cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

N. J. A. Sloane, Oct 31 2013

Keywords

Links

  • 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.

Crossrefs

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.

Programs

  • Maple
    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)];

A006064 Smallest junction number with n generators.

Original entry on oeis.org

0, 101, 10000000000001, 1000000000000000000000102
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Strictly speaking, a junction number is a number n with more than one solution to x+digitsum(x) = n. However, it seems best to start this sequence with n=0, for which there is just one solution, x=0. - N. J. A. Sloane, Oct 31 2013.
a(3) = 10^13 + 1 was found by Narasinga Rao, who reports that Kaprekar verified that it is the smallest term. No details of Kaprekar's proof were given.
a(4) = 10^24 + 102 was conjectured by Narasinga Rao.
a(5) = 10^1111111111124 + 102. - Conjectured by Narasinga Rao, confirmed by Max Alekseyev and N. J. A. Sloane.
a(6) = 10^2222222222224 + 10000000000002. - Max Alekseyev
a(7) = 10^( (10^24 + 10^13 + 115) / 9 ) + 10^13 + 2. - Max Alekseyev
a(8) = 10^( (2*10^24 + 214)/9 ) + 10^24 + 103. - Max Alekseyev

Examples

			a(2) = 101 since 101 is the smallest number with two generators: 101 = A062028(91) = A062028(100).
a(4) = 10^24 + 102 = 1000000000000000000000102 has exactly four inverses w.r.t. A062028, namely 999999999999999999999893, 999999999999999999999902, 1000000000000000000000091 and 1000000000000000000000100.

References

  • M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
  • D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
  • Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003052, A230093, A230100, A230303, A230857 (highest power of 10).
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.

Formula

a(n) = the smallest m such that there are exactly n solutions to A062028(x)=m.

Extensions

Edited, a(5)-a(6) added by Max Alekseyev, Jun 01 2011
a(1) added, a(5) corrected, a(7)-a(8) added by Max Alekseyev, Oct 26 2013

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

Views

Author

N. J. A. Sloane, Oct 31 2013

Keywords

Comments

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.

Examples

			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}.

Links

Crossrefs

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.

Extensions

a(8) from Max Alekseyev, Oct 31 2013

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

Views

Author

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

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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.

Programs

  • Maple
    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)];

Formula

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)).

Extensions

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

Views

Author

N. J. A. Sloane, Nov 05 2013

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

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.

Extensions

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

Views

Author

N. J. A. Sloane, Mar 19 2014

Keywords

Comments

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

Links

Crossrefs

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

Views

Author

N. J. A. Sloane, Mar 19 2014

Keywords

Comments

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

Links

Crossrefs

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.

A238843 Smallest number m such that u + (sum of base-9 digits of u) = m has exactly n solutions.

Original entry on oeis.org

0, 10, 82, 740, 282429536492, 109418989131512359292, 193632597890512706847971583764083347958511186984324587565465147107798425867049291402906445603076812
Offset: 1

Views

Author

N. J. A. Sloane, Mar 19 2014

Keywords

Links

Crossrefs

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.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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