A017341 -id:A017341 - cp's OEIS frontend

A168460 a(n) = 6 + 10*floor((n-1)/2).

6, 6, 16, 16, 26, 26, 36, 36, 46, 46, 56, 56, 66, 66, 76, 76, 86, 86, 96, 96, 106, 106, 116, 116, 126, 126, 136, 136, 146, 146, 156, 156, 166, 166, 176, 176, 186, 186, 196, 196, 206, 206, 216, 216, 226, 226, 236, 236, 246, 246, 256, 256, 266, 266, 276, 276, 286

Offset: 1

Vincenzo Librandi, Nov 26 2009

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

Cf. A017341, A168283.

[6+10*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013

A168460:=n->6 + 10*floor((n-1)/2): seq(A168460(n), n=1..100); # Wesley Ivan Hurt, Jul 24 2016

RecurrenceTable[{a[1]==6,a[n]==10n-a[n-1]-8},a,{n,80}] (* or *) LinearRecurrence[{1,1,-1},{6,6,16},80] (* Harvey P. Dale, Apr 25 2011 *)
Table[6 + 10 Floor[(n - 1)/2], {n, 70}] (* or *) CoefficientList[Series[2 (3 + 2 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)

a(n) = 10*n - a(n-1) - 8, with n>1, a(1)=6.
From R. J. Mathar, Jan 04 2011: (Start)
a(n) = 2*A168283(n).
a(n+1) = A017341(floor(n/2)).
G.f.: 2*x*(3 + 2*x^2) / ( (1+x)*(x-1)^2 ). (End)
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - Vincenzo Librandi, Sep 19 2013
From G. C. Greubel, Jul 23 2016: (Start)
a(n) = (10*n - 5*(-1)^n - 3)/2.
E.g.f.: (1/2)*(-5 + 8*exp(x) + (10*x - 3)*exp(2*x))*exp(-x). (End)
a(n) = a(n-2) + 10 for n>2. - Wesley Ivan Hurt, Jul 24 2016

New definition by Vincenzo Librandi, Sep 19 2013

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

A347254 Positive integers k such that 10*k+6 is equal to the product of two integers ending with 4 (A347253).

Original entry on oeis.org

1, 5, 9, 13, 17, 19, 21, 25, 29, 33, 37, 41, 45, 47, 49, 53, 57, 61, 65, 69, 73, 75, 77, 81, 85, 89, 93, 97, 101, 103, 105, 109, 113, 115, 117, 121, 125, 129, 131, 133, 137, 141, 145, 149, 153, 157, 159, 161, 165, 169, 173, 177, 181, 183, 185, 187, 189, 193, 197

Offset: 1

Stefano Spezia, Aug 24 2021

Since an integer 10*k + 6 = (10*a + 4)*(10*b + 4) implies that k = 10*a*b + 4*(a + b) + 1, all the terms of this sequence are odd.

			13 is a term because 4*34 = 136 = 13*10 + 6.

Cf. A016873 (ending with 5), A017341, A324298 (ending with 6), A346951 (ending with 3), A347253.

a={}; For[n=0, n<=200, n++, For[k=0, k<=n, k++, If[Mod[10*n+6, 10*k+4]==0 && Mod[(10*n+6)/(10*k+4), 10]==4 && n>Max[a], AppendTo[a, n]]]]; a

isok(k) =  my(x=10*k+6); sumdiv(x, d, (Mod(d, 10)==4) && Mod(x/d, 10)==4); \\ Michel Marcus, Oct 04 2021

def aupto(lim): return sorted(set(a*b//10 for a in range(4, 10*lim//4+3, 10) for b in range(a, 10*lim//a+3, 10) if a*b//10 <= lim))
print(aupto(197)) # Michael S. Branicky, Aug 24 2021

a(n) = (A347253(n) - 6)/10.

Lim_{n->infinity} a(n)/a(n-1) = 1.

A347746 Positive integers that are equal both to the product of two integers ending with 4 and to that of two integers ending with 6.

Original entry on oeis.org

96, 216, 256, 336, 416, 456, 576, 696, 736, 756, 816, 896, 936, 1056, 1176, 1216, 1296, 1376, 1416, 1456, 1536, 1596, 1656, 1696, 1776, 1836, 1856, 1896, 1976, 2016, 2136, 2176, 2256, 2336, 2376, 2436, 2496, 2576, 2616, 2656, 2736, 2816, 2856, 2916, 2976, 3016

Offset: 1

Stefano Spezia, Sep 12 2021

Intersection of A324297 and A347253.

			96 = 4*24 = 6*16, 216 = 4*54 = 6*36, 256 = 4*64 = 16*16, 336 = 4*84 = 6*56, ...

Cf. A017341 (supersequence), A324297, A347253, A347748.

max=3050;Select[Intersection[Union@Flatten@Table[a*b, {a, 4, Floor[max/4], 10}, {b, a, Floor[max/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[max/6], 10}, {b, a, Floor[max/a], 10}]], 0<#

def aupto(lim): return sorted(set(a*b for a in range(4, lim//4+1, 10) for b in range(a, lim//a+1, 10)) & set(a*b for a in range(6, lim//6+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(3017)) # Michael S. Branicky, Sep 12 2021

Lim_{n->infinity} a(n)/a(n-1) = 1.

A038861 Numbers ending with '6' that are the difference of two positive cubes.

Original entry on oeis.org

26, 56, 296, 316, 386, 866, 936, 1016, 1206, 1216, 1736, 1946, 2196, 2646, 2716, 2736, 2906, 3096, 3176, 4376, 4706, 4816, 4886, 5256, 5616, 5886, 6146, 6516, 6536, 7516, 7936, 8216, 8666, 8766, 9136, 9576, 10136, 10586, 10816, 10836, 11096

Offset: 1

Jeff Burch

Intersection of A017341 and A181123.

Intersection with A017343 is empty.

Name edited by Michel Marcus, Aug 04 2021

A258187 Numbers m such that either m^k - 1 or m^k - 2 is prime for some positive k, but not both.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 32, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 77, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 101

Offset: 1

Juri-Stepan Gerasimov, May 23 2015

From R. J. Mathar, Jul 22 2015: (Start)
10 is not in the sequence because 10^k-2 is even and 10^k-1 is divisible by 3 (because 10^k mod 3 = 1 as 10 mod 3 = 1). 16 is not in the sequence because 16^k-2 is even and 16^k-1 is divisible by 3 (because 16^k mod 3 = 1 as 16 mod 3 = 1). For the same reason almost all even numbers of the form 3m+1 (A016957) are absent, the only exception being 4, as 4^1-1 is a prime.
36 is not in the sequence because 36^k-1 is even and 36^k-1 is divisible by 5 (because 36^k mod 5 = 1 as 36 mod 5 = 1). This reasoning excludes all terms of A017341 (except for 6, as 6^1-1 is prime) from this sequence. With the same methology we can fish for (and exclude) even numbers of the form m*p+1 for primes p >= 3. (End)

			2 is not in this sequence because 2^2 - 1 = 3 and 2^2 - 2 = 2 are both prime.
3 is in this sequence because 3^1 - 1 = 2 (prime) and 3^1 - 2 = 1 (nonprime) or 3^2 - 1 = 5 (prime) and 3^2 - 2 = 4 (nonprime).

Cf. A016957, A017341.

is(n)=n>2 && if(n%2,1,isprime(n-1)) \\ Charles R Greathouse IV, Jun 03 2015

A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

Original entry on oeis.org

3, 1, 13, 9, 23, 7, 33, 19, 43, 3, 53, 29, 63, 17, 73, 39, 83, 11, 93, 49, 103, 27, 113, 59, 123, 1, 133, 69, 143, 37, 153, 79, 163, 21, 173, 89, 183, 47, 193, 99, 203, 13, 213, 109, 223, 57, 233, 119, 243, 31, 253, 129, 263, 67, 273, 139, 283, 9, 293, 149, 303

Offset: 1

Michel Lagneau, Mar 27 2016

The odd-indexed terms a(2i+1) = 10i+3 = A017305(i), i>=0;
a(4i+4) = 10i+9 = A017377(i), i>=0;
a(8i+6) = 10i+7 = A017353(i), i>=0;
a(16i+2) = 10i+1 = A017281(i), i>=0.
Note that a(n) = a(16n-6) = a(6n-2)/3. No multiple of 5 is in this sequence.
a(n) = R(2n-1) < 2n-1 for n = 2, 6, 10, ..., 2+4i,...

			a(4)=9 because (2*4-1) = 7  -> (5*7+1)/2^2 = 9.

Cf. A075677, A017281, A017305, A017353, A017377, A133423, A133424, A174795, A133419, A133420, A232711, A259207, A267969.

nextOddK[n_] := Module[{m=5n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]

a(n) = my(m = 2*n-1, c = 5*m+1); c/2^valuation(c, 2); \\ Michel Marcus, Mar 27 2016

a(n) = A000265(A017341(n-1)). - Michel Marcus, Mar 27 2016

cp's OEIS Frontend

A168460 a(n) = 6 + 10*floor((n-1)/2).

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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A347254 Positive integers k such that 10*k+6 is equal to the product of two integers ending with 4 (A347253).

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A347746 Positive integers that are equal both to the product of two integers ending with 4 and to that of two integers ending with 6.

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A038861 Numbers ending with '6' that are the difference of two positive cubes.

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A258187 Numbers m such that either m^k - 1 or m^k - 2 is prime for some positive k, but not both.

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A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

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