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 1-6 of 6 results.

A263473 Total number of positive integers < 10^n with multiplicative digital root value 5.

Original entry on oeis.org

1, 7, 40, 172, 607, 2073, 7414, 26070, 84099, 243529, 636130, 1518166, 3354325, 6940831, 13579716, 25318372, 45270813, 78039555, 130259668, 211289368, 334074499, 516217405, 781284010, 1160386410, 1694081935, 2434633461, 3448679742, 4820368690, 6655010857
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

Partial sums of A263479.

Links

Crossrefs

Cf. A031347, A034052, A263479.

Programs

  • Mathematica
    lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 5 &]; Count[t, n_ /; n <= 10^#] & /@ Range@ lim (* Michael De Vlieger, Oct 21 2015 *)
  • PARI
    t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 5, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + a(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).
From Chai Wah Wu, Apr 17 2024: (Start)
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 10.
G.f.: x*(-235*x^7 + 205*x^6 + 161*x^5 + 57*x^4 - 33*x^3 + 15*x^2 - 3*x + 1)/(x - 1)^10. (End)

Extensions

a(9)-a(29) from Hiroaki Yamanouchi, Oct 25 2015

A263470 Total number of positive integers < 10^n with multiplicative digital root value 0.

Original entry on oeis.org

0, 24, 476, 6739, 82401, 902608, 9394517, 96122290, 975700392, 9854082822, 99180099587, 995679223590, 9977627937023, 99879659224379, 999321444658475, 9996118748668338, 99978099721506172, 999879067589400315, 9999346524827012003, 99996542810942397874
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

Partial sums of A263476. - Michel Marcus, Oct 22 2015

Links

Crossrefs

Cf. A031347, A034048, A263476.

Programs

  • Mathematica
    Length@ Select[Range[10^# - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 0 &] & /@ Range@ 6 (* Michael De Vlieger, Oct 19 2015 *)
  • PARI
    t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 0, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

a(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + A263473(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

Extensions

a(9)-a(20) from Hiroaki Yamanouchi, Oct 25 2015

A263471 Total number of positive integers < 10^n with multiplicative digital root value 2.

Original entry on oeis.org

1, 9, 77, 543, 3213, 16673, 86093, 503815, 3529057, 25402097, 162303510, 884504882, 4156234265, 17270407962, 65375131342, 232901619970, 807191392546, 2795912956450, 9796747697594, 34556445906044, 120898966116007, 413105921852769, 1363586516014222
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

Partial sums of A263477.

Links

Crossrefs

Cf. A031347, A034049, A051822, A263477.

Programs

  • Mathematica
    Length@ Select[Range[10^# - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 2 &] & /@ Range@ 6 (* Michael De Vlieger, Oct 19 2015 *)
  • PARI
    t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 2, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263470(n) + A000027(n) + a(n) + A000217(n) + A263472(n) + A263473(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

Extensions

a(8) from Michael De Vlieger, Oct 19 2015
a(9)-a(23) from Hiroaki Yamanouchi, Oct 25 2015

A263472 Total number of positive integers < 10^n with multiplicative digital root value 4.

Original entry on oeis.org

1, 10, 65, 279, 894, 2345, 6174, 66354, 1005399, 9737884, 66699415, 356586629, 1585685916, 6342292785, 30560724590, 264486626166, 2926013859615, 28611339267816, 232062904906327, 1588966488941437, 9421788721876388, 49444241961339027, 233673191793220620
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

Partial sums of A263478.

Links

Crossrefs

Cf. A031347, A034051, A263478.

Programs

  • Mathematica
    lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 4 &]; Count[t, n_ /; n <= 10^#] & /@ Range@ lim (* Michael De Vlieger, Oct 21 2015 *)
  • PARI
    t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 4, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263470(n) + A000027(n) + A263471(n) + A000217(n) + a(n) + A263473(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

Extensions

a(9)-a(23) from Hiroaki Yamanouchi, Oct 25 2015

A263475 Total number of positive integers < 10^n with multiplicative digital root value 8.

Original entry on oeis.org

1, 23, 161, 1050, 5971, 32658, 187197, 1057467, 5495088, 25862850, 112452321, 501114082, 2867532188, 21469965415, 164448147485, 1116524049413, 6550885669936, 33615367021792, 154093286995596, 651413912544125, 2703190211181211, 12293485890559055
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

Partial sums of A263481.

Links

Crossrefs

Cf. A031347, A034055, A263481.

Programs

  • Mathematica
    lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 8 &]; Count[t, n_ /; n <= 10^#] & /@ Range@ lim (* Michael De Vlieger, Oct 21 2015 *)
  • PARI
    t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 8, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + A263473(n) + A263474(n) + A000217(n) + a(n) + A000292(n) = A002283(n).

Extensions

a(9)-a(22) from Hiroaki Yamanouchi, Oct 25 2015

A263480 Total number of n-digit positive integers with multiplicative digital root value 6.

Original entry on oeis.org

1, 13, 141, 1017, 5671, 36695, 274919, 1905346, 11961897, 70484777, 393138130, 2099243511, 11182203514, 61491912211, 342906589613, 1849155959180, 9348828582989, 44293571012946, 201612389814511, 924729895792089, 4608967720196409, 26924559367277143
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

First differences of A263474.

Links

Crossrefs

Cf. A031347, A034053, A263474.

Programs

  • Mathematica
    Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 6 &] (* Michael De Vlieger, Oct 21 2015 *)
  • PARI
    t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 6, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263476(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + A263479(n) + a(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).

Extensions

a(9)-a(22) from Hiroaki Yamanouchi, Oct 25 2015
Showing 1-6 of 6 results.

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