A008589 -id:A008589 - cp's OEIS frontend

A180597 Digital root of 7n.

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

Offset: 0

Odimar Fabeny, Sep 10 2010

Period of 9. - Robert G. Wilson v, Sep 20 2010

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A010888, A008589.
Cf. A180592, A180594, A180595, A180596, A180592, A180593, A180594, A180595, A180596, A180598, A180599.
Column k = 7 in A353109.

f[n_] := Mod[7 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)

a(n)=7*n%9 \\ Charles R Greathouse IV, Aug 22 2014

G.f.: x*(7 + 5*x + 3*x^2 + x^3 + 8*x^4 + 6*x^5 + 4*x^6 + 2*x^7 + 9*x^8)/(1 - x^9). - Stefano Spezia, Apr 21 2022

a(n) = A010888(A008589(n)). - Michel Marcus, Apr 21 2022

More terms from Robert G. Wilson v, Sep 20 2010

A212702 Main transitions in systems of n particles with spin 7/2.

Original entry on oeis.org

7, 112, 1344, 14336, 143360, 1376256, 12845056, 117440512, 1056964608, 9395240960, 82678120448, 721554505728, 6253472382976, 53876069761024, 461794883665920, 3940649673949184, 33495522228568064, 283726776524341248, 2395915001761103872, 20176126330619822080

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=8 (see formula), corresponding to spin S=(b-1)/2=7/2.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212703, A212704 (b = 2, 3, 4, 5, 6, 7, 9, 10).

Cf. A001018, A008589, A053539.

LinearRecurrence[{16,-64},{7,112},30] (* Harvey P. Dale, Feb 11 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212702.txt", n, " ", mtrans(n, 8)))

Vec(7*x/(8*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=8.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 16*a(n-1) - 64*a(n-2) for n > 2.
G.f.: 7*x/(8*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 7*x*exp(8*x).
a(n) = 7*A053539(n) = A008589(n)*A001018(n-1). (End)

A225327 Partition numbers of the form 7k.

Original entry on oeis.org

7, 42, 56, 77, 231, 385, 490, 1575, 2436, 3010, 10143, 21637, 31185, 37338, 44583, 124754, 147273, 281589, 329931, 386155, 451276, 1121505, 3087735, 8118264, 9289091, 20506255, 23338469, 49995925, 118114304, 133230930, 271248950, 607163746

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008589 and A000041.

			42 is in the sequence because 7*6 = 42 and 42 is a partition number: p(10) = A000041(10) = 42.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000041, A008589, A052001, A072871, A087183, A225324, A225325, A225326, A225358, A225360, A127544, A225361.

Select[PartitionsP[Range[300]], Mod[#, 7] == 0 &]

a(n) = 7*A222175(n).

A028898 Map n = Sum c_i 10^i to a(n) = Sum c_i 3^i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 21, 22, 23, 24, 25, 26, 27

Offset: 0

N. J. A. Sloane

If n is a multiple of 7, then a(n) is also a multiple of 7. See the Bhattacharyya link. - Michel Marcus, May 11 2016

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Malay Bhattacharyya, Sanghamitra Bandyopadhyay, Ujjwal Maulik, Non-primes are recursively divisible, Acta Universitatis Apulensis, No 19/2009.

Cf. A008589 (multiples of 7).

Different from A081502 for n>=100.

a:= proc(n) option remember;   n mod 10 + 3*procname(floor(n/10))
end proc:
a(0):= 0:
seq(a(i),i=0..100); # Robert Israel, May 11 2016

a = {1}; Do[AppendTo[a, If[Mod[n, 10] == 0, 3 a[[n/10]], a[[n - 1]] + 1]], {n, 2, 76}]; {0}~Join~a (* Michael De Vlieger, May 10 2016 *)

a(n)=if(n<1,0,if(n%10,a(n-1)+1,3*a(n/10)))

a(n) = subst(Pol(digits(n)), x, 3); \\ Michel Marcus, May 10 2016

a(0)=0, a(n)=3*a(n/10) if n==0 (mod 10), a(n)=a(n-1)+1 otherwise. - Benoit Cloitre, Dec 21 2002

G.f.: G(x) = (1-x)^(-1) * Sum_{i>=0} 3^i*p(x^(10^i)) where p(t) = (t+2*t^2+3*t^3+4*t^4+5*t^5+6*t^6+7*t^7+8*t^8+9*t^9)/(1+t+t^2+t^3+t^4+t^5+t^6+t^7+t^8+t^9) satisfies (1-x)*G(x) = p(x) + 3*(1-x^10)*G(x^10). - Robert Israel, May 11 2016

More terms from Erich Friedman

Moved Wesley Ivan Hurt's formula to A081502 where it applies. - Kevin Ryde, Dec 03 2019

A113803 Numbers that are congruent to {3, 11} mod 14.

Original entry on oeis.org

3, 11, 17, 25, 31, 39, 45, 53, 59, 67, 73, 81, 87, 95, 101, 109, 115, 123, 129, 137, 143, 151, 157, 165, 171, 179, 185, 193, 199, 207, 213, 221, 227, 235, 241, 249, 255, 263, 269, 277, 283, 291, 297, 305, 311, 319, 325, 333, 339, 347, 353, 361, 367

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Third row A113807.

Cf. A008589, A113801, A113802, A113804, A113805, A113806.

{3+#,11+#}&/@(14*Range[0,30])//Flatten (* Harvey P. Dale, Jun 28 2020 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 3). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7-(-1)^n)/2.
O.g.f.: x*(3+8*x+3*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(3*Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cot(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = sin(Pi/7)*cosec(3*Pi/14). (End)

A113802 Numbers that are congruent to {2, 12} mod 14.

Original entry on oeis.org

2, 12, 16, 26, 30, 40, 44, 54, 58, 68, 72, 82, 86, 96, 100, 110, 114, 124, 128, 138, 142, 152, 156, 166, 170, 180, 184, 194, 198, 208, 212, 222, 226, 236, 240, 250, 254, 264, 268, 278, 282, 292, 296, 306, 310, 320, 324, 334, 338, 348, 352, 362, 366, 376, 380

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Second row A113807.

Cf. A008589, A113801, A113803, A113804, A113805, A113806.

Select[Range[400],MemberQ[{2,12},Mod[#,14]]&] (* Harvey P. Dale, Oct 30 2011 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 2). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7-3*(-1)^n)/2.
O.g.f.: 2*x*(1+5*x+x^2)/((1+x)*(1-x)^2).
See the contribution of Bruno Berselli under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(Pi/7)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(Pi/7)*sin(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/7)*sin(Pi/14). (End)

A113805 Numbers that are congruent to {5, 9} mod 14.

Original entry on oeis.org

5, 9, 19, 23, 33, 37, 47, 51, 61, 65, 75, 79, 89, 93, 103, 107, 117, 121, 131, 135, 145, 149, 159, 163, 173, 177, 187, 191, 201, 205, 215, 219, 229, 233, 243, 247, 257, 261, 271, 275, 285, 289, 299, 303, 313, 317, 327, 331, 341, 345, 355, 359, 369

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Cf. A113801, A113802, A113803, A113804, A113806, A008589, A113807.

Flatten[Table[14n + {5, 9}, {n, 0, 28}]] (* Alonso del Arte, Dec 15 2011 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 5). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7+3*(-1)^n)/2.
O.g.f.: x*(5+4*x+5*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/7)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 4*sin(Pi/7)*sin(3*Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/7). (End)

A113806 Numbers that are congruent to {6, 8} mod 14.

Original entry on oeis.org

6, 8, 20, 22, 34, 36, 48, 50, 62, 64, 76, 78, 90, 92, 104, 106, 118, 120, 132, 134, 146, 148, 160, 162, 174, 176, 188, 190, 202, 204, 216, 218, 230, 232, 244, 246, 258, 260, 272, 274, 286, 288, 300, 302, 314, 316, 328, 330, 342, 344, 356, 358, 370

Offset: 1

Giovanni Teofilatto, Jan 22 2006

Cf. A008589, A113801, A113802, A113803, A113804, A113805, A113807.

Flatten[# + {6, 8} &/@ (14 Range[0, 30])] (* Harvey P. Dale, Jan 11 2011 *)

a(n) = 14*n - a(n-1) - 14 (with a(1) = 6). - Vincenzo Librandi, Nov 13 2010
From Wolfdieter Lang, Dec 15 2011: (Start)
a(n) = 7*n-(7+5*(-1)^n)/2.
O.g.f.: 2*x*(3+x+3*x^2)/((1+x)*(1-x)^2).
See the Bruno Berselli contribution under A113801. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/14)*Pi/14. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/14).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(Pi/7)*cosec(3*Pi/14)/4. (End)

A047382 Numbers that are congruent to {0, 5} mod 7.

Original entry on oeis.org

0, 5, 7, 12, 14, 19, 21, 26, 28, 33, 35, 40, 42, 47, 49, 54, 56, 61, 63, 68, 70, 75, 77, 82, 84, 89, 91, 96, 98, 103, 105, 110, 112, 117, 119, 124, 126, 131, 133, 138, 140, 145, 147, 152, 154, 159, 161, 166, 168

Offset: 1

N. J. A. Sloane

Except for the first term, numbers m such that 36*m^2 + 72*m + 35 = (6*m+5)*(6*m+7) is not of the form p*(p+2), with p prime. - Vincenzo Librandi, Aug 05 2010

Nonnegative k such that k or 4*k + 1 is divisible by 7. - Bruno Berselli, Feb 13 2018

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

Cf. A008589, A017041.

&cat[[7*n,7*n+5]: n in [0..23]];  // Bruno Berselli, Oct 17 2011

{#, 5 + #} &/@ (7 Range[0, 30]) // Flatten (* or *) LinearRecurrence[{1, 1, -1}, {0, 5, 7}, 60] (* Harvey P. Dale, Dec 01 2016 *)

a(n) = (14*n + 3*(-1)^n - 11)/4 \\ David Lovler, Sep 11 2022

a(n) = 7*n - a(n-1) - 9 for n>1, with a(1)=0. - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=A005009(k-1)=7*2^(k-1) for k>0. - Philippe Deléham, Oct 17 2011
From Bruno Berselli, Oct 17 2011:  (Start)
G.f.: x^2*(5 + 2*x)/((1 + x)*(1 - x)^2).
a(n) = (14*n + 3*(-1)^n - 11)/4.
a(-n) = -A047352(n+2). (End)
a(n) = ceiling((7/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
E.g.f.: 2 + ((14*x - 11)*exp(x) + 3*exp(-x))/4. - David Lovler, Sep 11 2022

A072750 Counting factor 7 in the first n squarefree numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 1

Reinhard Zumkeller, Jul 08 2002

nonn

			The first 10 squarefree numbers are: 1, 2, 3, 5, 6=2*3, 7, 10=2*5, 11, 13 and 14=2*7: 7 and 14 are divisible by 7, therefore a(10)=2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005117, A008589, A072747, A072748, A072749, A072751.

a072750 n = a072750_list !! (n-1)
a072750_list = scanl1 (+) $ map ((0 ^) . (`mod` 7)) a005117_list
-- Reinhard Zumkeller, Mar 25 2013

N:= 1000: # to use the squarefree numbers <= N
M:= map(proc(t) if numtheory:-issqrfree(t) then if t mod 7 = 0 then 1 else 0 fi fi end proc, [$1..N]):
ListTools:-PartialSums(M); # Robert Israel, Aug 23 2015

With[{sf=Select[Range[200],SquareFreeQ]},Accumulate[If[Divisible[#,7],1,0]&/@sf]] (* Harvey P. Dale, Mar 21 2013 *)

n = 94; k = 0; bag = List(); a = vector(n);
until(n == 0, k++; if (issquarefree(k), listput(bag, k); n--));
for (i=2, #bag, a[i] = a[i-1] + (bag[i] % 7 == 0));
print(a); \\ Gheorghe Coserea, Aug 23 2015

a(n) - a(n-1) = 1 if A005117(n) is in A008589, otherwise 0. - Robert Israel, Aug 23 2015

a(n) ~ n/8. - Amiram Eldar, Feb 24 2021

Name clarified by Gheorghe Coserea, Aug 23 2015

cp's OEIS Frontend

A180597 Digital root of 7n.

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A212702 Main transitions in systems of n particles with spin 7/2.

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A225327 Partition numbers of the form 7k.

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A028898 Map n = Sum c_i 10^i to a(n) = Sum c_i 3^i.

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A113803 Numbers that are congruent to {3, 11} mod 14.

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A113802 Numbers that are congruent to {2, 12} mod 14.

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A113805 Numbers that are congruent to {5, 9} mod 14.

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A113806 Numbers that are congruent to {6, 8} mod 14.

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