A061384 -id:A061384 - cp's OEIS frontend

A285093 Corresponding values of arithmetic means of digits of numbers from A061383.

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

Offset: 0

Jaroslav Krizek, Apr 14 2017

Jaroslav Krizek, Table of n, a(n) for n = 0..3000

Cf. A061383 (numbers with integer arithmetic mean of digits in base 10).
Sequences of numbers n such that a(n) = k for k = 1 - 9: A061384 (k = 1), A061385 (k = 2), A061386 (k = 3), A061387 (k = 4), A061388 (k = 5), A061423 (k = 6), A061424 (k = 7), A061425 (k = 8), A002283 (k = 9).
Cf. A004426, A004427, A257295 (supersequences).

[0] cat [&+Intseq(n) / #Intseq(n): n in [1..100000] | &+Intseq(n) mod #Intseq(n) eq 0];

lista(nn) = {for (n=0, nn, if (n, d = digits(n), d = [0]); if (!( vecsum(d) % #d), print1(vecsum(d)/#d, ", ")););} \\ Michel Marcus, Apr 15 2017

a(n) = A007953(A061383(n)) / A055642(A061383(n)) for n >= 1.

A308335 Palindromic primes such that sum of digits = number of digits.

Original entry on oeis.org

11, 10301, 1201021, 3001003, 10000900001, 10002520001, 10013131001, 10111311101, 10301110301, 11012121011, 11020302011, 11030103011, 11100500111, 11120102111, 12000500021, 12110101121, 13100100131, 30000500003, 30011111003, 1000027200001, 1000051500001

Offset: 1

Bernard Schott, May 20 2019

Every palindrome with an even number of digits is divisible by 11, so 11 is the only term of the sequence with an even number of digits.
Every palindrome with a number of digits which is a multiple of 3 also has a sum of digits which is divisible by 3, so there is no term with 3*k digits.
So, except 11 with 2 digits, the terms of this sequence must have a number of digits that belongs to A007310.
For n > 1, the middle digit of a(n) is odd. - Chai Wah Wu, Jun 30 2019

			3001003 is a term because it is a palindromic prime that has 7 digits and its sum of its digits is 7.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!, 10201021.

Intersection of A000040 (primes), A002113 (palindromes) and A061384 (sum of digits = number of digits).
Intersection of A002385 and A061384.
Intersection of A069710 and A002113.

f[n_] := If[n==2, {11}, If[Mod[(n-1) (n-5), 6]>0, {}, Block[{h = (n - 1)/2, L={}, p}, Do[p = Select[ Flatten[ Permutations /@ IntegerPartitions[ (n - c)/2, {h}, Range[0, 9]], 1], MemberQ[{1, 3, 7, 9}, Last[#]] &]; L = Join[L, Select[ FromDigits /@ (Flatten[{Reverse[#], c, #}] & /@ p), PrimeQ]], {c, 1, n-2, 2}]; Sort[L]]]]; Join @@ (f /@ Range[13]) (* Giovanni Resta, Jun 06 2019 *)

isok(p) = isprime(p) && (d=digits(p)) && (Vecrev(d) == d) && (#d == vecsum(d)); \\ Michel Marcus, Jun 29 2019

a(6)-a(21) from Jon E. Schoenfield, May 20 2019

A321771 Numbers whose digit product equals the number of their digits.

Original entry on oeis.org

1, 12, 21, 113, 131, 311, 1114, 1122, 1141, 1212, 1221, 1411, 2112, 2121, 2211, 4111, 11115, 11151, 11511, 15111, 51111, 111116, 111123, 111132, 111161, 111213, 111231, 111312, 111321, 111611, 112113, 112131, 112311, 113112, 113121, 113211, 116111, 121113

Offset: 1

Ivan Stoykov, Nov 21 2018

Idea is similar to A061384, which uses addition instead of multiplication.

			12 has two digits, and their product is also 2, as 1*2=2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A061384.
Cf. A007954, A055642. Subsequence of A007602.
Subsequence of A052382 (zeroless numbers).

Select[Range[1000000], Length[IntegerDigits[#]] == Times @@ IntegerDigits[#] &] (* Amiram Eldar, Nov 21 2018 *)

isok(n) = my(d=digits(n)); vecprod(d) == #d; \\ Michel Marcus, Nov 22 2018

More terms from Amiram Eldar, Nov 21 2018

A321999 Sum of digits of n minus the number of digits of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 7, 8, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7

Offset: 0

M. F. Hasler, Dec 07 2018

Concerning 0 we use the convention that 0 has 0 digits, so a(0) = 0 - 0 = 0, a(1) = 1 - 1 = 0, and a(10) = 1 - 2 = -1 is the first negative terms of the sequence.

			a(0) = 0 - 0 = 0. (We consider 0 has 0 digits.)
a(1) = 1 - 1 = 0;
a(2) = 2 - 1 = 1, ...,
a(9) = 9 - 1 = 8. (General formula: a(10^k - 1) = 8*k.)
a(10) = 1 - 2 = -1. (General formula: a(10^k) = -k.)
a(11) = 1+1 - 2 = 0, ...,
a(19) = 1+9 - 2 = 8;
a(20) = 2+0 - 2 = 0. (General formula: a(m*10^k) = a(m) - k.)
a(29) = 2+9 - 2 = 9, ...,
a(99) = 9+9 - 2 = 16: cf. a(9);
a(100) = 1+0+0 - 3 = -2;
a(101) = 1+0+1 - 3 = -1;
a(102) = 1+0+2 - 3 = 0, ...,
a(109) = 1+0+9 - 3 = 7;
a(110) = 1+1+0 - 3 = -1, ...,
a(119) = 1+1+9 - 3 = 8, ...,
a(199) = 1+9+9 - 3 = 16,
a(200) = 2+0+0 - 3 = -1: cf. a(20), ...,
a(999) = 9+9+9 - 3 = 24: cf. a(9);
a(1000) = 1+0+0+0 - 4 = -3, ...,
a(1001) = 1+0+0+1 - 4 = -2, ....

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A007953 (digit sum of n), A004218 (ceiling(log_10(n))), A055642 (number of digits of n).

The zeroes of this sequence, except 0 itself, are in A061384.

a:= n-> add(i, i=convert(n, base, 10))-length(n):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 10 2018

Table[(Plus@@IntegerDigits[n]) - Length[IntegerDigits[n]] + KroneckerDelta[n, 0], {n, 0, 99}] (* Alonso del Arte, Dec 07 2018 *)
Table[Total[IntegerDigits[n]]-IntegerLength[n],{n,0,100}] (* Harvey P. Dale, Dec 27 2022 *)

A321999(n)=sumdigits(n)-if(n,logint(n,10)+1)

a(n) = A007953(n) - A004218(n+1) = A007953(n) - A055642(n) for all n > 0. a(m*10^k) = a(m) - k for all m > 0, k >= 0, in particular:
a(10^k) = -k for all k >= 0. a(m) = m - 1 for 0 < m < 10.
a(n+1) = a(n) + 1 unless n = 9 (mod 10), in which case a(n+1) = a((n+1)/10).
a(10^k-1) = 8*k.

A354410 Numbers with as many zeros as the sum of their digits.

Original entry on oeis.org

10, 200, 1001, 1010, 1100, 3000, 10002, 10020, 10200, 12000, 20001, 20010, 20100, 21000, 40000, 100003, 100011, 100030, 100101, 100110, 100300, 101001, 101010, 101100, 103000, 110001, 110010, 110100, 111000, 130000, 200002, 200020, 200200, 202000, 220000

Offset: 1

Tamas Sandor Nagy, May 25 2022

As is normal, there are no leading zeros. The places of k zeros and the nonzero digits that are partitions of k are combinatorial.

Numbers k such that A007953(k) = A055641(k). - Felix Fröhlich, May 26 2022

Rémy Sigrist, PARI program

Subsequence of A011540.
Cf. A007953 (sum of digits), A055641 (number of 0's).
Cf. A031443, A061384.

Select[Range[250000],DigitCount[#,10,0]==Total[IntegerDigits[#]]&] (* Harvey P. Dale, Jan 12 2023 *)

isok(m) = my(d=digits(m)); vecsum(d) == #select(x->(x==0), d); \\ Michel Marcus, May 26 2022

PARI
```
See Links section.
```

# after linked PARI by Rémy Sigrist
base, vv, nb = 10, [0], 0
def visit(v, s, z, r):
    global base, vv, nb
    if v and s==z:
        nb += 1
        if nb > len(vv): vv.append(len(vv))
        vv[nb-1] = v
    if s-z-r <= 0 and s-z+(base-1)*r >= 0:
        if v: visit(base*v, s, z+1, r-1)
        for d in range(1, base): visit(base*v+d, s+d, z, r-1)
def auptod(digits): visit(0, 0, 0, digits); return sorted(set(vv))
print(auptod(6)) # Michael S. Branicky, May 26 2022

A380725 Positive integers k whose sum of digits equals the square of their number of digits.

Original entry on oeis.org

1, 13, 22, 31, 40, 108, 117, 126, 135, 144, 153, 162, 171, 180, 207, 216, 225, 234, 243, 252, 261, 270, 306, 315, 324, 333, 342, 351, 360, 405, 414, 423, 432, 441, 450, 504, 513, 522, 531, 540, 603, 612, 621, 630, 702, 711, 720, 801, 810, 900, 1069, 1078, 1087, 1096, 1159, 1168, 1177, 1186, 1195

Offset: 1

Anwar Hahj Jefferson-George, Jan 30 2025

This is a finite sequence since if k is L digits long then its sum of digits is at most 9*L which is < L^2 for L > 9.

Anwar Hahj Jefferson-George, Table of n, a(n) for n = 1..74583

Cf. A007953 (sum of digits), A061384.

Select[Range[1200],DigitSum[#]==IntegerLength[#]^2&] (* James C. McMahon, Feb 18 2025 *)

isok(k) = my(d=digits(k)); vecsum(d) == sqr(#d); \\ Michel Marcus, Feb 08 2025

def ok(n): return sum(map(int, s:=str(n))) == len(s)**2
print([k for k in range(2000) if ok(k)]) # Michael S. Branicky, Feb 08 2025

/// Is the term a valid entry in a380725
pub fn a380725_predicate(mut k: usize) -> bool {
    let base = 10usize;
    let mut len = 0usize;
    let mut digit_sum = 0usize;
    while k > 0 {
        let digit = k.rem_euclid(base);
        k /= base;
        digit_sum += digit;
        len += 1;
    }
    digit_sum == len.pow(2u32)
}

cp's OEIS Frontend

A285093 Corresponding values of arithmetic means of digits of numbers from A061383.

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A308335 Palindromic primes such that sum of digits = number of digits.

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A321771 Numbers whose digit product equals the number of their digits.

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A321999 Sum of digits of n minus the number of digits of n.

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Maple

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A354410 Numbers with as many zeros as the sum of their digits.

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A380725 Positive integers k whose sum of digits equals the square of their number of digits.

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Mathematica

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Python

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