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.

A263479 Total number of n-digit positive integers with multiplicative digital root value 5.

Original entry on oeis.org

1, 6, 33, 132, 435, 1466, 5341, 18656, 58029, 159430, 392601, 882036, 1836159, 3586506, 6638885, 11738656, 19952441, 32768742, 52220113, 81029700, 122785131, 182142906, 265066605, 379102400, 533695525, 740551526, 1014046281, 1371688948, 1834642167, 2428304010
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

First differences of A263473.

Links

Crossrefs

Cf. A031347, A034052, A263473.

Programs

  • Mathematica
    Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 5 &] (* 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) == 5, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263476(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + a(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).
a(n) = (1/720)*(3*n^8 + 6*n^7 - 664*n^6 + 6270*n^5 - 25783*n^4 + 55164*n^3 - 57796*n^2 + 23520*n). - Sergio Pimentel, Mar 27 2024
From Chai Wah Wu, Apr 17 2024: (Start)
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 9.
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)^9. (End)

Extensions

a(9)-a(30) 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

A263476 Total number of n-digit positive integers with multiplicative digital root value 0.

Original entry on oeis.org

0, 24, 452, 6263, 75662, 820207, 8491909, 86727773, 879578102, 8878382430, 89326016765, 896499124003, 8981948713433, 89902031287356, 899441785434096, 8996797304009863, 89981980972837834, 899900967867894143, 8999467457237611688, 89997196286115385871
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

First differences of A263470.

Links

Crossrefs

Cf. A031347, A034048, A263470.

Programs

  • Mathematica
    Last /@ Tally@ IntegerLength@ Select[Range[0, 10^6 - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 0 &] (* 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) == 0, 1, 0));  \\ Altug Alkan, Oct 19 2015

Formula

a(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + A263479(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).

Extensions

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

A263477 Total number of n-digit positive integers with multiplicative digital root value 2.

Original entry on oeis.org

1, 8, 68, 466, 2670, 13460, 69420, 417722, 3025242, 21873040, 136901413, 722201372, 3271729383, 13114173697, 48104723380, 167526488628, 574289772576, 1988721563904, 7000834741144, 24759698208450, 86342520209963, 292206955736762, 950480594161453
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

First differences of A263471.

Links

Crossrefs

Cf. A031347, A034049, A051813, A263471.

Programs

  • Mathematica
    Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 2 &] (* 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) == 2, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263476(n) + A000012(n) + a(n) + A000027(n) + A263478(n) + A263479(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).

Extensions

a(9)-a(23) 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

A263481 Total number of n-digit positive integers with multiplicative digital root value 8.

Original entry on oeis.org

1, 22, 138, 889, 4921, 26687, 154539, 870270, 4437621, 20367762, 86589471, 388661761, 2366418106, 18602433227, 142978182070, 952075901928, 5434361620523, 27064481351856, 120477919973804, 497320625548529, 2051776298637086, 9590295679377844, 54933121828772931
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

First differences of A263475.

Links

Crossrefs

Cf. A031347, A034055, A263475.

Programs

  • Mathematica
    Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 8 &] (* 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) == 8, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263476(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + A263479(n) + A263480(n) + A000027(n) + a(n) + A000217(n) = A052268(n).

Extensions

a(9)-a(23) from Hiroaki Yamanouchi, Oct 25 2015
Showing 1-6 of 6 results.

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