A126783 -id:A126783 - cp's OEIS frontend

A130179 Largest k such that k <= 81(number of digits of k^n)(number of digits of k^(n+1)).

2268, 7776, 18954, 35397, 56376, 85050, 119556, 159894, 209952, 267300, 331047, 402084, 479520, 570807, 670032, 777195, 892296, 1015335, 1146312, 1285227, 1432080, 1586871, 1749600, 1932498, 2125035, 2312712, 2522340, 2741607

Offset: 1

Klaus Brockhaus, May 20 2007

a(n) is an upper bound for A130181(n) and all the more so for A126783(n); apparently even A130181(n) < a(n)/4.
All terms are divisible by 81; the quotients a(n)/81 are in A130085.
For some n (18, 34, 35, 38, 42, 58, 59, ...) the line y = x and the graph of the staircase function y = 81*(number of digits of x^n)*(number of digits of x^(n+1)) intersect twice; this possibility has to be taken into account by the program.

			Let D(n,k) = 81*(number of digits of k^n)*(number of digits of k^(n+1)).
D(2,k) > k for k = 1..4641, D(2,k) = 7776 for k = 4642..9999, D(2,k) < k for k >= 10000, hence a(2) = 7776.
D(18,k) > k for k = 1..885866, D(18,k) = 997272 for k = 885867..999999, D(18,k) = 1015335 for k = 1000000..1128837, D(18,k) < k for k >= 1128838, hence a(18) = 1015335.

Klaus Brockhaus, Table of n, a(n) for n=1..100

Cf. A126783, A130181, A130085.

{for(n=1, 28, s=30*n; k=s; while(k<81*length(Str(k^n))*length(Str(k^(n+1))), k+=s); r=0; g=0; k-=s; b=1; while(b, p=81*length(Str(k^n))*length(Str(k^(n+1))); if(rr, b=0, g=h)); k++); print1(g, ","))}

A130181 Largest k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k, or 0 if no such k exists.

Original entry on oeis.org

486, 1215, 4374, 4672, 12862, 12649, 23408, 32761, 47477, 56852, 59048, 90746, 116864, 112346, 139472, 149705, 190512, 234247, 254015, 0, 322322, 331775, 391238, 446512, 454951, 546121, 530145, 316250, 613927, 763795, 786664, 809936

Offset: 1

Klaus Brockhaus, May 14 2007

			For n = 2 the largest such k is 1215: 1215^2 = 1476225 and 1+4+7+6+2+2+5 = 27; 1215^3 = 1793613375and 1+7+9+3+6+1+3+3+7+5 = 45; 27*45 = 1215. Hence a(2) = 1215.

Klaus Brockhaus, Table of n, a(n) for n=1..54

Cf. A126783 (smallest k), A130179 (upper bound), A130180 (number of such k).

A130180 Number of numbers k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k.

Original entry on oeis.org

5, 3, 12, 2, 6, 8, 4, 1, 13, 8, 7, 14, 8, 3, 9, 1, 5, 12, 4, 0, 13, 4, 7, 7, 1, 4, 7, 2, 5, 8, 2, 4, 8, 7, 1, 10, 5, 2, 8, 4, 2, 10, 2, 6, 10, 2, 3, 6, 2, 4, 4, 2, 3, 9, 2, 3, 8, 1, 3, 8, 5, 3, 6, 4, 6, 8, 4, 3, 10, 0, 1, 6, 3, 6, 6, 4, 2, 7, 2, 1

Offset: 1

Klaus Brockhaus, May 14 2007

			80, 1036, 1215 are the only numbers k > 1 such that (sum of digits of k^2)*(sum of digits of k^3) = k, hence a(2) = 3.

Lars Blomberg, Table of n, a(n) for n = 1..116

Cf. A126783 (smallest k), A130179 (upper bound), A130181 (largest k).

a(55)-a(80) and b-file from Lars Blomberg, Dec 11 2011

cp's OEIS Frontend

A130179 Largest k such that k <= 81(number of digits of k^n)(number of digits of k^(n+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A130181 Largest k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k, or 0 if no such k exists.

Views

Author

Keywords

Examples

Links

Crossrefs

A130180 Number of numbers k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

cp's OEIS Frontend

A130179 Largest k such that k <= 81*(number of digits of k^n)*(number of digits of k^(n+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A130181 Largest k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k, or 0 if no such k exists.

Views

Author

Keywords

Examples

Links

Crossrefs

A130180 Number of numbers k > 1 such that (sum of digits of k^n)*(sum of digits of k^(n+1)) = k.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A130179 Largest k such that k <= 81(number of digits of k^n)(number of digits of k^(n+1)).