A257297 -id:A257297 - cp's OEIS frontend

A368062 Numbers k such that k = A257850(k) + A257297(k).

0, 36, 655, 1258, 6208, 12508, 45715, 65455, 75385, 125008, 235297, 1250008, 2857144, 3214288, 4210528, 6545455, 6792453, 12500008, 34615386, 47058824, 87671233, 125000008, 654545455, 1250000008, 9529411765, 12500000008, 39130434783, 45714285715, 65454545455, 75384615385

Offset: 1

Nicolas Bělohoubek, Dec 10 2023

			     0 = 0*0   +   0*0;
    36 = 3*6   +   3*6;
   655 = 6*55  +  65*5;
  6208 = 6*208 + 620*8;
  ...

Kevin Ryde, Table of n, a(n) for n = 1..700
Nicolas Bělohoubek, C# program
Nicolas Bělohoubek, Subsequences of form A-(B)-C
Kevin Ryde, PARI/GP Code

Cf. A257850, A257297.

Select[Range[0,10^6], Part[digits=IntegerDigits[#],1]FromDigits[Drop[digits,1]] + FromDigits[Drop[digits,-1]]Part[digits,Length[digits]] == # &] (* Stefano Spezia, Dec 10 2023 *)

PARI
```
\\ See links.
```

def ok(n):
    if n < 10: return n == 0
    s = str(n)
    return n == int(s[0])*int(s[1:]) + (n%10)*(n//10)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Dec 10 2023

# faster for generating initial segment of sequence
from itertools import count, islice
def agen(): # generator of terms
    yield 0
    for digits in count(2):
        for first in range(1, 10):
            base = first*10**(digits-1)
            for rest in range(10**(digits-1)):
                n = base + rest
                if first*rest + (n%10)*(n//10) == n:
                    yield n
            print("...", digits, first, time()-time0, alst)
print(list(islice(agen(), 18))) # Michael S. Branicky, Dec 10 2023

a(24)-a(30) from Michael S. Branicky, Dec 10 2023

A035930 Maximal product of any two numbers whose concatenation is n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 10, 20, 30, 40, 50, 60, 70

Offset: 0

Erich Friedman

Agrees up to a(100) = 0 with A088117, A171765 and A257297, but all of the four differ in a(101) and subsequent values. - M. F. Hasler, Sep 01 2021

			a(341) = max(34*1,3*41) = 123.

Alois P. Heinz, Table of n, a(n) for n = 0..9999
Eric Angelini, Surface of a number, Sep 01 2021.

Different from A007954, A088117, A171765 and A257297. Cf. A035931-A035935.

a035930 n | n < 10    = 0
          | otherwise = maximum $ zipWith (*)
            (map read $ init $ tail $ inits $ show n)
            (map read $ tail $ init $ tails $ show n)
-- Reinhard Zumkeller, Aug 14 2011

a:= proc(n) local l, m; l:= convert(n, base, 10); m:= nops(l);
      `if`(m<2, 0, max(seq(parse(cat(seq(l[m-i], i=0..j-1)))
       *parse(cat(seq(l[m-i], i=j..m-1))), j=1..m)))
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, May 22 2009

Flatten[With[{c=Range[0,9]},Table[c*n,{n,0,10}]]] (* Harvey P. Dale, Jun 07 2012 *)

apply( {A035930(n)=if(n>9,vecmax([vecprod(divrem( n,10^j))|j<-[1..logint(n,10)]]))}, [0..111]) \\ M. F. Hasler, Sep 01 2021

def a(n):
    s = str(n)
    return max((int(s[:i])*int(s[i:]) for i in range(1, len(s))), default=0)
print([a(n) for n in range(108)]) # Michael S. Branicky, Sep 01 2021

An erroneous formula was deleted by N. J. A. Sloane, Dec 23 2008

A257850 a(n) = floor(n/10) * (n mod 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 10, multiply the last digit by the number with its last digit removed.
See A142150(n-1) for the base 2 analog and A257843 - A257849 for the base 3 - base 9 variants.
The first 100 terms coincide with those of A035930 (maximal product of any two numbers whose concatenation is n), A171765 (product of digits of n, or 0 for n<10), A257297 ((initial digit of n)*(n with initial digit removed)), but the sequence is of course different from each of these three.
The terms a(10) - a(100) also coincide with those of A007954 (product of decimal digits of n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257849.

Cf. also A007954, A035930, A171765, A257297.

[Floor(n/10)*(n mod 10): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Table[Floor[n/10] Mod[n, 10], {n, 100}] (* Vincenzo Librandi, May 11 2015 *)

PARI
```
a(n,b=10)=(n=divrem(n,b))[1]*n[2]
```

def A257850(n): return n//10*(n%10) # M. F. Hasler, Sep 01 2021

a(n) = 2*a(n-10)-a(n-20). - Colin Barker, May 11 2015

G.f.: x^11*(9*x^8+8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^4-x^3+x^2-x+1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, May 11 2015

A330633 The concatenation of the products of every pair of consecutive digits of n (with a(n) = 0 for 0 <= n <= 9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0

Offset: 0

Scott R. Shannon, Dec 21 2019

If the decimal expansion of n is d_1 d_2 ... d_k then a(n) is the number formed by concatenating the decimal numbers d_1*d_2, d_2*d_3, ..., d_{k-1}*d_k.

Due to the fact that for two digit numbers the sequence is simply the multiplication of those two numbers, this sequence matches numerous others for the first 100 terms. See the sequences in the cross references. The terms begin to differ beyond n = 100.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A007954, A088117, A040115, A053392, A035930, A257850, A080464, A171765, A169669, A257297.

read("transforms") :
A330633 := proc(n)
    local dgs,L,i ;
    if n <=9 then
        0;
    else
        dgs := ListTools[Reverse](convert(n,base,10)) ;
        L := [] ;
        for i from 2 to nops(dgs) do
            L := [op(L), op(i-1,dgs)*op(i,dgs)] ;
        end do:
        digcatL(L) ;
    end if;
end proc: # R. J. Mathar, Jan 11 2020

Array[If[Or[# == 0, IntegerLength@ # == 1], 0, FromDigits[Join @@ IntegerDigits[Times @@ # & /@ Partition[IntegerDigits@ #, 2, 1]]]] &, 81, 0] (* Michael De Vlieger, Dec 23 2019 *)

a(n) = my(d=digits(n), s="0"); for (k=1, #d-1, s=concat(s, d[k]*d[k+1])); eval(s); \\ Michel Marcus, Apr 28 2020

a(10) = 0 as 1 * 0 = 0.
a(29) = 18 as 2 * 9 = 18.
a(100) = 0 as 1 * 0 = 0 and 0 = 0 = 0, and '00' is reduced to 0.
a(110) = 10 as 1 * 1 = 1 and 1 * 0 = 0. This is the first term that differs from A007954 and A171765, the multiplication of all digits of n.

A347470 Least product of any two numbers whose concatenation is n, excluding 0*n for n > 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 0, 11, 12

Offset: 0

M. F. Hasler, Sep 03 2021

Leading zeros are not allowed: e.g., 101 = concat(10,1) but not concat(1,01). Although 0 is a valid number, we don't allow the trivial decomposition n = concat(0, n) except for the single-digit n < 10, otherwise the minimal product would always be 0.

For n < 111, this sequence coincides with A035930 (same with "largest"), because there is only one possible concatenation, but it differs for n > 111, cf. examples.

			The number n = 112 is the concatenation of 1 and 12, or of 11 and 2, with respective products 1*12 = 12 and 11*2 = 22. Hence, a(112) = 12, while A035930(112) = 22.

Eric Angelini, Surface of a number, Sep 01 2021.

Cf. A035930, A007954, A088117, A171765 and A257297.

apply( {A347470(x,t(b,c)=if(c\10<=b%c,b\c*(b%c),c>10,oo))= if(x>9,vecmin(vector(logint(x,10),j,t(x,10^j))))}, [0..112])

cp's OEIS Frontend

A368062 Numbers k such that k = A257850(k) + A257297(k).

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A035930 Maximal product of any two numbers whose concatenation is n.

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A257850 a(n) = floor(n/10) * (n mod 10).

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A330633 The concatenation of the products of every pair of consecutive digits of n (with a(n) = 0 for 0 <= n <= 9).

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Author

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A347470 Least product of any two numbers whose concatenation is n, excluding 0*n for n > 9.

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Author

Keywords

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Examples

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Crossrefs

Programs

PARI