A034052 -id:A034052 - cp's OEIS frontend

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

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

Martin Renner, Oct 19 2015

Partial sums of A263479.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A031347, A034052, A263479.

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 *)

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

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)

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

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

Martin Renner, Oct 19 2015

First differences of A263473.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A031347, A034052, A263473.

Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 5 &] (* Michael De Vlieger, Oct 21 2015 *)

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

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)

a(9)-a(30) from Hiroaki Yamanouchi, Oct 25 2015

A277061 Numbers with multiplicative digital root > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 57, 61, 62, 63, 64, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 97, 98, 99, 111, 112, 113, 114, 115

Offset: 1

J. Lowell, Sep 26 2016

Question: when will numbers not in this sequence outnumber numbers in this sequence? Up to n = 1249, there are 524 terms, so 525 terms not in this sequence. Up to n = 1522, there are n/2 terms. No n > 1522 has that property. Up to 10^10, only about 1.46% of numbers are a term.

To find how many terms there are up to 10^n, see if A009994(i) is for 2 <= i <= binomial(n + 9, 9). If it is then that adds A047726(A009994(i)) to the total (we don't have to worry about digits 0 in A009994(i) as there aren't any for the specified i). One may put further constraints on i. For example, A009994(i) can't contain an even digit and a 5 in the same number. - David A. Corneth, Sep 27 2016

			25 is not in this sequence because 2*5 = 10 and 1*0 = 0.

Index entries for 10-automatic sequences.

Cf. A009994, A047726.
Cf. A031347, A034048 (complement).
Cf. A028843 (a subsequence).
Union of A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 1-9).
Cf. A052382 (a supersequence).

Select[Range@ 112, FixedPoint[Times @@ IntegerDigits@ # &, #] > 0 &] (* Michael De Vlieger, Sep 26 2016 *)

is(n) = n=digits(n); while(#n>1,n=digits(prod(i=1,#n,n[i]))); #n>0 \\ David A. Corneth, Sep 27 2016

More terms from Michael De Vlieger, Sep 26 2016

A199985 Numbers with digital product = 5.

Original entry on oeis.org

5, 15, 51, 115, 151, 511, 1115, 1151, 1511, 5111, 11115, 11151, 11511, 15111, 51111, 111115, 111151, 111511, 115111, 151111, 511111, 1111115, 1111151, 1111511, 1115111, 1151111, 1511111, 5111111, 11111115, 11111151, 11111511, 11115111, 11151111, 11511111

Offset: 1

Jaroslav Krizek, Nov 13 2011

Subsequence of A034052.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

f:= proc(d) local b,i;
  b:= (10^d-1)/9;
  seq(b+4*10^i,i=0..d-1);
end proc:
seq(f(d),d=1..9);

Sort[FromDigits/@Flatten[Table[Permutations[PadRight[{5},n,1]],{n,9}],1]] (* Harvey P. Dale, Sep 03 2016 *)

A201018 Composite numbers whose multiplicative digital root is 5.

Original entry on oeis.org

15, 35, 51, 57, 75, 115, 135, 153, 175, 315, 351, 355, 395, 511, 513, 517, 531, 535, 539, 553, 575, 579, 597, 715, 755, 759, 795, 935, 957, 975, 1115, 1135, 1157, 1175, 1315, 1351, 1355, 1359, 1395, 1513, 1517, 1535, 1539, 1557, 1575, 1593, 1715, 1751, 1755, 1795

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A201017 with respect to A034052.

			Composite number 153 is in the sequence because 1*5*3=15, 1*5=5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root

Cf. A201017 (primes whose multiplicative digital root is 5), A034052 (numbers whose multiplicative digital root is 5).

mdr5Q[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]==5; Select[Range[1800], CompositeQ[ #] &&mdr5Q[#]&] (* Harvey P. Dale, Dec 19 2023 *)

A201017 Primes whose multiplicative digital root is 5.

Original entry on oeis.org

5, 53, 151, 157, 359, 557, 571, 593, 751, 953, 1151, 1153, 1511, 1531, 1553, 1571, 1579, 1597, 1759, 3511, 3533, 5113, 5153, 5171, 5179, 5197, 5333, 5351, 5531, 5711, 5791, 7151, 7159, 7559, 7577, 7591, 7757, 7951, 9157, 11351, 11579, 11593, 11597, 11953

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A201018 with respect to A034052.

			Prime 157 is in the sequence because 1*5*7=35, 3*5=15, 1*5=5.

Eric Weisstein's World of Mathematics, Multiplicative Digital Root

Cf. A201018 (composite numbers whose multiplicative digital root is 5), A034052 (numbers whose multiplicative digital root is 5).

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

Sergio Pimentel, Mar 27 2024

Conjectured to be complete.

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

Cf. A034052, A263473, A263479, A031347.

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 *)

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

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

A201014 Composite numbers (include 0) whose product of digits is 0.

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 102, 104, 105, 106, 108, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301, 302, 303, 304, 305, 306, 308, 309, 310, 320

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A056709 with respect to A011540. Subsequence of A199978 (nonprime numbers (including 0) whose multiplicative digital root is 0).

			Number 102 is in sequence because 1*0*2=0.

Cf. A056709 (primes whose product of digits is 0), A034052 (numbers whose product of digits is 0).

Join[{0},Select[Range[400],CompositeQ[#]&&DigitCount[#,10,0]>0&]] (* Harvey P. Dale, Jul 27 2022 *)

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

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

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A277061 Numbers with multiplicative digital root > 0.

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A199985 Numbers with digital product = 5.

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A201018 Composite numbers whose multiplicative digital root is 5.

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A201017 Primes whose multiplicative digital root is 5.

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

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A201014 Composite numbers (include 0) whose product of digits is 0.

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