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.

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

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

A263474 Total number of positive integers < 10^n with multiplicative digital root value 6.

Original entry on oeis.org

1, 14, 155, 1172, 6843, 43538, 318457, 2223803, 14185700, 84670477, 477808607, 2577052118, 13759255632, 75251167843, 418157757456, 2267313716636, 11616142299625, 55909713312571, 257522103127082, 1182251998919171, 5791219719115580, 32715779086392723
Offset: 1

Views

Author

Martin Renner, Oct 19 2015

Keywords

Comments

Partial sums of A263480.

Links

Crossrefs

Cf. A031347, A034053, A263480.

Programs

  • Mathematica
    lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 6 &]; 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) == 6, 1, 0)); \\ Altug Alkan, Oct 19 2015

Formula

A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + A263473(n) + a(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

Extensions

a(9)-a(22) 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

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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