A176995 -id:A176995 - cp's OEIS frontend

A337139 Indices m of repunits R_m that are not Colombian (or self) numbers.

2, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

Michel Marcus, Aug 19 2020

Note that 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, 5794777, 8177207 (see A004023) are terms. [Last 2 terms added by Serge Batalov, Aug 24 2021]

While all currently known A004023 terms are in this sequence, there is no clear argument that it would hold for all future values. - Serge Batalov, Aug 24 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A002275 (repunits), A004022 (repunit primes), A004023 (indices of repunit primes), A176995 (not Colombian).

Cf. A337208 (complement).

upto(n)= {my(res = List()); for(i = 1, n, if(is(i), listput(res, i); print1(i", "))); res}
is(n) = {if(n < 8, return(isprime(n))); qd = n; n = 10^n\9; r = 1 + (n-1)%9; h = (r + 9 * (r%2))/2; ld = 10; while(h + 9*qd >= n % ld, ld*=10); vs = qd - valuation(ld, 10); n %= ld; for(i = 0, qd, if(vs + vecsum(digits(n - h - 9*i)) == h + 9*i, return(1))); 0} \\ David A. Corneth, Aug 20 2020

A336985 Colombian numbers that are not Bogotá numbers.

Original entry on oeis.org

3, 5, 7, 20, 31, 53, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 266, 277, 288, 299, 310, 323, 334, 345, 356, 367, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 536, 547, 558, 569, 580, 591, 602, 613

Offset: 1

Bernard Schott, Aug 26 2020

Equivalently, numbers m that are not of the form k + sum of digits of k for any k (A003052), and that are not of the form q * product of digits of q for any q (complement of A336826).
As repunits are trivially Bogotá numbers, there are not repunits in the data.
A336983, A336984, A336986 and this sequence form a partition of the set of positive integers N*

			7 is a term because there are not k < 7  such that 7 = k + sum of digits of k, and that are not q such that 7 = q * product of digits of q.
13 is not of the form q * product of digits of q for any q <= 13, so 13 is not a Bogotá number, but 13 = 11 + (1+1) is not Colombian, hence 13 is not a term.
42 is Colombian because there does not exist m < 42 such that 42 = m + sum of digits of m, but as 42 = 21 * (2*1) is a Bogota number, 42 is not a term.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Index to sequences related to Colombian or self numbers.

Cf. A003052 (Colombian), A176995 (not Colombian), A336826 (Bogotá numbers), A336983 (Bogotá not Colombian), A336984 (Bogotá and Colombian), this sequence (Colombian not Bogotá), A336986 (not Colombian and not Bogotá).

m = 600; Intersection[Complement[Range[m], Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &]], Complement[Range[m], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]]] (* Amiram Eldar, Aug 26 2020 *)

lista(nn) = Vec(setintersect(setminus([1..nn], Set(vector(nn, k, k+sumdigits(k)))), setminus([1..nn], Set(vector(nn, k, k*vecprod(digits(k))))))); \\ Michel Marcus, Aug 26 2020

A336986 Numbers that are not Colombian and not Bogotá.

Original entry on oeis.org

2, 6, 8, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85, 87, 89, 90, 91

Offset: 1

Bernard Schott, Aug 22 2020

Equivalently, numbers m that are of the form k + sum of digits of k for some k (A176995), but are not of the form q * product of digits of q for any q.
As repunits are trivially Bogotá numbers, there are not repunits in the data.
A336983, A336984, A336985 and this sequence form a partition of the set of positive integers N*.

			13 = 11 + (1+1) is not Colombian and 13 is not of the form q * product of digits of q for any q <= 13, so 13 is not a Bogotá number, hence 13 is a term.
39 = 33 + (3+3) is not Colombian but 39 = 13 * (1*3) is a Bogotá number, hence 39 is not a term.
42 = 21 * (2*1) is a Bogotá number but there does not exist k < 42 such that 42 = k + sum of digits of k, hence 42 is a Colombian number and 42 is not a term.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Index to sequences related to Colombian numbers.

Cf. A003052 (Colombian), A176995 (not Colombian), A336826 (Bogotá), A336983 (Bogotá and not Colombian), A336984 (Bogotá and Colombian), A336985 (Colombian not Bogotá), this sequence (not Colombian and not Bogotá).

m = 100; Intersection[Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &], Complement[Range[m], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]]] (* Amiram Eldar, Aug 22 2020 *)

lista(nn) = Vec(setintersect(Set(vector(nn, k, k+sumdigits(k))), setminus([1..nn], Set(vector(nn, k, k*vecprod(digits(k))))))); \\ Michel Marcus, Aug 23 2020

A337816 Numbers that can be written as (m * sum of digits of m) for some m.

Original entry on oeis.org

0, 1, 4, 9, 10, 16, 22, 25, 36, 40, 49, 52, 63, 64, 70, 81, 88, 90, 100, 112, 115, 124, 136, 144, 160, 162, 175, 190, 198, 202, 205, 208, 220, 238, 243, 250, 252, 280, 301, 306, 319, 324, 333, 352, 360, 364, 370, 400, 405, 412, 418, 424, 427, 448, 460, 468, 484, 486, 490

Offset: 1

Bernard Schott, Sep 23 2020

If 3 divides a(n), then 9 divides a(n).

			10 = 10 * (1+0);
22 = 11 * (1+1).

Range of A057147 and of A117570.
Similar sequences: A176995 (m + sum of digits of m), A336826 (m * product of digits of m), A337718 (m + product of digits of m).
Cf. A337817.
Some subsequences: A011557, A052268, A093141.

m = 500; Select[Union @ Table[k * Plus @@ IntegerDigits[k], {k, 0, m}], # <= m &] (* Amiram Eldar, Sep 23 2020 *)

is(k)={if(k==0, return(1)); fordiv(k, d, if(d*sumdigits(d)==k, return(1))); 0} \\ Andrew Howroyd, Sep 23 2020

A332240 Palindromes that are the sum of a number and the sum of its digits.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 333, 343, 353, 363, 373, 383, 393, 404, 414, 434, 444, 454, 464, 474, 484, 494, 505, 515, 535, 545, 555, 565, 575

Offset: 1

Eric Fox, Feb 07 2020

			196 + 1 + 9 + 6 = 212, so 212 is in this sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Intersection of A002113 and A176995.

Cf. A062028, A229545.

pal:=func; [k:k in [0..600]| pal(k) and exists(m){s:s in [0..k]| s+&+Intseq(s) eq k}]; // Marius A. Burtea, Feb 08 2020

palQ[n_] := Block[{d = IntegerDigits[n]}, Reverse[d] == d]; Select[ Union[(# + Plus @@ IntegerDigits@#) & /@ Range[0, 600]], # <= 600 && palQ[#] &] (* Giovanni Resta, Feb 07 2020 *)

a(n) in { A062028(A229545(i)) : i >= 1 }. - Amiram Eldar, Feb 12 2020

More terms from Giovanni Resta, Feb 07 2020

A336143 Integers that are Brazilian and not Colombian.

Original entry on oeis.org

8, 10, 12, 13, 14, 15, 16, 18, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 43, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Bernard Schott, Jul 10 2020

There are no squares of primes in the data (all squares of primes are not Brazilian except for 121 that is Brazilian, but 121 is Colombian).

			15 is a term because 15 = 12 + (sum of digits of 12), so 15 is not Colombian and 15 = 33_4, so 15 is Brazilian.

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Wikipédia, Nombre brésilien (in French).
Index to sequences related to Brazilian Numbers.
Index to sequences related to Colombian Numbers.

Intersection of A125134 (Brazilian) and A176995 (not Colombian).

Cf. A003052 (Colombian), A333858 (Brazilian and Colombian), this sequence (Brazilian not Colombian), A336144 (Colombian not Brazilian).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[ IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; n = 100; Select[Union@Table[Plus @@ IntegerDigits[k] + k, {k, 1, n}], # <= n && brazQ[#] &] (* Amiram Eldar, Jul 10 2020 *)

A225065 Numbers of the form n^2 plus the sum of squared digits of n^2.

Original entry on oeis.org

2, 20, 53, 54, 81, 90, 101, 116, 127, 146, 177, 258, 287, 314, 321, 353, 407, 416, 438, 474, 580, 639, 686, 690, 797, 863, 913, 922, 981, 1045, 1079, 1219, 1235, 1259, 1418, 1493, 1496, 1552, 1637, 1783, 1866, 2011, 2058, 2063, 2158, 2298, 2333, 2422, 2529

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 26 2013

Note that consecutive terms are not necessarily generated by consecutive values of n.

It appears that 146 is the only term that can be generated by two values of n (7 and 9). There are no other duplicates in the first 10000 terms.

			For n=11: 11^2=121; 121 + 1^2 + 2^2 + 1^2 = 127.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A176995, A062028, A209303.

sort(unique((1:101)^2+sapply((1:101)^2,function(x) sum(as.numeric(unlist(strsplit(as.character(x),split="")))^2))))

A282473 Multiples of 9 which cannot be expressed as the difference between a natural number k and its digit sum s(k).

Original entry on oeis.org

90, 189, 288, 387, 486, 585, 684, 783, 882, 981, 990, 1089, 1188, 1287, 1386, 1485, 1584, 1683, 1782, 1881, 1980, 1989, 2088, 2187, 2286, 2385, 2484, 2583, 2682, 2781, 2880, 2979, 2988, 3087, 3186, 3285, 3384, 3483, 3582, 3681, 3780, 3879, 3978, 3987, 4086, 4185, 4284, 4383, 4482, 4581, 4680, 4779, 4878, 4977, 4986

Offset: 1

Sharvil Kesarwani, Feb 16 2017

Based on empirical observations, it can be noted that the difference between consecutive terms is usually 99. However, this breaks down when the hundreds digit is 9, in which case the difference between consecutive terms is 9. This changes back, however.

			If k=90, then k-s(k)=81. If k=100, then k-s(k)=99. This is an increasing function, so k-s(k)=90 is unachievable.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Brilliant.org, Those are some nifty numbers
Australian Mathematics Olympiad, Question 2, 2015, p. 84.

Cf. A003052, A176995.

okQ[n_] := Catch[ Do[ If[ x- Total@ IntegerDigits@ x == n, Throw@ False], {x, n, n+ 9 IntegerLength[n]}]; True]; Select[9 Range[1000], okQ[#] &] (* Giovanni Resta, Feb 27 2017 *)

is(n)=for(k=n,n+9*#Str(n)+9, if(k-sumdigits(k)==n, return(0))); n%9==0 \\ Charles R Greathouse IV, Feb 27 2017

from math import ceil
def a(n): #Outputs all numbers less than n which are in the sequence
    def s(n):
        r = 0
        while n:
            r, n = r + n% 10, n//10
        return r
    mult9=[]
    if n%9==0:
        for x in range(1, ceil(n/9)+1):
            mult9.append(9*x)
    else:
        for x in range(1, ceil(n/9)):
            mult9.append(9*x)
    for y in range(1, ceil(n/10)+1):
        mult9.remove(10*y-s(10*y))
    return mult9

A337733 Numbers that can be written as (k + sum of digits of k) for some k, also as (m + product of digits of m) for some m, and finally as (q * product of digits of q) for some q.

Original entry on oeis.org

4, 16, 24, 56, 81, 88, 138, 144, 192, 242, 250, 297, 366, 408, 456, 516, 520, 522, 564, 575, 704, 744, 777, 795, 819, 884, 900, 912, 966, 1008, 1053, 1071, 1080, 1104, 1134, 1250, 1312, 1316, 1375, 1512, 1520, 1608, 1644, 1680, 1712, 1778, 1928, 1950, 2025, 2048, 2072

Offset: 1

Bernard Schott, Sep 18 2020

Equivalently, Bogotá numbers that are not Colombian and that can be written as (m + product of digits of m) for some m.

The only primes that can belong to this sequence are repunits > 11 whose indices are in A004023. It is known that these primes belong to A336983, but do they belong also to A337718?

			4 = 2 + 2 = 2 + 2 = 2 * 2;
16 = 8 + 8 = 8 + 8 = 4 * 4;
24 = 21 + (2+1) = 17 + (1*7) = 12 * (1*2);
56 = 46 + (4+6) = 51 + (5*1) = 14 * (1*4);
81 = 72 + (7+2) = 63 + (6*3) = 9 * 9.

Index to sequences related to Colombian numbers.

Intersection of A176995, A336826 and A337718.
Intersection of A336983 and A337718.
Cf. A004023, A337139.

m = 2100; Select[Intersection @@ Union /@ Transpose[Table[{n + Plus @@ (d = IntegerDigits[n]), n + (p = Times @@ d), n*p}, {n, 1, m}]], # <= m &] (* Amiram Eldar, Sep 18 2020 *)

isok(m) = {if (m==0, return (1)); for (k=1, m,  if (k+vecprod(digits(k)) == m, return (1)); ); } \\ A337718
listb(nn) = Vec(setintersect(Set(vector(nn, k, k+sumdigits(k))), Set(vector(nn, k, k*vecprod(digits(k)))))); \\ A336983
lista(nn) = select(x->isok(x), listb(nn)); \\ Michel Marcus, Sep 18 2020

More terms from Michel Marcus, Sep 18 2020

A337839 Numbers that can be written as (k + sum of digits of k) for some k, then as (m + product of digits of m) for some m, also as (q * product of digits of q) for some q, and finally as (t * sum of digits of t) for some t.

Original entry on oeis.org

4, 16, 81, 88, 144, 250, 520, 900, 1008, 1053, 1134, 2025, 2304, 2655, 3726, 4680, 6408, 6624, 9928, 12024, 12150, 12510, 13608, 14256, 15480, 16408, 17128, 17172, 18304, 19152, 19288, 19602, 23310, 24336, 25110, 26550, 29358, 32896, 32968, 36864, 37485, 38592

Offset: 1

Bernard Schott, Sep 25 2020

Equivalently, Bogotá numbers that are not Colombian and that can be written as (m + product of digits of m) for some m and also as (t * sum of digits of t) for some t.
The only primes that can belong to this sequence are repunits > 11 whose indices are in A004023. It is known that these primes belong to A336983 but do they belong also to A337718 and A337816?
Observation: 7 of the first 13 terms are perfect squares: 4, 16, 81, 144, 900, 2025, 2304 (see examples).

			4 = 2 + 2 = 2 + 2 = 2 * 2 = 2 * 2;
16 = 8 + 8 = 8 + 8 = 4 * 4 = 4 * 4;
81 = 72 + (7+2) = 63 + (6*3) = 9 * 9 = 9 * 9;
144 = 135 + (1+3+5) = 128 + (1*2*8) = 18 * (1*8) = 24 * (2+4).

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Index to sequences related to Colombian numbers.

Intersection of A176995, A336826, A337718 and A337816.
Intersection of A336983, A337718 and A337816.
Intersection of A337733 and A337816.

m = 40000; Select[Intersection @@ Union /@ Transpose[Table[{n + (s = Plus @@ (d = IntegerDigits[n])), n + (p = Times @@ d), n*s, n*p}, {n, 1, m}]], # <= m &] (* Amiram Eldar, Sep 25 2020 *)

lista(nn) = {my(vd = vector(nn, k, digits(k)), vs = vector(nn, k, vecsum(vd[k])), vp = vector(nn, k, vecprod(vd[k])), vsp = Set(vector(nn, k, k+vp[k])), vss = Set(vector(nn, k, k+vs[k])), vps = Set(vector(nn, k, k*vs[k])), vpp = Set(vector(nn, k, k*vp[k])), vk = vector(nn, k, k)); Vec(setintersect(vk, setintersect(vsp, setintersect(vss, setintersect(vps, vpp)))));} \\ Michel Marcus, Oct 01 2020

Terms a(7) and beyond from Amiram Eldar, Sep 25 2020

cp's OEIS Frontend

A337139 Indices m of repunits R_m that are not Colombian (or self) numbers.

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A336985 Colombian numbers that are not Bogotá numbers.

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A336986 Numbers that are not Colombian and not Bogotá.

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A337816 Numbers that can be written as (m * sum of digits of m) for some m.

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A332240 Palindromes that are the sum of a number and the sum of its digits.

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Magma

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A336143 Integers that are Brazilian and not Colombian.

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A225065 Numbers of the form n^2 plus the sum of squared digits of n^2.

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A282473 Multiples of 9 which cannot be expressed as the difference between a natural number k and its digit sum s(k).

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