A336826 -id:A336826 - cp's OEIS frontend

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

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

A336879 a(n) is the product of the decimal digits of A336876(n).

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 2, 5, 6, 3, 2, 7, 4, 8, 5, 9, 4, 3, 6, 1, 7, 6, 8, 4, 9, 8, 2, 2, 10, 5, 9, 12, 8, 3, 6, 14, 3, 12, 2, 16, 4, 4, 7, 12, 10, 18, 15, 4, 5, 18, 6, 16, 6, 12, 5, 21, 6, 15, 9, 4, 4, 20, 24, 3, 8, 6, 8, 14, 27, 9, 20, 24, 1, 18, 8, 9, 10, 6, 16

Offset: 1

Rémy Sigrist, Aug 06 2020

			For n = 26:
- A336876(26) = 24,
- so a(26) = 2*4 = 8.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A336879

Cf. A007954, A336826, A336876.

C
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See Links section.
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a(n) = A007954(A336876(n)).

a(n) * A336876(n) = A336826(n).

A336864 Bogotá numbers k such that k + 1 is also Bogotá number.

Original entry on oeis.org

0, 24, 2510, 5210, 8991, 56384, 348732, 460719, 867839, 28997919, 193889375, 254181375, 419321664, 1018179999, 2654951424, 1297015971839, 62061633644031

Offset: 1

Seiichi Manyama, Aug 06 2020

a(18) > 10^15 if it exists. - David A. Corneth, Aug 06 2020
From Chai Wah Wu, Aug 17 2020: (Start)
The following numbers are terms:
2805402158142975 = 153931531311*(1*5*3*9*3*1*5*3*1*3*1*1) = 111822471227*(1*1*1*8*2*2*4*7*1*2*2*7) - 1.
8748948067725824 = 2441112742111*(2*4*4*1*1*1*2*7*4*2*1*1*1) = 53339113353*(5*3*3*3*9*1*1*3*3*5*3) - 1.
(End)

			   n | a(n)                               a(n)+1
-----+------------------------------------------------------------------
   1 | 0 = 0 * 0                          1 = 1 * 1
   2 | 24 = 12 * (1*2)                    25 = 5 * 5
   3 | 2510 = 251 * (2*5*1)               2511 = 93 * (9*3)
   4 | 5210 = 521 * (5*2*1)               5211 = 193 * (1*9*3)
   5 | 8991 = 333 * (3*3*3)               8992 = 1124 * (1*1*2*4)
   6 | 56384 = 881 * (8*8*1)              56385 = 537 * (5*3*7)
   7 | 348732 = 3229 * (3*2*2*9)          348733 = 7117 * (7*1*1*7)
   8 | 460719 = 7313 * (7*3*1*3)          460720 = 11518 * (1*1*5*1*8)
   9 | 867839 = 17711 * (1*7*7*1*1)       867840 = 5424 * (5*4*2*4)
  10 | 28997919 = 119333 * (1*1*9*3*3*3)  28997920 = 51782 * (5*1*7*8*2)

Puzzling Stackexchange, Pairs of Bogotá numbers, 2020.

Cf. A336826.

a(11)-a(17) from David A. Corneth, Aug 06 2020

A337100 Numbers that have at least 4 different representations as the product of a number and of its decimal digits.

Original entry on oeis.org

0, 549504, 1578092544, 12276847296, 28961412480, 35998381440, 87012926784, 118082893824, 259456659840, 335449175040, 397315715328, 579305502720, 672777778176, 712539265536, 741360356352, 863562591360, 1138944651264, 1264664088576, 1276070713344, 1300488037632

Offset: 1

Chai Wah Wu, Aug 15 2020

Subsequence of A337054. a(61) = 20150684596224 is the smallest positive number with 5 representations. Other terms with 5 representations include 242374224347136, 1461825635235840, 1761950567301120, 3194185120277760, 3415710732779520.

			a(3) = 12276847296 = 676634*(6*7*6*6*3*4) = 773296*(7*7*3*2*9*6) = 2368219*(2*3*6*8*2*1*9) = 12179412*(1*2*1*7*9*4*1*2).

Chai Wah Wu, Table of n, a(n) for n = 1..113

Cf. A098736, A336826, A336876, A336944, A337051, A337054.

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

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

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A336879 a(n) is the product of the decimal digits of A336876(n).

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A336864 Bogotá numbers k such that k + 1 is also Bogotá number.

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A337100 Numbers that have at least 4 different representations as the product of a number and of its decimal digits.

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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.

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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.

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