A322001 -id:A322001 - cp's OEIS frontend

A085215 Square array A(x,y) = the number whose factorial expansion A007623 is that of x and y concatenated; zero expanded as empty string; read by ascending antidiagonals: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

0, 1, 1, 2, 3, 2, 3, 7, 8, 3, 4, 9, 26, 9, 4, 5, 13, 32, 27, 10, 5, 6, 15, 50, 33, 28, 11, 6, 7, 25, 56, 51, 34, 29, 30, 7, 8, 27, 122, 57, 52, 35, 126, 31, 8, 9, 31, 128, 123, 58, 53, 150, 127, 32, 9, 10, 33, 146, 129, 124, 59, 246, 151, 128, 33, 10, 11, 37, 152, 147, 130, 125, 270, 247, 152, 129, 34, 11

Offset: 0

Antti Karttunen, Jun 23 2003

			From _M. F. Hasler_, Nov 27 2018: (Start)
The array starts:
   0   1   2   3   4   5  6 ...
   1   3   8   9  10  11 30 ...
   2   7  26  27  28  29 ...
   3   9  32  33  34  ...
   4  13  50  51  ...
  (...)  (End)
A(4,3) = 51 which has a factorial expansion '2011' (2*24+0*6+1*2+1*1), a concatenation of factorial expansions of 4, '20' and of 3, '11'. Similarly, A(3,4) = 34 which has a factorial expansion '1120' (1*24+1*6+2*2+0*1). See A085217 for the corresponding factorial expansions.

Transpose: A085216. Variant: A085219. Can be used to compute A085201. Cf. A000142, A007623, A084558, A025581, A002262.

A085215(x,y)=A322001(eval(Str(A007623(x),A007623(y)))) \\ M. F. Hasler, Nov 27 2018. N.B. This has the same caveat as Hasler's formula. See below for program that is correct for all x, y >= 0. - Antti Karttunen, Mar 24 2025

up_to = 78;
A085215sq(x,y) = { my(i=2,j=2,z=0,f=1); while(y>0, z += (y%i)*f; f *= i; y \= i; i++); while(x>0, z += (x%j)*f; f *= i; x \= j; i++; j++); (z); };
A085215list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A085215sq(a-col, col))); (v); };
v085215 = A085215list(up_to);
A085215(n) = v085215[1+n]; \\ Antti Karttunen, Mar 24 2025

A(x,y) = A322001(concat(A007623(x), A007623(y))), where A322001 is a left inverse of A007623. - M. F. Hasler, Nov 27 2018. Note: this formula is valid only with x and y such that A322001(A007623(x)) = x and A322001(A007623(y)) = y, i.e., at least for all x,y <= 36287999. See N. J. A. Sloane's Jun 04 2012 comment in A007623. - Antti Karttunen, Feb 23 2025

More terms added by Antti Karttunen, Mar 24 2025

A156230 Sum of the products of the digits of n and the positions of the digits m in n starting from the last digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 3

Offset: 1

Cino Hilliard, Feb 06 2009

Starts to differ from A081594 when n>=100. [From R. J. Mathar, Feb 19 2009]

			a(19) = 9*1 + 1*2 = 11.

A156230 := proc(n)
    local dgs;
    dgs := convert(n,base,10) ;
    add(i*op(i,dgs),i=1..nops(dgs)) ;
end proc:
seq(A156230(n),n=1..100) ; # R. J. Mathar, Dec 02 2018

pdn[n_]:=Module[{idn=IntegerDigits[n]},Total[idn Range[Length[idn],1,-1]]]; Array[pdn,80] (* Harvey P. Dale, Aug 22 2012 *)

gr(n) = v=Vec((rev(n)));sum(x=1,length(v),x*eval(v[x]))
gr1(n) = for(j=1,n,print1(gr(j)","))
rev(str) = /* Get the reverse of the input string
*/ {
local(tmp,s,j);
tmp = Vec(Str(str));
s="";
forstep(j=length(tmp),1,-1,
s=concat(s,tmp[j]));
return(s)
}

Let n = d(1)d(2)...d(m) where d(1),d(2),...,d(m) are the digits of n. Then a(n) = m*d1+(m-1)*d2+...+d(m).

Changed the description, formula and Pari code Cino Hilliard, Feb 08 2009

More terms to separate from A322001. - R. J. Mathar, Dec 02 2018

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A085215 Square array A(x,y) = the number whose factorial expansion A007623 is that of x and y concatenated; zero expanded as empty string; read by ascending antidiagonals: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

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A156230 Sum of the products of the digits of n and the positions of the digits m in n starting from the last digit.

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