A123384 -id:A123384 - cp's OEIS frontend

A371470 Triangle read by rows: for 1 <= k <= n, T(n,k) is the least sum of decimal digits of numbers with n binary digits and binary weight k.

1, 2, 3, 4, 5, 7, 8, 1, 2, 6, 7, 2, 3, 3, 4, 5, 4, 5, 6, 7, 9, 10, 8, 1, 2, 2, 3, 10, 11, 4, 2, 3, 4, 5, 7, 12, 13, 5, 4, 3, 4, 5, 6, 6, 7, 8, 7, 8, 6, 7, 1, 2, 3, 4, 6, 7, 5, 4, 2, 3, 2, 3, 3, 4, 6, 13, 14, 6, 5, 3, 4, 4, 3, 4, 6, 7, 8, 18, 19, 5, 6, 6, 5, 6, 6, 6, 7, 8, 9, 14, 19, 20, 7, 8, 5

Offset: 1

Robert Israel, May 31 2024

			T(5,3) = 3 because the numbers with 5 binary digits of which 3 are 1 are 19, 21, 22, 25, 26 and 28, and the least sum of decimal digits of these is 3 (for 21).
Triangle starts:
  1;
  2, 3;
  4, 5, 7;
  8, 1, 2, 6;
  7, 2, 3, 3, 4;
  5, 4, 5, 6, 7, 9;

Robert Israel, Table of n, a(n) for n = 1..325 (rows 1 to 25, flattened)

Cf. A000120, A007953, A118738, A123384, A373289.

M:= proc(n,k) local i,R,m,r,v;
   R:= ListTools:-Reverse(map(t -> 2^(n-1)+add(2^(n-1-t[i]),i=1..k-1), combinat:-choose(n-1,k-1 )));
   m:= infinity;
   for r in R do
     v:= convert(convert(r,base,10),`+`);
     if v < m then m:= v fi;
   od;
   m
end proc:
for n from 1 to 12 do
  seq(M(n,k),k=1..n)
od;

a(n) = A007953(A373289(n)).

T(A123384(i),A118738(i)) = 1.

A373289 Triangle read by rows: for 1 <= k <= n, T(n,k) is a number with n binary digits and binary weight k whose sum of decimal digits is least; in case of a tie, choose the least such number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 15, 16, 20, 21, 30, 31, 32, 40, 41, 51, 61, 63, 64, 80, 100, 101, 110, 111, 127, 128, 130, 200, 201, 211, 221, 223, 255, 256, 320, 400, 300, 301, 311, 501, 510, 511, 512, 520, 521, 600, 601, 1000, 1001, 1011, 1021, 1023, 1024, 1040, 1030, 1100, 1101, 2000, 2001, 2010

Offset: 1

Robert Israel, May 30 2024

			T(5,3) = 21 because 21 = 10101_2 has 5 binary digits of which 3 are 1, and it has a lower sum of decimal digits, 3, than the other numbers (19, 22, 25, 26 and 28) with 5 binary digits of which 3 are 1.
Triangle starts
   1;
   2,  3;
   4,  5,  7;
   8, 10, 11, 15;
  16, 20, 21, 30, 31;
  32, 40, 41, 51, 61, 63;

Robert Israel, Table of n, a(n) for n = 1..325 (rows 1 to 25, flattened)

Cf. A000120, A007953, A118738, A123384, A371470.

M:= proc(n,k) local i,R,m,b,r,v;
   R:= ListTools:-Reverse(map(t -> 2^(n-1)+add(2^(n-1-t[i]),i=1..k-1), combinat:-choose(n-1,k-1 )));
   m:= infinity;
   for r in R do
     v:= convert(convert(r,base,10),`+`);
     if v < m then b:= r; m:= v fi;
   od;
   b
end proc:
for n from 1 to 12 do
  seq(M(n,k),k=1..n)
od;

T(A123384(i),A118738(i)) = 10^i.
T(m,1) = 2^(m-1).
T(m,m) = 2^m - 1.

A025740 Index of 10^n within sequence of numbers of form 2^i*10^j.

Original entry on oeis.org

1, 5, 12, 22, 36, 53, 73, 97, 124, 154, 188, 225, 265, 309, 356, 406, 460, 517, 577, 641, 708, 778, 852, 929, 1009, 1093, 1180, 1270, 1364, 1461, 1561, 1664, 1771, 1881, 1994, 2111, 2231, 2354, 2481, 2611, 2744, 2881, 3021, 3164, 3311, 3461, 3614, 3771, 3931

Offset: 1

David W. Wilson

nonn

Positions of zeros in A025639. - R. J. Mathar, Jul 06 2025

Partial sums of A123384.

cp's OEIS Frontend

A371470 Triangle read by rows: for 1 <= k <= n, T(n,k) is the least sum of decimal digits of numbers with n binary digits and binary weight k.

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A373289 Triangle read by rows: for 1 <= k <= n, T(n,k) is a number with n binary digits and binary weight k whose sum of decimal digits is least; in case of a tie, choose the least such number.

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Examples

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Programs

Maple

Formula

A025740 Index of 10^n within sequence of numbers of form 2^i*10^j.

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