A337816 -id:A337816 - cp's OEIS frontend

A117570 Numbers of the form k * (sum of digits of k) listed sorted with multiplicity.

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

Offset: 1

Ernesto Estrada (estrada66(AT)yahoo.com), Jan 02 2008

			2008 is a term since 2008 = 251*(2+5+1).

David A. Corneth, Table of n, a(n) for n = 1..10000
Ömer Eğecioğlu and Bünyamin Şahin, On twin EP numbers, Transact. Comb. (2025) Vol. 14, Iss. 4, Art. No. 4, 261-270. See p. 262.
Ernesto Estrada, Puri Pereira-Ramos, Spatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts, Complexity, Vol. 2018 (2018), Article ID 9893867.

Sorted terms of A057147.

Terms without duplicates are given by A337816.

With[{nn = 400}, TakeWhile[#, # <= nn &] &@ Union@ Array[# Total@ IntegerDigits[#] &, nn + 1, 0]] (* Michael De Vlieger, Apr 19 2018 *)

A337817 Smallest nonnegative number that has exactly n different representations as the product of a number and the sum of its decimal digits.

Original entry on oeis.org

2, 0, 36, 900, 138600, 25336080, 3732276240, 240277237200

Offset: 0

Bernard Schott, Sep 23 2020

With "positive" instead "nonnegative", a(1) would be equal to 1, and other terms would not change.

a(8) <= 425616965373600. - Giovanni Resta, Oct 13 2022

			2 is the smallest number that is not possible to write as (m * sum of digits of m) for some m, hence a(0) = 2.
0 = 0 * 0, hence a(1) = 0
36 = 6 * 6 = 12 * (1+2) and 36 is the smallest number with 2 such representations, hence a(2) = 36.

Cf. A057147, A337816.

Cf. A337051 (similar for Bogotá numbers), A337732.

f[n_] := n*Plus @@ IntegerDigits[n]; m = 2*10^5; v = Table[0, {m}]; Do[i = f[n] + 1; If[i <= m, v[[i]]++], {n, 0, m}]; s = {}; k = 0; While[(p = Position[v, k]) != {}, AppendTo[s, p[[1, 1]] - 1]; k++]; s (* Amiram Eldar, Sep 23 2020 *)

a(n)={if(n==1, 0, for(k=1, oo, if(sumdiv(k, d, d*sumdigits(d)==k) == n, return(k))))} \\ Andrew Howroyd, Sep 23 2020

a(3)-a(5) from Amiram Eldar, Sep 23 2020

a(6)-a(7) from Bert Dobbelaere, Sep 27 2020, matching upper bounds from David A. Corneth

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

A117570 Numbers of the form k * (sum of digits of k) listed sorted with multiplicity.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A337817 Smallest nonnegative number that has exactly n different representations as the product of a number and the sum of its decimal digits.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions