) is the number of steps needed for b_n to reach a term in one of b_0, b_1, b_3 or b_9, or a(n) = -1 if b_n never joins one of these four sequences.

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Extensions

A243259 Numbers n such that n appears in the sequence x(i) = x(i-1) +/- digitsum(x(i-1)), where even digitsums are added, odd digitsums are subtracted and x(0) = n.

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