A037124 -id:A037124 - cp's OEIS frontend

A193460 Numbers m such that A193459(m) = A000005(m).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 111, 200, 222, 300, 333, 400, 444, 500, 555, 600, 666, 700, 777, 800, 888, 900, 999, 1000, 1111, 2000, 2222, 3000, 3333, 4000, 4444, 5000, 5555, 6000

Offset: 1

Reinhard Zumkeller, Jul 26 2011

Union of A037124 and A010785, apart from initial 0.

import Data.List (elemIndices)
a193460 n = a193460_list !! (n-1)
a193460_list = elemIndices 0 $ 1 : zipWith (-) a193459_list a000005_list

A051596 Numerical values or Gematriot of Hebrew letters {aleph, bet, ..., tav}.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400

Offset: 1

Asher Auel

First 22 terms of A037124. - Michael Turniansky, Mar 26 2014

Encyclopedia Judaica, Keter Publishing House, Jerusalem, circa 1972, Volume 7, Page 369.

Gal Einai Institute, Gematria Hebrew Letter Chart
Wikipedia, Gematria
E. V. Yatziv, Overview of Hebrew Language Structure [broken link]

Definition corrected by Daniel Sterman, Nov 24 2020

A084050 Numbers n such that at least one permutation of the digits of n yields a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166, 171, 177, 181, 188, 191, 199, 200, 202, 211, 212, 220, 221

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 26 2003

Union of A037124 and numbers with at most one decimal digit occurring an odd number of times. a(281)=1001 is the first term greater than 90 not also a term of A044959. The terms greater than 90 are a subsequence of A109303. - Rick L. Shepherd, Jun 24 2005

Cf. A044959, A084051.

Cf. A037124 (numbers with only one nonzero digit), A109303 (numbers with at least one duplicate digit).

Corrected by Rick L. Shepherd, Jun 24 2005

A242948 Numbers k such that A = k+DigitProd(k) is divisible by the largest power of 10 <= A.

Original entry on oeis.org

1, 2, 3, 4, 5, 15, 35, 42, 44, 46, 48, 55, 91, 191, 272, 337, 692, 919, 1919, 2488, 5775, 7795, 7984, 17795, 17984, 28464, 29424, 394816, 489632, 683872, 3992224, 4996112, 7872992, 29927424, 89668224, 7991871488, 9968647168, 9991319797

Offset: 1

Derek Orr, May 27 2014

Trivially, numbers in A037124 satisfy this but are not included in this sequence.
It is not known if a(n) is infinite.
a(54) (if it exists) is > 10^49. - Hiroaki Yamanouchi, Jul 10 2014

			272+2*7*2 = 300 is divisible by the highest power of 10 lower than 300 (in this case, 100). Thus 272 is a member of this sequence.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..53

Cf. A037124, A055642, A007954.

DP(n)={p=1; d= digits(n); for(i=1,#d,p*=d[i]);return(p)}
for(n=1,10^7,if(n%10!=0&&(n+DP(n))%10^(#Str(n+DP(n))-1)==0,print1(n,", ")))

a(34)-a(38) from Hiroaki Yamanouchi, Jul 10 2014

A273737 Number of permutations of n elements divided by the number of 9-ary heaps on n+1 elements.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 20000, 30000, 40000, 50000, 60000, 70000, 80000, 90000, 100000

Offset: 0

Alois P. Heinz, May 28 2016

nonn

The first term with more than one nonzero digit (and thus not a member of A037124) is a(91) = 2200000000.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, D-ary heap

Column k=9 of A273730.

Cf. A000142, A037124, A273696.

a(n) = A000142(n)/A273696(n+1).

A274206 a(n) = the last nonzero digit of n followed by all the trailing zeros of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 1, 2, 3, 4, 5, 6, 7, 8, 9, 40, 1, 2, 3, 4, 5, 6, 7, 8, 9, 50, 1, 2, 3, 4, 5, 6, 7, 8, 9, 60, 1, 2, 3, 4, 5, 6, 7, 8, 9, 70, 1, 2, 3, 4, 5, 6, 7, 8, 9, 80

Offset: 1

A. D. Skovgaard, Jun 13 2016

a(n) is the number formed by the rightmost A160094(n) digits -- only the position(s) that changed -- of a decimal counter (e.g., an odometer) after it increments from n - 1 to n. - Rick L. Shepherd, Jun 29 2016

			a(1) = 1 because when 1 is added to 1 - 1 = 0, the units digit changes so the units digit of 1 is shown.
a(110) = 10 because when 1 is added to 109, the tens digit and the units digit change, so the last two digits of 110 are shown.
a(1000) = 1000 because when 1 is added to 999, all the digits change so they are all shown.

A. D. Skovgaard, Table of n, a(n) for n = 1..10000

Cf. A010879, A037124 (these increasing distinct terms), A006519 (binary equivalent shown in decimal), A160094.

f:= n -> n mod 10^(1+min(padic:-ordp(n,2), padic:-ordp(n,5))):
map(f, [$1..100]); # Robert Israel, Aug 08 2016

Table[FromDigits@ Join[{Last@ #}, Table[0, {Log10[n/FromDigits@ #]}]] &@ Select[IntegerDigits@ n, # != 0 &], {n, 120}] (* Michael De Vlieger, Jun 29 2016 *)

a(n) = n%10^(valuation(n,10)+1); \\ David A. Corneth, Jun 29 2016

a(n) = n mod 10 if n is not a multiple of 10.
From Robert Israel, Aug 08 2016: (Start)
a(10*n) = 10*a(n).
a(10*n+k) = k for 1 <= k <= 9.
G.f. g(x) satisfies g(x) = (x+2x^2+...+9x^9)/(1-x^10) + 10 g(x^10). (End)

A355675 a(0) = 0, and for any n > 0 and d = 1..9, a(10n) = 10a(n), a(10n + d) = d - 10a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 20, -19, -18, -17, -16, -15, -14, -13, -12, -11, 30, -29, -28, -27, -26, -25, -24, -23, -22, -21, 40, -39, -38, -37, -36, -35, -34, -33, -32, -31, 50, -49, -48, -47, -46, -45, -44, -43, -42

Offset: 0

Rémy Sigrist, Jul 14 2022

This sequence establishes a bijection from the nonnegative integers (N) to the integers (Z).
This sequence is to base 10 what A065620 is to base 2.
To compute a(n): write n a sum of terms of A037124 with distinct 10-adic valuations and take the alternating sum.
This sequence has similarities with A073835; they first differ in absolute value for n = 101: a(101) = 99 whereas A073835(101) = 101.
This sequence has similarities with A334387; they first differ in absolute value for n = 111: a(111) = 91 whereas A334387(111) = 81.

			For n = 17039:
  17039 = 10000 + 7000 + 30 + 9,
  so a(17039) = -10000 + 7000 - 30 + 9 = -3021.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to decimal expansion of n

Cf. A037124, A065620, A073835, A334387, A355624.

a(n, base=10) = { my (d=digits(n, base), s=1); forstep (k=#d, 1, -1, if (d[k], d[k]*=s; s=-s)); return (fromdigits(d, base)) }

a(n) = 0 iff n = 0 or n belongs to A037124.

A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 22, 24, 30, 33, 36, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 150, 200, 210, 220, 240, 250, 300, 330, 360, 400, 420, 440, 480, 500, 510, 520, 550, 600, 630, 660, 700, 770, 800, 840, 880

Offset: 1

Rémy Sigrist, May 01 2025

This sequence is infinite as ten times a term is also a term.
All terms are of the form A037124(k) or A037124(k) + d where k > 0 and d divides A037124(k) while having strictly less decimal digits as A037124(k).
Empirically, all terms have either one or two nonzero decimal digits.

			The number 240 is divisible by 2, 24, 240, 4 and 40, so 240 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10314
Rémy Sigrist, PARI program

Cf. A037124, A078546, A175381 (binary variant), A178157, A218978.

Select[Range[880],AllTrue[#/Select[FromDigits/@Subsequences[IntegerDigits[#]],#>0&],IntegerQ]&] (* James C. McMahon, May 13 2025 *)

is(n, base = 10) = {
    my (d = digits(n, base));
    for (i = 1, #d,
        if (d[i],
            for (j = i, #d,
                if (n % fromdigits(d[i..j], base),
                    return (0);););););
    return (1); }

PARI
```
\\ See Links section.
```

def ok(n):
    s = str(n)
    subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
    return n and all(n%v == 0 for ss in subs if (v:=int(ss)) > 0)
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, May 09 2025

A343132 a(n) is the quotient obtained when integer A343131(n) = k is divided by A061486(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 10, 1, 10, 1, 10, 3, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 100, 6, 5, 24, 11, 10, 1, 100, 9, 8, 42, 13, 13, 10, 1, 100, 43, 16, 22, 10, 1, 100, 30, 9, 2, 10, 1, 100, 4, 3, 10, 1, 100, 31, 6, 5, 10, 1, 100, 15, 10, 1, 100, 13, 11, 10, 1, 100, 10, 1, 1000

Offset: 1

Bernard Schott, Apr 07 2021

The first 9 terms corresponding to the 1-digit numbers k = u are the quotients u/u = 1.
The next 19 terms from a(10) = 10 to a(28) = 1 corresponding to 2-digit numbers k = du are the quotients du/(d+u + d*u).
The next 50 terms from a(29) = 100 to a(78) = 1 corresponding to 3-digit numbers k = hdu (in A328864) are the quotients hdu/f_3(h,d,u) where f_3(h,d,u) = (h+d+u) + (h*d+d*u+u*h) + (h*d*u).
The next 87 terms, from a(79) = 1000 to a(165) = 1, corresponding to 4-digit numbers k = thdu are the quotients thdu/f_4(t,h,d,u) where f_4(t,h,d,u) = (t+h+d+u) + (t*h+t*d+t*u+h*d+h*u+d*u) + (t*h*d+t*h*u+t*d*u+h*d*u) + (t*h*d*u).
When A343131(n) = z*10^q = A037124(r) is a number that contains only one nonzero digit z, then A061486(A037124(r)) = this nonzero digit z and a(n) = 10^q.

			For A343131(7) = 7, A061486(7) = 7 and a(7) = 7/7 = 1.
For A343131(17) = 42, A061486(42) = 4+2 + 4*2 = 14 and a(17) = 42/14 = 3.
For A343131(58) = 573, A061486(573) = 5+7+3 + 5*7+7*3+3*5 + 5*7*3 = 191 and a(58) = 573/191 = 3.

Eric Weisstein's World of Mathematics, Symmetric polynomial.
Wikipedia, Elementary symmetric polynomial.

Cf. A005349, A007602, A038366, A061486, A328864, A343131.

sympol(X, n) = my(s=0); forvec(i=vector(n, j, [1, #X]), s+=prod(k=1, n, X[i[k]]), 2); s ;
f(n) = my(d=digits(n)); sum(k=1, #d, sympol(d, k));
lista(nn) = {for (n=1, nn, my(q = n/f(n)); if (denominator(q) == 1, print1(q, ", ")););} \\ Michel Marcus, Apr 08 2021

a(n) = A343131(n)/A061486(A343131(n)).

A376129 Run lengths of the most significant decimal digit in the primes (A077648).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 2, 3, 2, 2, 3, 2, 1, 21, 16, 16, 17, 14, 16, 14, 15, 14, 135, 127, 120, 119, 114, 117, 107, 110, 112, 1033, 983, 958, 930, 924, 878, 902, 876, 879, 8392, 8013, 7863, 7678, 7560, 7445, 7408, 7323, 7224, 70435, 67883, 66330, 65367, 64336, 63799, 63129, 62712, 62090

Offset: 1

Stuart Coe, Sep 11 2024

			The primes and the run lengths of their initial digits begin
  primes   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...
  runs                 \------------/  \----/  \----/
  lengths  1  1  1  1         4          2       2     ...

Cf. A000720, A037124, A077648.

from sympy import primepi
def A376129(n):
    if n<5: return 1
    def f(m): return (lambda x:primepi(10**x[0]*(x[1]+1)))(divmod(m,9))
    return int(f(n+5)-f(n+4)) # Chai Wah Wu, Oct 16 2024

a(n) = A000720(A037124(n+6)) - A000720(A037124(n+5)) for n >= 5. - Pontus von Brömssen, Oct 07 2024

cp's OEIS Frontend

A193460 Numbers m such that A193459(m) = A000005(m).

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Haskell

A051596 Numerical values or Gematriot of Hebrew letters {aleph, bet, ..., tav}.

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A084050 Numbers n such that at least one permutation of the digits of n yields a palindrome.

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A242948 Numbers k such that A = k+DigitProd(k) is divisible by the largest power of 10 <= A.

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A273737 Number of permutations of n elements divided by the number of 9-ary heaps on n+1 elements.

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A274206 a(n) = the last nonzero digit of n followed by all the trailing zeros of n.

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A355675 a(0) = 0, and for any n > 0 and d = 1..9, a(10*n) = 10*a(n), a(10*n + d) = d - 10*a(n).

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A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

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A343132 a(n) is the quotient obtained when integer A343131(n) = k is divided by A061486(k).

A355675 a(0) = 0, and for any n > 0 and d = 1..9, a(10n) = 10a(n), a(10n + d) = d - 10a(n).