A002796 -id:A002796 - cp's OEIS frontend

A078548 a(n) = lcm(n, A052429(n)) - A052429(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 10, 10, 36, 24, 10, 42, 112, 64, 162, 18, 40, 20, 132, 20, 40, 72, 364, 48, 504, 27, 90, 90, 30, 192, 90, 30, 756, 432, 108, 36, 160, 80, 504, 40, 160, 264, 1288, 40, 1728, 45, 250, 250, 780, 520, 50, 810, 1960, 1120, 2610, 54, 360, 180

Offset: 1

Labos Elemer, Dec 05 2002

Cf. A052429, A078546, A078547, A002796.

lc[x_] := Apply[LCM, DeleteCases[IntegerDigits[x], 0]] Table[LCM[lc[w], w]-lc[w], {w, 1, 128}]

lcnzd(n) = lcm(select(x->(x!=0), digits(n)));
a(n) = my(lc=lcnzd(n)); lcm(n, lc) - lc; \\ Michel Marcus, Mar 18 2018

A342650 Numbers divisible both by their nonzero individual digits and by the sum of their digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 24, 30, 36, 40, 48, 50, 60, 70, 80, 90, 100, 102, 110, 111, 112, 120, 126, 132, 135, 140, 144, 150, 162, 200, 204, 210, 216, 220, 222, 224, 240, 264, 280, 288, 300, 306, 312, 315, 324, 330, 333, 336, 360, 396, 400, 408, 420, 432, 440, 444, 448, 480, 500

Offset: 1

Bernard Schott, Mar 18 2021

Equivalently, Niven numbers that are divisible by their nonzero digits. A Niven number (A005349) is a number that is divisible by the sum of its digits.
Niven numbers without zero digit that are divisible by their individual digits are in A051004.
Differs from super Niven numbers, the first 25 terms are the same, then A328273(26) = 120 while a(26) = 111.
This sequence is infinite since if m is a term, then 10*m is another term.

			102 is divisible by its nonzero digits 1 and 2, and 102 is also divisible by the sum of its digits 1 + 0 + 2 = 3, then 102 is a term.

Intersection of A002796 and A005349.
Supersequence of A051004.
Cf. A034838, A328273.

q[n_] := AllTrue[(d = IntegerDigits[n]), # == 0 || Divisible[n, #] &] && Divisible[n, Plus @@ d]; Select[Range[500], q] (* Amiram Eldar, Mar 18 2021 *)

isok(m) = if (!(m % sumdigits(m)), my(d=select(x->(x>0), Set(digits(m)))); setintersect(d, divisors(m)) == d); \\ Michel Marcus, Mar 18 2021

A055639 Number of nonzero digits of n which are not factors of n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jun 06 2000

Marius A. Burtea, Table of n, a(n) for n = 1..5000

Cf. A002796, A038769.

[#[c:c in Intseq(k) |not IsZero(c) and k mod c ne 0]:k in [1..105]]; // Marius A. Burtea, Dec 23 2019

Array[Count[Position[Most@ DigitCount@ #, ?(# > 0 &)][[All, 1]], k /; Mod[#, k] != 0] &, 105] (* Michael De Vlieger, Dec 23 2019 *)

A342445 Numbers that are divisible by their nonzero digits but are not divisible by the product of their nonzero digits.

Original entry on oeis.org

22, 33, 44, 48, 55, 66, 77, 88, 99, 122, 124, 126, 155, 162, 168, 184, 202, 204, 222, 244, 248, 264, 280, 288, 303, 324, 330, 333, 336, 366, 396, 404, 408, 412, 420, 424, 440, 444, 448, 488, 505, 515, 555, 606, 636, 648, 660, 666, 707, 728, 770, 777, 784, 808, 824, 840

Offset: 1

Bernard Schott, Mar 20 2021

Numbers that are divisible by the product of their nonzero digits (A055471) are trivially divisible by each of their nonzero digits (A002796), but the converse is false. This sequence = A002796 \ A055471 and consists of these counterexamples.

This sequence differs from A337163: the first sixteen terms are the same but a(17) = 202 while A337163(17) = 222.

			204 is divisible by 2 and 4 but 204 is not divisible by 2*4 = 8, hence 204 is a term.
248 is divisible by 2, by 4 and by 8 but 248 is not divisible by 2*4*8 = 64, hence 248 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Equals A002796 \ A055471.

Cf. A337163 = A034838 \ A007602 (subsequence of zeroless numbers).

q[n_] := AllTrue[(d = Select[IntegerDigits[n], # > 0 &]), Divisible[n, #] &] && ! Divisible[n, Times @@ d]; Select[Range[840], q] (* Amiram Eldar, Mar 21 2021 *)
dnzQ[n_]:=With[{c=DeleteCases[IntegerDigits[n],0]},Union[Boole[Divisible[n,c]]]=={1}&&!Divisible[n,Times@@c]]; Select[ Range[ 1000],dnzQ] (* Harvey P. Dale, Jan 16 2025 *)

isok(m) = my(d=select(x->(x != 0), digits(m))); (m % vecprod(d)) && (sum(k=1, #d, m % d[k]) == 0); \\ Michel Marcus, Mar 22 2021

A339439 Number of n-digit numbers (in base 10) that are divisible by each of their nonzero digits.

Original entry on oeis.org

1, 9, 23, 105, 614, 3826, 25373, 177247, 1288440, 9679469, 74720569, 590162149, 4753928352, 38956270302, 324107031605, 2733414294539, 23338817531394, 201537057920381, 1758534957457627, 15493080163102862, 137729286979112861, 1234692470638216407, 11155826203456047704

Offset: 0

Phuc H. G. Trinh, Dec 04 2020

			a(2) = 23: 10, 11, 12, 15, 20, 22, 24, 30, 33, 36, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99.

Cf. A002796.

C(digs,n,m)={my(v=vector(m)); v[1]=1; for(i=1, n, v=vector(m, j, sum(k=1, #digs, my(r=10^(i-1)*digs[k]%m); v[1+(j-1-r)%m]))); v}
a(n)={if(n==1, 10, my(m=lcm([2..9])); sumdiv(m, d, my(digs=select(t->t<=1||d%t==0, [0..9])); my(v=C(digs,n,m)-C(digs,n-1,m)); sum(i=1, m/d, if(gcd(m/d,i)==1, v[1+i*d%m]))))} \\ Andrew Howroyd, Dec 04 2020

Terms a(18) and beyond from Andrew Howroyd, Dec 04 2020

cp's OEIS Frontend

A078548 a(n) = lcm(n, A052429(n)) - A052429(n).

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A342650 Numbers divisible both by their nonzero individual digits and by the sum of their digits.

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A055639 Number of nonzero digits of n which are not factors of n.

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A342445 Numbers that are divisible by their nonzero digits but are not divisible by the product of their nonzero digits.

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A339439 Number of n-digit numbers (in base 10) that are divisible by each of their nonzero digits.

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