A006064 -id:A006064 - cp's OEIS frontend

A230857 Leading power of 10 in A006064(n).

Original entry on oeis.org

2, 13, 24, 1111111111124, 2222222222224, 111111111112222222222235, 222222222222222222222246

Offset: 2

N. J. A. Sloane, Oct 31 2013

a(9) = ( 10^1111111111124 + 10^24 + 214 ) / 9.

a(10) = ( 2*10^1111111111124 + 214 ) / 9.

Max Alekseyev, Table n, expression for a(n) for n=2..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.

Cf. A006064.

Terms a(9) onward from Max Alekseyev, Oct 31 2013

A230093 Number of values of k such that k + (sum of digits of k) is n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1

Offset: 0

N. J. A. Sloane, Oct 10 2013

a(n) is the number of times n occurs in A062028.

For n>=1, a(10^n) = a(9*n-1). - Max Alekseyev, Feb 23 2021

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A006064, A007953 (sum of digits), A062028 (n + sum of its digits), A004207, A228085, A003052, A176995, A225793, A230094, A055642.

Cf. A107740 (this applied to primes).

a230093 n = length $ filter ((== n) . a062028) [n - 9 * a055642 n .. n]  -- Reinhard Zumkeller, Oct 11 2013

# Maple code for A062028, A230093, A003052, A225793, A230094.
with(LinearAlgebra):
read transforms; # to get digsum
M := 1000; A062028 := Array(0..M); A230093 := Array(0..M);
for n from 0 to M do
   m := n+digsum(n);
   A062028[n] := m;
   if m <= M then A230093[m] := A230093[m]+1; fi;
od:
t1:=[seq(A062028[i],i=0..M)]; # A062028 as list (but incorrect offset 1)
t2:=[seq(A230093[i],i=0..M)]; # A230093 as list, but then a(0) has index 1
# A003052 := COMPl(t1); # COMPl has issues, may be incorrect for M <> 1000
ctmax:=4;
for h from 0 to ctmax do ct[h] := []; od:
for i from 1 to M do
   h := lis2[i];
   if h <= ctmax then ct[h] := [op(ct[h]),i]; fi;
od:
A225793 := ct[1]; A230094 := ct[2]; # A003052 := ct[0]; # see there for better code

Module[{nn=110,a,b,c,d},a=Tally[Table[x+Total[IntegerDigits[x]],{x,0,nn}]];b=a[[All,1]];c={#,0}&/@Complement[Range[nn],b];d=Sort[Join[a,c]];d[[All, 2]]] (* Harvey P. Dale, Jun 12 2019 *)

apply( A230093(n)=sum(i=n>0,min(9*logint(n+!n,10)+8,n\2),sumdigits(n-i)==i), [1..150]) \\ M. F. Hasler, Nov 08 2018

Edited by M. F. Hasler, Nov 08 2018

A230640 Let M(1)=0 and for n>1, B(n)=(M(ceiling(n/2))+M(floor(n/2))+2)/2, M(n)=3^B(n)+M(floor(n/2))+1. This sequence gives M(n).

Original entry on oeis.org

0, 4, 28, 248, 129140168, 68630377364912, 2088595827392656793085408064780643444068898148936888424953199350296

Offset: 1

N. J. A. Sloane, Oct 31 2013

Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.

Cf. A230639.
Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)
Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0, 0]);
else
B:=(f(ceil(n/2))[2] + f(floor(n/2))[2] + 2)/2;
M:=3^B+f(floor(n/2))[2]+1; RETURN([B, M]); fi;
end proc;
[seq(f(n)[2], n=1..7)];

A230638 Smallest number m such that u + (sum of base-4 digits of u) = m has exactly n solutions.

Original entry on oeis.org

0, 17, 16385, 16777234

Offset: 1

N. J. A. Sloane, Oct 31 2013

Indices of records in A230632: a(n) is the index of the first n in A230632.
The terms are a(1)=0, a(2)=4^2+1, a(3)=4^7+1, a(4)=4^12+17+1, a(5)=4^5368+17+1, a(6)=4^10924+16385+1, a(7)=4^5597880+16385+20. Note that a(7) breaks the pattern of the first six terms.
a(8) = 4^16777229 + 4^12 + 19.
For the leading power of 4 see A230637.

			n=2: the two solutions to u+(base-4 digit-sum of u) = 17 are 13 and 16.
n=3: the three solutions to u+(base-4 digit-sum of u) = 4^7+1 are 4^7, 4^7-15, 4^7-18.
n=4: the four solutions to u+(base-4 digit-sum of u) = 4^12+17+1 are 4^12+{16, 13, -14, -17}.

Max Alekseyev, Table of expressions for a(n), for n=1..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
Index entries for Colombian or self numbers and related sequences

Cf. A230637.
Related base-4 sequences:  A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)
Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

a(8) from Max Alekseyev, Oct 31 2013

A107740 Number of numbers m such that prime(n) = m + (digit sum of m).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1

Offset: 1

Reinhard Zumkeller, May 23 2005

a(A049084(A006378(n))) = 0; a(A049084(A048521(n))) > 0. [Corrected by Reinhard Zumkeller, Sep 27 2014]
a(n) <= 2 for n <= 10^5. Conjecture: sequence is bounded.
I would rather conjecture the opposite. Of course a(n) >= m implies n >= A006064(m), having more than A230857(m) digits, i.e., 14, 25 and 1111111111125 digits of n, for a(n) = 3, 4, 5. - M. F. Hasler, Nov 09 2018

			A000040(26) = 101 = 91 + (9 + 1) = 100 + (1 + 0 + 0): a(26) = # {91, 100} = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007953, A000040, A062028, A047791, A107741.

Cf. A006378, A048521, A049084, A055642, A062028.

a107740 n = length [() | let p = a000040 n,
                         m <- [max 0 (p - 9 * a055642 p) .. p - 1],
                         a062028 m == p]
-- Reinhard Zumkeller, Sep 27 2014

Table[p=Prime[n];c=0;i=1;While[iJayanta Basu, May 03 2013 *)

apply( A107740(n)=A230093(prime(n)), [1..150]) \\ M. F. Hasler, Nov 08 2018

a(n) = A230093(prime(n)), i.e.: A107740 = A230093 o A000040. - M. F. Hasler, Nov 08 2018

A230303 Let M(1)=0 and for n >= 2, let B(n)=M(ceiling(n/2))+M(floor(n/2))+2, M(n)=2^B(n)+M(floor(n/2))+1; sequence gives M(n).

Original entry on oeis.org

0, 5, 129, 4102, 87112285931760246646623899502532662132742, 1852673427797059126777135760139006525652319754650249024631321344126610074239106

Offset: 1

N. J. A. Sloane, Oct 24 2013; Mar 26 2014

nonn

M(n) is the smallest value of k such that A228085(k) = n. For example, 129 is the first time a 3 appears in A228085 (and is therefore the first term in A230092). M(4) = 4102 is the first time a 4 appears in A228085 (and is therefore the first term in A227915).

			The terms are a(1) = 0, a(2) = 2^2+0+1, a(3) = 2^7+0+1, a(4) = 2^12+5+1, a(5) = 2^136+5+1, a(6) = 2^160+129+1, a(7) = 2^4233+129+1, a(8) = 2^8206+4102+1, a(9) = 2^k+4102+1 with k=2^136+4110, ... .
The length (in bits) of the n-th term is A230302(n)+1.

Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
Index entries for Colombian or self numbers and related sequences

Cf. A228085, A230092, A227915, A230093, A230302 (for B(n)), A230864.

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0,0]);
else
if (n mod 2) = 0 then B:=2*f(n/2)[2]+2;
else B:=f((n+1)/2)[2]+f((n-1)/2)[2]+2; fi;
M:=2^B+f(floor(n/2))[2]+1; RETURN([B,M]); fi;
end proc;
[seq(f(n)[2],n=1..6)];

Define i by 2^(i-1) < n <= 2^i. Then it appears that
a(n) = 2^2^2^...^2^x
a tower of height i+3, containing i+2 2's, where x is in the range 0 < x <= 1.
For example, if n=7, i=3, and
a(7) = 2^4233+130 = 2^2^2^2^2^.88303276...
Note also that i+2 = A230864(a(n)).

a(1)-a(8) were found by Donovan Johnson, Oct 22 2013.

A230867 Smallest number m such that u + (sum of base-5 digits of u) = m has exactly n solutions.

Original entry on oeis.org

0, 6, 26, 632, 1953134, 30517578152

Offset: 1

N. J. A. Sloane, Nov 05 2013

Indices of records in A230866: a(n) is the index of the first n in A230866.

The next two terms are a(7) = 5^165 + 27, a(8) = 5^317 + 633.

			a(5) = 1953134 corresponds to the five solutions:
1953099 (base-5: 444444344)
1953103 (base-5: 444444403)
1953105 (base-5: 444444410)
1953129 (base-5: 1000000004)
1953131 (base-5: 1000000011).

Pontus von Brömssen, Table of n, a(n) for n = 1..8 (based on comment about a(7) and a(8))
Max Alekseyev, Table of expressions for a(n), for n=1..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.

A230868 gives the leading power of 5 in a(n).
Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.
Cf. A230865, A230866.

a(5) corrected by Donovan Johnson, Nov 05 2013

A238840 Smallest number m such that u + (sum of base-6 digits of u) = m has exactly n solutions.

Original entry on oeis.org

0, 37, 10077697, 2821109907494

Offset: 1

N. J. A. Sloane, Mar 19 2014

The next term is a(5) = 6^((6^9+6^2+8)/5) + 38 = 6^2015548 + 38 and is too large to display.

Max Alekseyev, Table of expressions for a(n), for n=1..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

A238841 Smallest number m such that u + (sum of base-7 digits of u) = m has exactly n solutions.

Original entry on oeis.org

0, 8, 50, 352, 282475258, 232630513987258, 418377847259091645147530834859099334519176045887014771594

Offset: 1

N. J. A. Sloane, Mar 19 2014

Max Alekseyev, Table of expressions for a(n), for n=1..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

A238842 Smallest number m such that u + (sum of base-8 digits of u) = m has exactly n solutions.

Original entry on oeis.org

0, 65, 8589934593, 1152921504606847042

Offset: 1

N. J. A. Sloane, Mar 19 2014

The next term is a(5) = 8^((8^11+76)/7) + 66 = 8^1227133524 + 66 and is too large to display.

Max Alekseyev, Table of expressions for a(n), for n=1..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

cp's OEIS Frontend

A230857 Leading power of 10 in A006064(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A230093 Number of values of k such that k + (sum of digits of k) is n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Extensions

A230640 Let M(1)=0 and for n>1, B(n)=(M(ceiling(n/2))+M(floor(n/2))+2)/2, M(n)=3^B(n)+M(floor(n/2))+1. This sequence gives M(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A230638 Smallest number m such that u + (sum of base-4 digits of u) = m has exactly n solutions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A107740 Number of numbers m such that prime(n) = m + (digit sum of m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A230303 Let M(1)=0 and for n >= 2, let B(n)=M(ceiling(n/2))+M(floor(n/2))+2, M(n)=2^B(n)+M(floor(n/2))+1; sequence gives M(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A230867 Smallest number m such that u + (sum of base-5 digits of u) = m has exactly n solutions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A238840 Smallest number m such that u + (sum of base-6 digits of u) = m has exactly n solutions.

Views

Author

Keywords

Comments

Links

Crossrefs