A034837 -id:A034837 - cp's OEIS frontend

A277804 Numbers n such that first digit of n divides n, last digit of n divides n, number of divisors of n divides n and phi(n) divides n, where phi(n) is the Euler totient function.

1, 2, 8, 12, 24, 36, 128, 288, 384, 864, 972, 1152, 1944, 3456, 6144, 6912, 13122, 18432, 26244, 31104, 62208, 69984, 209952, 279936, 294912, 497664, 839808, 884736, 1679616, 3538944, 4478976, 13436928, 22674816, 25165824, 31850496, 45349632

Offset: 1

Ilya Gutkovskiy, Nov 01 2016

Numbers n such that A000030(n)|n, A010879(n)|n, A000005(n)|n and A000010(n)|n.

Intersection of A007694, A034709, A033950 and A034837.

			a(5) = 24 because 24/2 = 12, 24/4 = 6, 24 has 8 divisors {1,2,3,4,6,8,12,24}, 24/8 = 3, phi(24) = 8 {1,5,7,11,13,17,19,23} and 24/8 = 3 (all are an integers).

Robert G. Wilson v, Table of n, a(n) for n = 1..43
Index to divisibility sequences

Cf. A000005, A000010, A000030, A007694, A034709, A033950, A034837, A010879.

Select[Range[50000000], Divisible[#1, First[IntegerDigits[#1]]] && Divisible[#1, Last[IntegerDigits[#1]]] && Divisible[#1, DivisorSigma[0, #1]] && Divisible[#1, EulerPhi[#1]] & ]

a(24) - a(36) added by G. C. Greubel, Nov 02 2016

A176659 Partial sums of A038770.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 277, 302, 328, 356, 386, 417, 449, 482, 517, 553, 592, 632, 673, 715, 759, 804, 852, 902, 953, 1005, 1060, 1120, 1181, 1243, 1306, 1370, 1435, 1501, 1571, 1642

Offset: 1

Jonathan Vos Post, Apr 23 2010

Partial sums of numbers divisible by at least one of their digits. Identical to triangular numbers A000217 until a(23), then differs, because 23 is the smallest natural number in the complement of A038770 (A038772, i.e., not divisible by at least one of its digits). Hence this partial sum is the triangular numbers minus the partial sums of A038772, properly offset. The subsequence of primes (of course 3 is the largest prime triangular number) in the partial sum begins: 3, 277, 449, 673, 953, 1181, 1571, 1789, 2027. What are the equivalents in bases other than 10?

			a(23) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 24 = 277 is prime.

Cf. A038770, A034709, A034837, A038769, A038772.

Accumulate[Select[Range[110],MemberQ[Divisible[#,Cases[ IntegerDigits[ #], Except[ 0]]], True]&]] (* Harvey P. Dale, May 11 2017 *)

a(n) = Sum_{i=1..n} A038770(i).

A188454 Numbers n whose decimal digits are distinct and no digit divides n.

Original entry on oeis.org

23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 207, 209, 239, 247, 249, 253, 257, 259, 263, 267, 269, 283, 289, 293, 307, 308, 329, 346, 347, 349, 356, 358, 359, 367, 370, 374

Offset: 1

Andy Edwards, Mar 31 2011

These may not contain 1 or have a 2 or 5 as the last digit. They include prime numbers not containing the digit 1 and composites with a smallest prime factor > 10 and obeying the other constraints (e.g. the largest case is 987654203 = 31*31859813).
The first even case is 34. The first consecutive pair is {37, 38}. {56,57,58,59} is a consecutive quadruple which is the maximal size for such a subset.
There are 202623 terms in this sequence. - Nathaniel Johnston, May 19 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A038772, A034709, A034837, A038769, A038770.

dddQ[n_]:=Module[{dcn=DigitCount[n]},Max[dcn]==1&&First[dcn]==0 && Union[ Divisible[n,Select[IntegerDigits[n],#!=0&]]]=={False}]; Select[Range[ 400],dddQ] (* Harvey P. Dale, May 01 2012 *)

A247884 Number of positive integers < 10^n divisible by their first digit.

Original entry on oeis.org

9, 41, 327, 3158, 31450, 314349, 3143320, 31433005, 314329833, 3143298089, 31432980631, 314329806030, 3143298060001, 31432980599686, 314329805996514, 3143298059964770, 31432980599647312, 314329805996472711, 3143298059964726682, 31432980599647266367

Offset: 1

Derek Orr, Sep 25 2014

a(n)/10^n seems to converge to a number around .3143...

a(n)/10^n converges to 7129/22680. - Hiroaki Yamanouchi, Sep 26 2014

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100
Beyond Solutions Blog, Multiple of the first digit
J. J. O'Connor and E. F. Robertson, Pietro Mengoli

Cf. A034837, A028569, A119947, A119948.

a(n)=c=0;for(k=1,10^n-1,d=digits(k);if(k%d[1]==0,c++));c
n=1;while(n<10,print1(a(n),", ");n++)

count = 9 # Start with the first 9 digits
print(1, 9)
n = 2
while n < 101:
    for a in range(1, 10):
        count += 10**(n-1)//a
        if 10**(n-1) % a != 0:
            count += 1
    print(n, count)
    n += 1
# David Consiglio, Jr., Sep 26 2014

G.f.: x*(9 - 67*x + 24*x^2 + 14*x^3 - 56*x^4 + 21*x^5 + 7*x^6 + 5*x^7)/((1 - x)^2*(1 + x)*(1 - 10*x)*(1 - x + x^2)). - Robert Israel, Mar 10 2025

a(9)-a(20) from Hiroaki Yamanouchi, Sep 26 2014

A308237 Numbers m not ending with 0 that contain a digit, other than the leftmost digit, that can be removed such that the resulting number d divides m.

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, 99, 105, 108, 121, 132, 135, 143, 154, 165, 176, 187, 192, 195, 198, 225, 231, 242, 253, 264, 275, 286, 297, 315, 341, 352, 363, 374, 385, 396, 405, 451, 462, 473, 484, 495, 561, 572, 583, 594, 671, 682, 693

Offset: 1

Bernard Schott, May 16 2019

When m is a term, then, necessarily, the digit that is removed is the second from the left.
This sequence is finite with 95 integers and the greatest term is 180625. The number of terms with respectively 2, 3, 4, 5, 6 digits is 23, 44, 10, 17, 1.
The obtained quotients m/d belong to: { 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19 } (all proofs in Diophante link).

			264 is a term because 264/24 = 11.
34875 is a term because 34875/3875 = 9.

Marius A. Burtea, Table of n, a(n) for n = 1..95
Diophante, A333. Un chiffre à la trappe, Oct. 2011 (in French).

Cf. A034837, A178157.

m=1;
for u=10:700 digit=dec2base(u,10)-'0';
   if digit(length(digit))~=0 aa=str2num(strrep(num2str(digit), ' ', ''));
      digit(2)=[]; a=str2num(strrep(num2str(digit), ' ', ''));
if mod(aa,a)==0 sol(m)=u;  m=m+1;  end; end; end;
sol % Marius A. Burtea, May 16 2019

Select[Range[700], With[{m = #}, And[Mod[#, 10] != 0, AnyTrue[FromDigits@ Delete[IntegerDigits[m], #] & /@ Range[2, IntegerLength@ m], Mod[m, #] == 0 &]]] &] (* Michael De Vlieger, Jun 09 2019 *)

isok(m) = {if (m % 10, my(d=digits(m)); for (k=2, #d, mk = fromdigits(vector(#d-1, i, if (iMichel Marcus, Jun 21 2019

A337184 Numbers divisible by their first digit and their last digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 101, 102, 104, 105, 111, 112, 115, 121, 122, 123, 124, 125, 126, 128, 131, 132, 135, 141, 142, 144, 145, 147, 151, 152, 153, 155, 156, 161, 162, 164, 165, 168, 171, 172, 175, 181, 182

Offset: 1

Bernard Schott, Jan 29 2021

The first 23 terms are the same first 23 terms of A034838 then a(24) = 101 while A034838(24) = 111.
Terms of A034709 beginning with 1 and terms of A034837 ending with 1 are terms.
All positive repdigits (A010785) are terms.
There are infinitely many terms m for any of the 53 pairs (first digit, last digit) of m described below: when m begins with {1, 3, 7, 9} then m ends with any digit from 1 to 9; when m begins with {2, 4, 6, 8}, then m must also end with {2, 4, 6, 8}; to finish, when m begins with 5, m must only end with 5. - Metin Sariyar, Jan 29 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Intersection of A034709 and A034837.
Subsequences: A010785\{0}, A034838, A043037, A043040, A208259, A066622.
Cf. A139138.

Select[Range[175], Mod[#, 10] > 0 && And @@ Divisible[#, IntegerDigits[#][[{1, -1}]]] &] (* Amiram Eldar, Jan 29 2021 *)

is(n) = n%10>0 && n%(n%10)==0 && n % (n\10^logint(n,10)) == 0 \\ David A. Corneth, Jan 29 2021

def ok(n): s = str(n); return s[-1] != '0' and n%int(s[0])+n%int(s[-1]) == 0
print([m for m in range(180) if ok(m)]) # Michael S. Branicky, Jan 29 2021

(10n-9)/9 <= a(n) < 45n. (I believe the liminf of a(n)/n is 3.18... and the limsup is 6.18....) - Charles R Greathouse IV, Nov 26 2024

Conjecture: 3n < a(n) < 7n for n > 75. - Charles R Greathouse IV, Dec 02 2024

A357769 Positive numbers with decimal expansion d_1, ..., d_w that are divisible by d_1 + ... + d_k for k = 1..w.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 24, 30, 36, 40, 48, 50, 60, 70, 80, 90, 100, 102, 108, 110, 112, 114, 120, 126, 132, 140, 150, 156, 180, 190, 200, 204, 210, 216, 220, 224, 228, 230, 240, 252, 264, 270, 280, 300, 306, 312, 330, 336, 360, 396, 400

Offset: 1

Rémy Sigrist, Oct 12 2022

Leading zeros are ignored (d_1 > 0).
In other words, this sequence corresponds to numbers that are divisible by the sum of digits of all their prefixes.
All terms belong to A005349 (Niven numbers), A034837 and to A328273.
If t is a term, then 10*t is also a term (see A356350 for the primitive terms).
Contains no odd terms > 9. Else, d_1 and all d_1 + ... + d_k for k = 2..w-1 would have to be odd, but then d_1 + ... + d_w would be even. - Michael S. Branicky, Oct 15 2022

			180 is a term as it is divisible by 1, 1+8 and 1+8+0.
111 is not a term as it is divisible by 1 and 1+1+1 but not by 1+1.

Index entries for sequences related to decimal expansion of n

Cf. A005349, A034837, A259433, A328273, A356350.

Select[Range@400, And @@ IntegerQ /@ (#/Accumulate@ IntegerDigits@ #) &] (* Giovanni Resta, Oct 15 2022 *)

is(n, base=10) = { my (d=digits(n, base), s=0); for (k=1, #d, if (n % (s+=d[k]), return (0));); return (1); }

def ok(n):
    s = str(n); sk = int(s[0])
    for k in range(len(s)-1):
        if n%sk != 0: return False
        sk += int(s[k+1])
    return n%sk == 0
print([k for k in range(1, 401) if ok(k)]) # Michael S. Branicky, Oct 12 2022

A359841 Integers Xd which are divisible by X, where d is the last decimal digit.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420

Offset: 1

Bernard Schott, Jan 15 2023

Integers k such that k is divisible by A059995(k).

This sequence consists of {the thirty-two 2-digit terms of A034837 (from 10 up to 99)} Union {the positive multiples of 10 (A008592\{0})}.

N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics, problem 15, pp. 11 and 102, Dover publications, Inc., New York, 1993.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A034837, A059995, A178157, A292683 (similar but with dX).

Subsequence: A008592\{0}.

Select[Range[10, 500], Divisible[#, Floor[#/10]] &] (* Amiram Eldar, Jan 15 2023 *)

isok(k) = (k>9) && (k % (k \ 10) == 0); \\ Michel Marcus, Jan 20 2023

def ok(n): return n > 9 and n%(n//10) == 0
print([k for k in range(421) if ok(k)]) # Michael S. Branicky, Jan 15 2023

def A359841(n): return (10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99)[n-1] if n <= 32 else (n-23)*10 # Chai Wah Wu, Jan 20 2023

A176660 Partial sums of A038772.

Original entry on oeis.org

23, 50, 79, 113, 150, 188, 231, 277, 324, 373, 426, 480, 536, 593, 651, 710, 777, 845, 914, 987, 1061, 1137, 1215, 1294, 1377, 1463, 1550, 1639, 1733, 1830, 1928, 2131, 2338, 2547, 2770, 2997, 3226, 3459, 3698, 3945, 4194, 4447, 4704, 4963, 5226, 5493

Offset: 1

Jonathan Vos Post, Apr 23 2010

Partial sums of numbers not divisible by any of their digits. The partial sums of the complement (A038770, numbers divisible by at least one of their digits) is A176659. Hence the sum of the two partial sums (properly offset) is triangular numbers A000217. The subsequence of primes in the partial sum begins: 23, 79, 113, 277, 373, 593, 1061, 1733, 2131, 4447, 9479. The smallest square in the sequence is 324.

			a(4) = 23 + 27 + 29 + 34 = 113 is prime.

Cf. A038772, 034709, A034837, A038769, A038770, A176659.

a(n) = SUM[i=1..n] A038772(i).

cp's OEIS Frontend

A277804 Numbers n such that first digit of n divides n, last digit of n divides n, number of divisors of n divides n and phi(n) divides n, where phi(n) is the Euler totient function.

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A176659 Partial sums of A038770.

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A188454 Numbers n whose decimal digits are distinct and no digit divides n.

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A247884 Number of positive integers < 10^n divisible by their first digit.

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A308237 Numbers m not ending with 0 that contain a digit, other than the leftmost digit, that can be removed such that the resulting number d divides m.

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A337184 Numbers divisible by their first digit and their last digit.

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A357769 Positive numbers with decimal expansion d_1, ..., d_w that are divisible by d_1 + ... + d_k for k = 1..w.

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A359841 Integers Xd which are divisible by X, where d is the last decimal digit.