A055641 -id:A055641 - cp's OEIS frontend

A193238 Number of prime digits in decimal representation of n.

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 0, 0, 1, 1, 0, 1

Offset: 0

Reinhard Zumkeller, Jul 19 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A010051.

Cf. A211681, A046034, A052382, A055640, A055641, A055642, A102669 - A102685, A122640, A117804, A196563, A196564.

a193238 n = length $ filter (`elem` "2357") $ show n

Count[IntegerDigits[#],?PrimeQ]&/@Range[0,100] (* _Harvey P. Dale, Nov 13 2022 *)

a(n)=n=eval(Vec(Str(n)));sum(i=1,#n,isprime(n[i])) \\ Charles R Greathouse IV, Jul 29 2011

a(A084984(n))=0; a(A118950(n))>0; a(A092620(n))=1; a(A092624(n))=2; a(A092625(n))=3; a(A046034(n))=A055642(A046034(n));
a(A000040(n)) = A109066(n).
From Hieronymus Fischer, May 30 2012: (Start)
a(n) = sum_{j=1..m+1} (floor(n/10^j+0.3) + floor(n/10^j+0.5) + floor(n/10^j+0.8) - floor(n/10^j+0.2) - floor(n/10^j+0.4) - floor(n/10^j+0.6)), where m=floor(log_10(n)), n>0.
a(10n+k) = a(n) + a(k), 0<=k<10, n>=0.
a(n) = a(floor(n/10)) + a(n mod 10), n>=0.
a(n) = sum_{j=0..m} a(floor(n/10^j) mod 10), n>=0.
a(A046034(n)) = floor(log_4(3n+1)), n>0.
a(A211681(n)) = 1 + floor((n-1)/4), n>0.
G.f.: g(x) = (1/(1-x))*sum_{j>=0} (x^(2*10^j) + x^(3*10^j)+ x^(5*10^j) + x^(7*10^j))*(1-x^10^j)/(1-x^10^(j+1)).
Also: g(x) = (1/(1-x))*sum_{j>=0} (x^(2*10^j)- x^(4*10^j)+ x^(5*10^j)- x^(6*10^j)+ x^(7*10^j)- x^(8*10^j))/(1-x^10^(j+1)). (End)

A085974 Number of 0's in the decimal expansion of prime(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jason Earls, Jul 06 2003

			prime(26) = 101, so a(26) = 1 and prime(1230) = 10007, so a(1230) = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. 1's A085975, 2's A085976, 3's A085977, 4's A085978, 5's A085979, 6's A085980, 7's A085981, 8's A085982, 9's A085983.

Cf. A055641.

a085974 = count0 0 . a000040 where
   count0 c x | d == 0    = if x < 10 then c + 1 else count0 (c + 1) x'
              | otherwise = if x < 10 then c else count0 c x'
              where (x', d) = divMod x 10
-- Reinhard Zumkeller, Apr 08 2014

DigitCount[Prime[Range[100]],10,0] (* Paolo Xausa, Oct 30 2023 *)

A100910 Table of number of occurrences in n of each decimal digit from 0 to 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0

Offset: 0

Rick L. Shepherd, Nov 21 2004

Each row of this table has length 10 and corresponds to one term of A100909. n = 0 is normally represented as the single digit 0, so the first row here is 1, 0, 0, 0, 0, 0, 0, 0, 0, 0.

Robert Israel, Table of n, a(n) for n = 0..10009 (rows 0 to 1000, flattened)

Cf. A100909 (similar but each row of A100910 provides one A100909 term).
Cf. A055642 (row sums), A055641 (column 0), A268643 (column 1), A316863 (column 2), A316864 (column 3), A316865 (column 4), A316866 (column 5), A316867 (column 6), A316868 (column 7), A316869 (column 8), A102683 (column 9).
Cf. A059995, A010879.

seq(seq(numboccur(k, convert(n,base,10)),k=0..9),n=0..100); # Robert Israel, Jul 08 2016

A100910row[n_] := RotateRight[DigitCount[n]];
Array[A100910row, 10, 0] (* Paolo Xausa, Jul 16 2025 *)

T(n, k) = #select(x->x==k, digits(n))+!(n+k); \\ Jinyuan Wang, Mar 01 2020

From Robert Israel, Jul 08 2016: (Start)
a(n,k) = a(A059995(n),k) + (1 if A010879(n)=k, otherwise 0).
G.f. g(x,y) satisfies g(x,y) = ((1-x^10)/(1-x))*g(x^10,y) + (x^10-x)/(1-x) + x^10/(1-x^10) + x*y*(1-x^9*y^9)/((1-x^10)*(1-x*y)). (End)

A316863 Number of times 2 appears in the decimal expansion of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 2.
a(2) = 1 since 2 appears once in the decimal expansion of 2.
a(22) = 2 since 2 appears twice in the decimal expansion of 22.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A043497, A043498, A043499, A043500.

Cf. A055641, A268643, A316864, A316865, A316866, A316867, A316868, A316869, A102683.

f:= n -> numboccur(2,convert(n,base,10)):
map(f, [$0..200]); # Robert Israel, Apr 21 2020

Mathematica
```
Array[ DigitCount[#, 10, 2] &, 105, 0]
```

a(n) = #select(x->x==2, digits(n)); \\ Michel Marcus, Jul 20 2018

G.f.: (1-x)^(-1)*Sum_{k>=0} (x^(2*10^k)-x^(3*10^k))/(1-x^(10^(k+1))). - Robert Israel, Apr 21 2020

A316864 Number of times 3 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 3.
a(3) = 1 since 3 appears once in the decimal expansion of 3.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A043501, A043502, A043503, A043504.

Cf. A055641, A268643, A316864, A316865, A316866, A316867, A316868, A316869, A102683.

f:= proc(n) option remember;
procname(floor(n/10)) + `if`(n mod 10 = 3, 1, 0)
end proc:
for i from 0 to 9 do f(i):= `if`(i=3,1,0) od:
map(f, [$0..100]); # Robert Israel, Dec 10 2019

Mathematica
```
Array[ DigitCount[#, 10, 3] &, 105, 0]
```

a(n) = #select(x->x==3, digits(n)); \\ Michel Marcus, Jul 20 2018

From Robert Israel, Dec 10 2019: (Start)
a(10*n+3) = a(n)+1, a(10*n+i)=a(i) for i = 0,1,2,4..9.
G.f. g(z) satisfies g(z) = z^3/(1-z^10) + ((1-z^10)/(1-z))*g(z^10). (End)

A316868 Number of times 7 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 7.
a(7) = 1 since 7 appears once in the decimal expansion of 7.

Cf. A043517, A043518, A043519, A043520.
Cf. A055641, A268643, A316863, A316864, A316865, A316866, A316867, A316869, A102683.
Cf. A079692.

Mathematica
```
Array[ DigitCount[#, 10, 7] &, 105, 0]
```

a(n) = #select(x->x==7, digits(n)); \\ Michel Marcus, Jul 20 2018

A316869 Number of times 8 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 8.
a(8) = 1 since 8 appears once in the decimal expansion of 8.

Cf. A043521, A043522, A043523, A043524.

Cf. A055641, A268643, A316863, A316864, A316865, A316866, A316867, A316868, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 8] &, 105, 0]
```

a(n) = #select(x->x==8, digits(n)); \\ Michel Marcus, Jul 20 2018

A257510 Number of nonleading zeros in factorial base representation of n (A007623).

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 3, 2, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 4

Offset: 1

Antti Karttunen, Apr 27 2015

Sequence starts from n=1, because 0 is an ambiguous case.

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A227157 (numbers n such that a(n) = 0), A227187 (n for which a(n) > 0).
Cf. A007623, A060130, A084558, A230403.
Cf. also A257511.
Cf. also A023416, A080791 (analogous sequences for base-2), A055641 (for base-10).

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 120}]; Last@ DigitCount[#] & /@ s (* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)

(define (A257510 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (floor->exact (/ n i)) (+ 1 i) (+ s (if (zero? (modulo n i)) 1 0)))))))

a(n) = A084558(n) - A060130(n).
Other identities and observations:
For all n >= 0, a(A000142(n+1)) = n. [(n+1)! gives the position where n first appears.]
For all n, a(n) >= A230403(n).

A316865 Number of times 4 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 4.
a(4) = 1 since 4 appears once in the decimal expansion of 4.

Cf. A043505, A043506, A043507, A043508.

Cf. A055641, A268643, A316863, A316864, A316866, A316867, A316868, A316869, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 4] &, 105, 0]
```

a(n) = #select(x->x==4, digits(n)); \\ Michel Marcus, Jul 20 2018

A316866 Number of times 5 appears in decimal expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Robert G. Wilson v, Jul 15 2018

			a(0) = 0 since the decimal representation of 0 does not contain the digit 5.
a(5) = 1 since 5 appears once in the decimal expansion of 5.

Cf. A043509, A043510, A043511, A043512.

Cf. A055641, A268643, A316863, A316864, A316865, A316867, A316868, A316869, A102683.

Mathematica
```
Array[ DigitCount[#, 10, 5] &, 105, 0]
```

a(n) = #select(x->x==5, digits(n)); \\ Michel Marcus, Jul 20 2018

cp's OEIS Frontend

A193238 Number of prime digits in decimal representation of n.

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A085974 Number of 0's in the decimal expansion of prime(n).

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A100910 Table of number of occurrences in n of each decimal digit from 0 to 9.

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A316863 Number of times 2 appears in the decimal expansion of n.

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A316864 Number of times 3 appears in decimal expansion of n.

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A316868 Number of times 7 appears in decimal expansion of n.

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A316869 Number of times 8 appears in decimal expansion of n.

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A257510 Number of nonleading zeros in factorial base representation of n (A007623).

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