A008589 -id:A008589 - cp's OEIS frontend

A166394 Multiples of 7 whose reversal - 1 is also a multiple of 7.

63, 112, 119, 182, 189, 203, 273, 364, 455, 546, 630, 637, 721, 728, 791, 798, 812, 819, 882, 889, 903, 973, 1036, 1120, 1127, 1190, 1197, 1211, 1218, 1281, 1288, 1302, 1309, 1372, 1379, 1463, 1554, 1645, 1736, 1820, 1827, 1890, 1897, 1911, 1918, 1981, 1988

Offset: 1

Claudio Meller, Oct 13 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Subsequence of A008589.

Select[7 Range[5!], Divisible[FromDigits[Reverse[IntegerDigits[#]]] - 1, 7] &] (* G. C. Greubel, May 12 2016 *)

A168622 Triangle read by rows: T(n, k) = [x^k]( 7(1+x)^n - 6(1+x^n) ) with T(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 21, 21, 1, 1, 28, 42, 28, 1, 1, 35, 70, 70, 35, 1, 1, 42, 105, 140, 105, 42, 1, 1, 49, 147, 245, 245, 147, 49, 1, 1, 56, 196, 392, 490, 392, 196, 56, 1, 1, 63, 252, 588, 882, 882, 588, 252, 63, 1, 1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

			Triangle begins as:
  1;
  1,  1;
  1, 14,   1;
  1, 21,  21,   1;
  1, 28,  42,  28,    1;
  1, 35,  70,  70,   35,    1;
  1, 42, 105, 140,  105,   42,    1;
  1, 49, 147, 245,  245,  147,   49,   1;
  1, 56, 196, 392,  490,  392,  196,  56,   1;
  1, 63, 252, 588,  882,  882,  588, 252,  63,  1;
  1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A022090, A131115, A132047, A168620, A168623.

Columns (essentially): A008589 (k=1), A024966 (k=2).

A168622:= func< n,k | k eq 0 or k eq n select 1 else 7*Binomial(n,k) >;
[A168622(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2025

(* First program *)
p[x_, n_]:= With[{m=3}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
(* Second program *)
A168622[n_, k_]:= If[k==0 || k==n, 1, 7*Binomial[n,k]];
Table[A168622[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2025 *)

def A168622(n,k):
    if k==0 or k==n: return 1
    else: return 7*binomial(n,k)
print(flatten([[A168622(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 10 2025

From G. C. Greubel, Apr 10 2025: (Start)
T(n, k) = 7*binomial(n, k), with T(n, 0) = T(n, n) = 1.
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = 2*A048489(n-1) + 6*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = -6*(1 + (-1)^n) + 13*[n=0].
Sum_{k=0..floor(n/2)} T(n-k, k) = A022090(n+1) - 3*(3 + (-1)^n) + 6*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (14/sqrt(3))*(-1)^n*cos((4*n+1)*Pi/6) - 6*(1 + (-1)^n*cos(n*Pi/2)) + 6*[n=0]. (End)

A182208 Carmichael numbers divisible by 7.

Original entry on oeis.org

1729, 2821, 6601, 8911, 15841, 41041, 52633, 63973, 101101, 126217, 172081, 188461, 670033, 748657, 825265, 838201, 997633, 1033669, 1082809, 1773289, 2628073, 4463641, 4909177, 6840001, 7995169, 8719921, 8830801, 9585541, 9890881

Offset: 1

Marius Coman, Apr 18 2012

nonn

Conjecture: Any Carmichael number C divisible by 7 can be written in one of two ways: (1) C=7*(6m+1)*(6n+1), where m and n are natural numbers or (2) C=7*(6m-1)*(6n-1), where m and n are natural numbers. In other words, there aren’t Carmichael numbers divisible by 7 of the form C=7*(6m+1)*(6n-1). Checked for the first 27 Carmichael numbers divisible by 7. Note: a Carmichael number with more than 3 prime divisors can be written (sometimes) in both ways: 41041 = 7*11*13*41 = 7*13*451 (form 1) = 7*11*533 = 7*41*143 (form 2).
Observation: in the first 100 Carmichael numbers with three prime divisors (not divisible by 3) there is no one to can be written as (6x+1)(6y+1)(6z-1), they are all of the form (6x+1)(6y+1)(6z+1), (6x-1)(6y-1)(6z-1) or (6x+1)(6y-1)(6z-1). Would not that be enough to make an assumption that there are no such Carmichael numbers with three prime divisors, or even more, that aren't Carmichael numbers even with more than three divisors to can be written this way?
The conjecture follows from Korselt's criterion. - Charles R Greathouse IV, Oct 02 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Carmichael Number

Intersection of A002997 (Carmichael) and A008589 (multiples of 7). - Michel Marcus, Oct 11 2016

CarmichaelNbrQ[n_] := ! PrimeQ[n] && Mod[n, CarmichaelLambda@ n] == 1; 7 Select[ Range[2500000], CarmichaelNbrQ[ 7#] &] (* Robert G. Wilson v, Aug 24 2012 *)

Korselt(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1
forstep(n=49,1e6,42,if(Korselt(n),print1(n", "))) \\ Charles R Greathouse IV, Oct 02 2012

Corrected by Robert G. Wilson v, Aug 24 2012

A182341 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 7.

Original entry on oeis.org

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 128, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

The prime tower factorization of a number can be recursively defined as follows:
(0) The prime tower factorization of 1 is itself
(1) To find the prime tower factorization of an integer n>1, let n = p1^e1 * p2^e2 * ... * pk^ek be the usual prime factorization of n. Then the prime tower factorization is given by p1^(f1) * p2^(f2) * ... * pk^(fk), where fi is the prime tower factorization of ei.
Observation: the union of 128 and the first 54 nonzero multiples of 7 (cf. A008589) gives the first 55 terms of this sequence. - Omar E. Pol, Feb 01 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251 [math.NT], 2012-2014.

Cf. A008589, A182318.

# The integer n is in this sequence if and only if
# containsPrimeInTower(7, n) returns true
containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
for i to nops(L) do currentExponent := L[i][2];
if containsPrimeInTower(q, currentExponent) then return true: end if
end do;
return false:
end proc:

containsPrimeInTower[q_, n_] := containsPrimeInTower[q, n] = Module[{i, L, currentExponent}, If[n <= 1, Return[False]]; If[IntegerQ[n/q], Return[True]]; L = FactorInteger[n]; For[i = 1, i <= Length[L] , i++, currentExponent = L[[i, 2]]; If[containsPrimeInTower[q, currentExponent], Return[True]]]; Return[False]];
Select[Range[400], containsPrimeInTower[7, #]&] (* Jean-François Alcover, Jan 22 2019, from Maple *)

A328947 Numbers formed from decimal digits 0 and/or 1 which are divisible by 7.

Original entry on oeis.org

0, 1001, 10010, 10101, 11011, 100100, 101010, 101101, 110110, 111111, 1000111, 1001000, 1010100, 1011010, 1011101, 1100001, 1101100, 1110011, 1111110, 10000011, 10001110, 10010000, 10011001, 10100111, 10101000, 10110100, 10111010, 10111101, 11000010, 11000101, 11001011, 11010111, 11011000

Offset: 1

Robert Israel, Oct 31 2019

If x and y are members of the sequence and 10^k > y, then 10^k*x+y is a member.

The number of terms of up to k digits is A263366(k-1).

			a(3)=10010 is in the sequence because it is divisible by 7 and each of its decimal digits is 0 or 1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Divisibility by 7 of a number consisting of 0 and 1s

Intersection of A007088 and A008589.

Cf. A263366.

a:=[]; f:=func; for k in [0..220] do if f(k) mod 7 eq 0 then Append(~a,f(k)); end if; end for; a; // Marius A. Burtea, Nov 01 2019

bintodec:= proc(n) local L,i; L:= convert(n,base,2); add(10^(i-1)*L[i],i=1..nops(L)) end proc:
select(t -> t mod 7 = 0, map(bintodec,[$0..1000]));

A328947_list = [n for n in (int(bin(m)[2:]) for m in range(10**4)) if not n % 7] # Chai Wah Wu, Nov 01 2019

A335774 Numbers k such that in prime factorization of k the second smallest factor is 7.

Original entry on oeis.org

14, 21, 28, 35, 56, 63, 98, 112, 147, 154, 175, 182, 189, 196, 224, 231, 238, 245, 266, 273, 308, 322, 357, 364, 385, 392, 399, 406, 434, 441, 448, 455, 476, 483, 518, 532, 567, 574, 595, 602, 609, 616, 644, 651, 658, 665, 686, 693, 728, 742, 777, 784, 805, 812, 819, 826, 854, 861

Offset: 1

Zak Seidov, Jun 22 2020

nonn

Trivially, all terms are multiples of 7. Also terms are divisible by 2 or 3 or 5, and by any number of primes > 7.

			14 = 2*7, 28 = 2*2*7, 35 = 5*7, 56 = 2^3*7, 63 = 3*3*7, 147 = 3*7*7, 154 = 2*7*11.

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A008589 (multiples of 7).

Subsequence of A080671.

Select[Range[14, 1000], 1 < Length[fi = FactorInteger[#]] && 7 == fi[[2, 1]] &]

A351663 Perfect powers that are divisible by 7.

Original entry on oeis.org

49, 196, 343, 441, 784, 1225, 1764, 2401, 2744, 3136, 3969, 4900, 5929, 7056, 8281, 9261, 9604, 11025, 12544, 14161, 15876, 16807, 17689, 19600, 21609, 21952, 23716, 25921, 28224, 30625, 33124, 35721, 38416, 41209, 42875, 44100, 47089, 50176, 53361, 56644

Offset: 1

Marco Ripà, May 04 2022

Terms are multiples of 49, since no perfect power divisible by 7 can have a 7-adic valuation below 2.

			196 is a term since 196 = (2*7)^2 is a power of a multiple of 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A001597 and A008589.

Other perfect powers: A075090, A075109, A353238, A353152.

q:= n-> igcd(seq(i[2], i=ifactors(n)[2]))>1:
select(q, [49*i$i=1..2000])[];  # Alois P. Heinz, May 05 2022

Select[49*Range[1200], GCD @@ FactorInteger[#][[All, 2]] > 1 &]

PARI
```
isok(k) = ispower(k) && !(k % 7)
```

a(n) has the form (7*m)^k for some m > 0 and k > 1.

Sum_{n>=1} 1/a(n) = -Sum_{k>=2} mu(k)*zeta(k)/7^k = 0.0371288923... - Amiram Eldar, Jul 02 2022

A362792 Numbers k such that 3k and 7k share the same set of digits.

Original entry on oeis.org

0, 45, 75, 423, 445, 450, 513, 750, 891, 1089, 1305, 2382, 2497, 4230, 4445, 4450, 4488, 4491, 4500, 4505, 4513, 4878, 5013, 5045, 5130, 5133, 5868, 7317, 7500, 7686, 8360, 8703, 8891, 8901, 8910, 8911, 8955, 8991, 9756, 9891, 10089, 10449, 10889, 10890, 10891

Offset: 1

Alexandru Petrescu, May 04 2023

The sequence is infinite because if k is a term, then 10*k is also a term.

Every number k of the form 44...45 (one of more 4's followed by 5, cf. A093140) is a term because 3*k = 133...35 and 7*k = 311...15.

			k = 75 is a term because 3*k = 225 and 7*k = 525 share the same set of digits, namely {2,5}.
k = 423 is a term because 3*k = 1269 and 7*k = 2961 share the same set of digits, namely {1,2,6,9}.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A008585, A008589, A093140.

Select[Range[0, 11000], Union[IntegerDigits[3*#]] == Union[IntegerDigits[7*#]] &] (* Amiram Eldar, May 18 2023 *)

isok(k) = Set(digits(3*k)) == Set(digits(7*k));

def ok(n): return set(str(3*n)) == set(str(7*n))
print([k for k in range(11000) if ok(k)]) # Michael S. Branicky, May 04 2023

A242570 a(n) = 252 * n.

Original entry on oeis.org

0, 252, 504, 756, 1008, 1260, 1512, 1764, 2016, 2268, 2520, 2772, 3024, 3276, 3528, 3780, 4032, 4284, 4536, 4788, 5040, 5292, 5544, 5796, 6048, 6300, 6552, 6804, 7056, 7308, 7560, 7812, 8064, 8316, 8568, 8820, 9072, 9324, 9576, 9828, 10080, 10332, 10584, 10836, 11088, 11340

Offset: 0

Derek Orr, May 17 2014

As lcm(1,2,3,...,9) = 2520, 10*a(n) + k is divisible by each k from 1 through 9.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008600, A008603, A008589, A008591, A008594, A008596.

Cf. A044102, A135628.

252*Range[0, 49] (* Alonso del Arte, May 17 2014 *)
LinearRecurrence[{2,-1},{0,252},50] (* Harvey P. Dale, Mar 25 2025 *)

PARI
```
for(n=0,50,print(252*n))
```

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 252*x/(x-1)^2.
E.g.f.: 252*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A044102(n) = 9*A135628(n) = 12*A008603(n) = 14*A008600(n) = 18*A008596(n) = 21*A008594(n) = 28*A008591(n) = 36*A008589(n) = 252*A001477(n). (End)

A336483 a(n) = floor(n/10) + (5 times last digit of n).

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 7

Offset: 0

Michel Marcus, Aug 11 2020

If the resulting number is divisible by 7, then n is divisible by 7; (re)discovered by 12-year-old Nigerian Chika Ofili.

L. E. Dickson, History of the theory of numbers. Vol. I: Divisibility and primality. Chelsea Publishing Co., New York 1966.

D. B. Eperson, Puzzles, Pastimes, Problems, Mathematics in School Vol. 16, No. 5 (Nov., 1987), pp. 18-19, 34-35.
OB360 Media, 12-year-old Nigerian Chika Ofili wins special award for discovering a new Mathematics formula, November 2019.
Skeptics Stack Exchange, Is Chika Ofili's method for checking divisibility for 7 a “new discovery” in math?.
A. Zbikowski, Note sur la divisibilité des nombres, Bull. Acad. Sci. St. Petersbourg 3 (1861) pp. 151-153.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A008589, A076309.

Table[Floor[n/10]+5Mod[n,10],{n,0,80}] (* or  *) LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,5,10,15,20,25,30,35,40,45,1},80] (* Harvey P. Dale, Nov 01 2023 *)

PARI
```
a(n) = 5*(n % 10) + (n\10);
```

From Stefano Spezia, Aug 11 2020: (Start)
O.g.f.: x*(5 + 5*x + 5*x^2 + 5*x^3 + 5*x^4 + 5*x^5 + 5*x^6 + 5*x^7 + 5*x^8 - 44*x^9)/(1 - x - x^10 + x^11).
a(n) = a(n-1) + a(n-10) - a(n-11) for n > 10. (End)

cp's OEIS Frontend

A166394 Multiples of 7 whose reversal - 1 is also a multiple of 7.

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A168622 Triangle read by rows: T(n, k) = [x^k]( 7*(1+x)^n - 6*(1+x^n) ) with T(0, 0) = 1.

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A182208 Carmichael numbers divisible by 7.

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A182341 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 7.

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Maple

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A328947 Numbers formed from decimal digits 0 and/or 1 which are divisible by 7.

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A335774 Numbers k such that in prime factorization of k the second smallest factor is 7.

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Mathematica

A351663 Perfect powers that are divisible by 7.

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A362792 Numbers k such that 3*k and 7*k share the same set of digits.

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A168622 Triangle read by rows: T(n, k) = [x^k]( 7(1+x)^n - 6(1+x^n) ) with T(0, 0) = 1.

A362792 Numbers k such that 3k and 7k share the same set of digits.