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 1-7 of 7 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

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

A263478 Total number of n-digit positive integers with multiplicative digital root value 4.

Original entry on oeis.org

1, 9, 55, 214, 615, 1451, 3829, 60180, 939045, 8732485, 56961531, 289887214, 1229099287, 4756606869, 24218431805, 233925901576, 2661527233449, 25685325408201, 203451565638511, 1356903584035110, 7832822232934951, 40022453239462639, 184228949831881593
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

First differences of A263472.

Links

Crossrefs

Cf. A031347, A034051, A263472.

Programs

  • Mathematica
    Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 4 &] (* 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) == 4, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263476(n) + A000012(n) + A263477(n) + A000027(n) + a(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

A371561 Numbers with multiplicative digital root of 5 that are free of 1s and have their digits in ascending order.

Original entry on oeis.org

5, 35, 57, 355, 359, 557, 579, 3335, 3357, 5579, 5777, 33557, 35559, 333555, 357799, 557779, 3335779, 3355777, 33333577
Offset: 1

Views

Author

Sergio Pimentel, Mar 27 2024

Keywords

Comments

Conjectured to be complete.
If it exists, a(20) > 10^500. - Michael S. Branicky, Apr 18 2024

Crossrefs

Cf. A034052, A263473, A263479, A031347.

Programs

  • Mathematica
    A031347 = Table[NestWhile[Times @@ IntegerDigits[#] &, n, # > 9 &], {n, 1, 100000}]; Select[Range[100000], A031347[[#]] == 5 && DigitCount[#, 10, 1] == 0 && Sort[IntegerDigits[#]] == IntegerDigits[#] &] (* Vaclav Kotesovec, Apr 17 2024 *)
  • Python
    from math import prod
from itertools import count, islice, combinations_with_replacement as mc
def A031347(n):
    while n > 9: n = prod(map(int, str(n)))
    return n
def bgen(): yield from (m for d in count(1) for m in mc((3,5,7,9), d))
def agen(): yield from (int("".join(map(str, t))) for t in bgen() if A031347(prod(t)) == 5)
print(list(islice(agen(), 19))) # Michael S. Branicky, Apr 17 2024, edited Apr 18 2024 after Chai Wah Wu
  • Python
    from math import prod
from itertools import count, islice
def A371561_gen(): # generator of terms
    for l in count(1):
        for a in range(l,-1,-1):
            a3 = 3**a
            for b in range(l-a,-1,-1):
                b3 = a3*5**b
                for c in range(l-a-b,-1,-1):
                    d = l-a-b-c
                    d3 = b3*7**c*9**d
                    while d3 > 9:
                        d3 = prod(int(x) for x in str(d3))
                    if d3==5:
                        yield (10**(a+b+c+d)-1)//3+(10**d*(10**c*(10**b+1)+1)-3)*2//9
A371561_list = list(islice(A371561_gen(),19)) # Chai Wah Wu, Apr 17 2024
Showing 1-7 of 7 results.

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