A093136 -id:A093136 - cp's OEIS frontend

A132026 Decimal expansion of Product_{k>=0} (1 - 1/(2*10^k)).

4, 7, 2, 3, 6, 2, 4, 4, 3, 8, 1, 6, 5, 7, 2, 2, 3, 6, 5, 5, 1, 4, 1, 3, 3, 8, 3, 3, 3, 2, 3, 2, 7, 3, 5, 3, 3, 4, 9, 6, 6, 4, 2, 9, 5, 8, 5, 0, 2, 2, 1, 9, 4, 6, 2, 1, 8, 8, 9, 0, 9, 6, 1, 1, 7, 7, 8, 7, 1, 9, 9, 4, 4, 2, 6, 0, 1, 3, 0, 7, 7, 9, 5, 4, 2, 9, 4, 3, 2, 5, 3, 0, 7, 2, 3, 0, 7, 8, 1, 1, 8, 1, 2

Offset: 0

Hieronymus Fischer, Jul 28 2007

			0.472362443816572236551413383332...

Cf. A000217, A067080, A093136, A098844, A132019.

digits = 103; Product[1-1/(2*10^k), {k, 0, Infinity}] // N[#, digits+1]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/10], 10, 100][[1]] (* Jan Mangaldan, Jan 04 2017 *)

prodinf(k=0, 1 - 1/(2*10^k)) \\ Amiram Eldar, May 09 2023

Equals lim inf_{n->oo} Product_{k=0..floor(log_10(n))} floor(n/10^k)*10^k/n.
Equals lim inf_{n->oo} A067080(n)/n^(1+floor(log_10(n)))*10^(1/2*(1+floor(log_10(n)))*floor(log_10(n))).
Equals lim inf_{n->oo} A067080(n)/n^(1+floor(log_10(n)))*10^A000217(floor(log_10(n))).
Equals lim inf_{n->oo} A067080(n)/A067080(n+1).
Equals 1/2*exp(-Sum_{n>0} 10^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals Product_{n>=1} (1 - 1/A093136(n)). - Amiram Eldar, May 09 2023

A169965 Numbers whose decimal expansion contains only 0's and 2's.

Original entry on oeis.org

0, 2, 20, 22, 200, 202, 220, 222, 2000, 2002, 2020, 2022, 2200, 2202, 2220, 2222, 20000, 20002, 20020, 20022, 20200, 20202, 20220, 20222, 22000, 22002, 22020, 22022, 22200, 22202, 22220, 22222, 200000, 200002, 200020, 200022, 200200, 200202, 200220, 200222

Offset: 1

N. J. A. Sloane, Aug 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to carryless arithmetic

Cf. A007088, A014263, A169964, A169966, A169967.

Cf. A078241, A204093, A204094, A204095, A097256.

a169965 n = a169965_list !! (n-1)
a169965_list = map (* 2) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

Map[FromDigits,Tuples[{0,2},6]] (* Paolo Xausa, Oct 30 2023 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*2)))) \\ Charles R Greathouse IV, Nov 16 2011

lista(N) = vector(N, i, fromdigits(binary(i-1)*2)); \\ Ruud H.G. van Tol, Oct 26 2024

a(n+1) = Sum_{k>=0} A030308(n,k)*A093136(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 2 * A007088(n-1).

A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 49, 56, 63, 70, 77

Offset: 1

Malo David, Nov 10 2021

The only primes in the sequence are 2, 3, 5 and 7. - Bernard Schott, Nov 23 2021

			For n=256, a(256) = 2*(2+5)*(2+5+6) = 182.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Malo David, About the Product Sum Digit Sequence

Cf. A055642, A284001 (binary analog), A349190 (fixed points).
Cf. A007953 (sum of digits), A059995 (floor(n/10)).
Cf. A349278 (similar, with the last digits).
Cf. A349898, A349899.

f:=func; [f(n):n in [1..100]]; // Marius A. Burtea, Nov 23 2021

Table[Product[Sum[Part[IntegerDigits[n],j],{j,i}],{i,Length[IntegerDigits[n]]}],{n,74}] (* Stefano Spezia, Nov 10 2021 *)

a(n) = my(d=digits(n)); prod(i=1, #d, sum(j=1, i, d[j])); \\ Michel Marcus, Nov 10 2021

first(n)=if(n<9,return([1..n])); my(v=vector(n)); for(i=1,9,v[i]=i); for(i=10,n, v[i]=sumdigits(i)*v[i\10]); v \\ Charles R Greathouse IV, Dec 04 2021

from math import prod
from itertools import accumulate
def a(n): return prod(accumulate(map(int, str(n))))
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Nov 10 2021

For n>10: a(n) = a(A059995(n))*A007953(n) where A059995(n) = floor(n/10).
In particular, for n<100: a(n) = floor(n/10)*A007953(n)
From Bernard Schott, Nov 23 2021: (Start)
a(n) = 1 iff n = 10^k, k >= 0 (A011557).
a(n) = 2 iff n = 10^k + 1, k >= 0 (A000533 \ {1}).
a(n) = 3 iff n = 10^k + 2, k >= 0 (A133384).
a(n) = 5 iff n = 10^k + 4, k >= 0.
a(n) = 7 iff n = 10^k + 6, k >= 0. (End)
From Marius A. Burtea, Nov 23 2021: (Start)
a(A002275(n)) = n! = A000142(n), n >= 1.
a(A090843(n - 1)) = (2*n - 1)!! = A001147(n), n >= 1.
a(A097166(n)) = (3*n - 2)!!! = A007559(n).
a(A093136(n)) = 2^n = A000079(n).
a(A093138(n)) = 3^n = A000244(n). (End)

A178501 Zero followed by powers of ten.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000, 100000000000000000000

Offset: 0

Reinhard Zumkeller, May 28 2010

The sequence S consisting of the nonnegative numbers arranged in lexicographic order according to their decimal expansion begins 0, 1, 10, 100, 1000, ..., 2, 20, 200, 2000, ..., 3, 30, ... does not have an OEIS entry, since there are uncountably many terms before 2 appears (or even before 100000010000 appears). However, S does begin with the present sequence. - N. J. A. Sloane, Dec 09 2024

a(n)^k + reverse(a(n))^k is a palindrome for any positive integer k. - Bui Quang Tuan, Mar 31 2015

Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A093136, A131577, A140429, A178500; subsequence of A029793.

The powers of 10, A011557, is a subsequence.

Join[{0}, 10^Range[0, 19]] (* Alonso del Arte, Mar 31 2015 *)

a(n)=10^n\10 \\ Charles R Greathouse IV, Oct 07 2015

concat(0, Vec(-x/(10*x-1) + O(x^22))) \\ Elmo R. Oliveira, Jul 21 2025

a(n+1) = A011557(n).
a(n) = A178500(n)/10.
From Paul Barry, Jul 09 2003: (Start)
a(n) = (10^n - 0^n)/10.
E.g.f.: exp(5*x)*sinh(5*x)/5.
Binomial transform of A015577. (End)
G.f.: x/(1 - 10*x). - Chai Wah Wu, Jun 17 2020
From Elmo R. Oliveira, Jul 21 2025: (Start)
a(n) = 10*a(n-1) for n > 1.
a(n) = A093136(n)/2 for n >= 1. (End)

More terms from Elmo R. Oliveira, Jul 21 2025

A352159 Numbers m such that the smallest digit in the decimal expansion of 1/m is 5, ignoring leading and trailing 0's.

Original entry on oeis.org

2, 18, 20, 132, 148, 180, 200, 1320, 1480, 1800, 2000, 13008, 13200, 14544, 14800, 18000, 20000, 130080, 132000, 145440, 148000, 180000, 200000, 1300800, 1320000, 1454400, 1480000, 1734375, 1800000, 2000000, 11521152, 12890625, 13008000, 13200000, 14544000, 14800000

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 25 2022

Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 (A003592) would be terms.
If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms: 2, 18, 132, 148, 14544, ...

			m = 148 is a term since 1/148 = 0.00675675675... and the smallest digit after the leading 0's is 5.
m = 1320 is a term since 1/1320 = 0.000075757575... and the smallest digit after the leading 0's is 5.

Cf. A351471.
Subsequence: A093136 \ {0}.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), this sequence (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 5 &]

is(n) = my(d=#digits(n-1), m=9, r=10^d, x=valuation(n, 2), y=valuation(n, 5)); for(k=1, max(x,y)-d*(n==x=2^x*5^y)+znorder(Mod(10, n/x)), if(5>m=min(m, r\n), return(0)); r=r%n*10); m==5; \\ Jinyuan Wang, Mar 27 2022

from itertools import count,islice
from sympy import multiplicity,n_order
def A352159_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,n), multiplicity(5,n)
        k, m = 10**max(m2,m5), 10**(t := n_order(10,n//2**m2//5**m5))-1
        c = k//n
        s = str(m*k//n-c*m).zfill(t)
        if s == '0' and min(str(c)) == '5':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '5':
                yield n
A352159_list = list(islice(A352159_gen(),10)) # Chai Wah Wu, Mar 28 2022

A352153(a(n)) = 5.

More terms from Jinyuan Wang, Mar 27 2022

A337855 Number of n-digit positive integers that are the product of two integers ending with 5.

Original entry on oeis.org

0, 2, 18, 180, 1800, 18000, 180000, 1800000, 18000000, 180000000, 1800000000, 18000000000, 180000000000, 1800000000000, 18000000000000, 180000000000000, 1800000000000000, 18000000000000000, 180000000000000000, 1800000000000000000, 18000000000000000000, 180000000000000000000

Offset: 1

Stefano Spezia, Sep 27 2020

a(n) is the number of n-digit numbers in A053742.

Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to digits.

Cf. A011557 (powers of 10), A052268 (number of n-digit integers), A053742 (product of two integers ending with 5), A093136, A337856.

Mathematica
```
LinearRecurrence[{10},{0,0,2,18},22]
```

O.g.f.: 2*(1 - x)*x^2/(1 - 10*x).
E.g.f.: (9*exp(10*x) - 9 - 90*x + 50*x^2)/500.
a(n) = 10*a(n-1) for n > 3 , with a(1) = 0, a(2) = 2 and a(3) = 18.
a(n) = 18*10^(n-3) for n > 2.
a(n) = 18*A011557(n - 3) for n > 2.
a(n) = 2*A052268(n - 2) for n > 2.
Sum_{i=2..n} a(n) = A093136(n - 1) for n > 1.
a(n) = 2*floor((k + 27*10^(n-2))/30), with 2 < k < 28. [This formula was found in the form k = 7 by Christian Krause's LODA miner] - Stefano Spezia, Dec 06 2021

A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

Original entry on oeis.org

0, 1, 2, 3, 10, 20, 30, 100, 200, 245, 247, 249, 251, 253, 283, 300, 448, 548, 949, 1000, 1249, 1253, 1416, 1747, 1749, 1751, 1753, 1755, 2000, 2245, 2247, 2249, 2251, 2253, 2429, 2450, 2451, 2470, 2490, 2498, 2510, 2530, 2647, 2830, 3000, 3747, 3751, 4480, 4899

Offset: 1

Bernard Schott, Oct 27 2019

The idea of this sequence comes from the 1st problem of the 28th British Mathematical Olympiad in 1992 (see the link).
This sequence is infinite because the family of integers {10^k, k >= 0} (A011557) belongs to this sequence.
The numbers m, m + 1, m + 2 where m = 49*10^k - 3, or m = 99*10^k - 3, k >= 3 are terms with all nonzero digits. - Marius A. Burtea, Dec 21 2020

			247^2 = 61009, hence 247 and 61009 both have 3 nonzero digits, 247 is a term.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 57 and 109 (1992)

Giovanni Resta, Table of n, a(n) for n = 1..10000.
British Mathematical Olympiad, 1992 - Problem 1.
Index to sequences related to Olympiads.

Subsequences: A011557, A093136, A093138.
Cf. A052040, A104315, A134844.
Cf. A328781, A328782, A328783.

nz:=func; [k:k in [0..5000] | nz(k) eq nz(k^2)]; // Marius A. Burtea, Dec 21 2020

q:= n->(f->f(n)=f(n^2))(t->nops(subs(0=[][], convert(t, base, 10)))):
select(q, [$0..5000])[];  # Alois P. Heinz, Oct 27 2019

Select[Range[0, 5000], Equal @@ Total /@ Sign@ IntegerDigits[{#, #^2}] &] (* Giovanni Resta, Feb 27 2020 *)

isok(k) = hammingweight(digits(k)) == hammingweight(digits(k^2)); \\ Michel Marcus, Dec 22 2020

More terms from Alois P. Heinz, Oct 27 2019

A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).

Original entry on oeis.org

1, 3, 23, 223, 2223, 22223, 222223, 2222223, 22222223, 222222223, 2222222223, 22222222223, 222222222223, 2222222222223, 22222222222223, 222222222222223, 2222222222222223, 22222222222222223, 222222222222222223, 2222222222222222223, 22222222222222222223, 222222222222222222223

Offset: 0

Paul Barry, Mar 24 2004

Second binomial transform of 2*A001045(3*n)/3 + (-1)^n.
Partial sums of A093136.
A convex combination of 10^n and 1.
In general the second binomial transform of k*Jacobsthal(3*n)/3 + (-1)^n is 1, 1+k, 1+11*k, 1+111*k, ... This is the case for k=2.
Essentially the same as A091628 (cf. 2nd formula). - Georg Fischer, Oct 06 2018
a(n) is 3^n represented in bijective base-3 numeration. - Alois P. Heinz, Aug 26 2019

Wikipedia, Bijective numeration.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A001045, A002279, A047855, A062397, A091628, A093136.

a(n) = (2*10^n + 7)/9.
a(n) = 10*a(n-1) - 7 (with a(0)=1). - Vincenzo Librandi, Aug 02 2010
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(7 + 2*exp(9*x))/9.
a(n) = 11*a(n-1) - 10*a(n-2).
a(n) = (A062397(n) - A002279(n))/2. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A351650 Integers m such that digsum(m) divides digsum(m^2) where digsum = sum of digits = A007953.

Original entry on oeis.org

1, 2, 3, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 24, 27, 30, 31, 33, 36, 42, 45, 46, 54, 55, 63, 72, 74, 81, 90, 92, 99, 100, 101, 102, 103, 108, 110, 111, 112, 113, 117, 120, 121, 122, 123, 126, 128, 130, 132, 135, 144, 145, 153, 162, 171, 180, 189, 190, 191, 198

Offset: 1

Bernard Schott, Feb 16 2022

This is a generalization of a problem proposed by French site Diophante in link.
The smallest term k such that the corresponding quotient = n is A280012(n).
The quotient is 1 iff m is in A058369 \ {0}.
If k is in A061909, then digsum(k^2) = digsum(k)^2.
If k is a term, 10*k is also a term.
There are infinitely many m such that both m and m+1 are in the sequence, for example subsequence A002283 \ {0}.
Corresponding quotients are in A351651.

			digit sum of 42 = 4+2 = 6; then 42^2 = 1764, digit sum of 1764 = 1+7+6+4 = 18; as 6 divides 18, 42 is a term.

Diophante, A1730 - Des chiffres à sommer pour un entier (in French).

Cf. A004159, A007953, A351651.

Subsequences: A002283, A011557, A052268, A058369, A061909, A093136, A093138, A254066, A280012.

Select[Range[200], Divisible[Total[IntegerDigits[#^2]], Total[IntegerDigits[#]]] &] (* Amiram Eldar, Feb 16 2022 *)

is(n)=sumdigits(n^2)%sumdigits(n) == 0 \\ David A. Corneth, Feb 16 2022

def sd(n): return sum(map(int, str(n)))
def ok(n): return sd(n**2)%sd(n) == 0
print([m for m in range(1, 200) if ok(m)]) # Michael S. Branicky, Feb 16 2022

A004159(a(n)) = A007953(a(n)) * A351651(n).

More terms from David A. Corneth, Feb 16 2022

A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

Original entry on oeis.org

72, 108, 200, 392, 432, 500, 648, 675, 968, 1125, 1323, 1352, 1372, 1800, 2000, 2312, 2592, 2700, 2888, 3087, 3267, 3528, 3888, 4232, 4500, 4563, 5000, 5292, 5324, 5400, 5488, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10125, 10584, 10952, 11979

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers k such that 2 <= A051904(k) = A051903(k) - 1.

Numbers that are product of two coprime nonsquarefree powers of squarefree numbers (A072777) with consecutive exponents.

			72 = 2^3 * 3^2 is a term since A051904(72) = 2 is larger than 1 and is 1 less than A051903(72) = 3.

Cf. A051903, A051904, A072777.
Subsequence of A001694.
Subsequences: A143610, A167747 \ {1, 2, 12}, A093136 \ {1, 2, 20}, A179666, A179702, A190472, A375073.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 2 <= Min[e] == Max[e] - 1]; Select[Range[12000], q]

is(k) = {my(e = factor(k)[,2]); k > 1 && 2 <= vecmin(e) && vecmin(e) + 1 == vecmax(e);}

Sum_{n>=1} 1/a(n) = Sum_{k>=2} f(k) = 0.053695635500385312854..., where f(k) = Product_{p prime} (1 + 1/p^k + 1/p^(k+1)) - zeta(k)/zeta(2*k) - zeta(k+1)/zeta(2*k+2) + 1 is the sum of reciprocals of the subset of numbers m with A051904(m) = k.

cp's OEIS Frontend

A132026 Decimal expansion of Product_{k>=0} (1 - 1/(2*10^k)).

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A169965 Numbers whose decimal expansion contains only 0's and 2's.

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A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

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A178501 Zero followed by powers of ten.

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A352159 Numbers m such that the smallest digit in the decimal expansion of 1/m is 5, ignoring leading and trailing 0's.

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A337855 Number of n-digit positive integers that are the product of two integers ending with 5.

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A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

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A093135 Expansion of g.f. (1-8x)/((1-x)(1-10*x)).