A017341 -id:A017341 - cp's OEIS frontend

A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

10, 16, 26, 22, 36, 50, 28, 46, 64, 82, 34, 56, 78, 100, 122, 40, 66, 92, 118, 144, 170, 46, 76, 106, 136, 166, 196, 226, 52, 86, 120, 154, 188, 222, 256, 290, 58, 96, 134, 172, 210, 248, 286, 324, 362, 64, 106, 148, 190, 232, 274, 316, 358, 400, 442, 70, 116, 162

Offset: 1

Vincenzo Librandi, Jan 21 2009

First column: A016957, second column: A017341, third column: 2*A017029, fourth column: A082286. - Vincenzo Librandi, Nov 21 2012
Conjecture: Let p = prime number. If 2^p belongs to the sequence, then 2^p-1 is not a Mersenne prime. - Vincenzo Librandi, Dec 12 2012
Conjecture is true because if T(n, k) = 2^p with p prime, then 2^p-1 = 4*n*k + 2*n + 2*k + 1 = (2*n+1)*(2*k+1) hence 2^p-1 is not prime. - Michel Marcus, May 31 2015
It appears that T(m,p) = 2^p for Lucasian primes (A002515) greater than 3. For instance: T(44, 11) = 2^11, T(89240, 23) = 2^23. - Michel Marcus, May 28 2015
For n > 1, ascending numbers along the diagonal are also terms of the even principal diagonal of a 2n X 2n spiral (A137928). - Avi Friedlich, May 21 2015

			Triangle begins
  10;
  16,  26;
  22,  36,  50;
  28,  46,  64,  82;
  34,  56,  78, 100, 122;
  40,  66,  92, 118, 144, 170;
  46,  76, 106, 136, 166, 196, 226;
  52,  86, 120, 154, 188, 222, 256, 290;
  58,  96, 134, 172, 210, 248, 286, 324, 362;
  64, 106, 148, 190, 232, 274, 316, 358, 400, 442;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A000043, A000668, A016957, A017029, A017341, A054723, A082286, A144650.

[4*n*k + 2*n + 2*k + 2: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

seq(seq( 2*(2*n*k+n+k+1), k=1..n), n=1..15) # G. C. Greubel, Mar 21 2021

T[n_,k_]:=4*n*k + 2*n + 2*k + 2; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*(2*n*k+n+k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 21 2021

T(n, k) = 2*A144650(n, k).

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n + 3) = n*A014105(n+2) =

Edited by Robert Hochberg, Jun 21 2010

A017343 a(n) = (10*n + 6)^3.

Original entry on oeis.org

216, 4096, 17576, 46656, 97336, 175616, 287496, 438976, 636056, 884736, 1191016, 1560896, 2000376, 2515456, 3112136, 3796416, 4574296, 5451776, 6434856, 7529536, 8741816, 10077696, 11543176, 13144256, 14886936, 16777216, 18821096, 21024576, 23393656, 25934336

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A017341.

[(10*n+6)^3: n in [0..35]]; // Vincenzo Librandi, Aug 03 2011

(10*Range[0,30]+6)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{216,4096,17576,46656},30] (* Harvey P. Dale, Nov 05 2019 *)

vector(40, n, n--; (10*n+6)^3) \\ Michel Marcus, Aug 04 2021

a(n) = A000578(A017341(n)). - Michel Marcus, Aug 04 2021
From Wesley Ivan Hurt, Jan 27 2022: (Start)
G.f.: 8*(27+404*x+311*x^2+8*x^3)/(x-1)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). (End)

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 3, 0, 1, 2, 4, 1, 0, 1, 3, 5, 4, 6, 0, 1, 4, 6, 7, 9, 3, 0, 1, 5, 7, 10, 12, 9, 10, 0, 1, 6, 8, 13, 15, 15, 16, 6, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, 0, 1, 8, 10, 19, 21, 27, 28, 26, 25, 10, 0, 1, 9, 11, 22, 24, 33, 34, 36, 35

Offset: 0

Omar E. Pol, Feb 05 2012

Note that a single formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 lists the generalized pentagonal numbers A001318. See some other examples in the cross-references section.

			Array begins:
(A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...
(A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...
A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...
A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...
A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...
A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...
A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...
A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...
A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...
A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...
A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...

Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818
Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).
Cf. A139600.

A262389 Numbers whose last digit is composite.

Original entry on oeis.org

4, 6, 8, 9, 14, 16, 18, 19, 24, 26, 28, 29, 34, 36, 38, 39, 44, 46, 48, 49, 54, 56, 58, 59, 64, 66, 68, 69, 74, 76, 78, 79, 84, 86, 88, 89, 94, 96, 98, 99, 104, 106, 108, 109, 114, 116, 118, 119, 124, 126, 128, 129, 134, 136, 138, 139, 144, 146, 148, 149

Offset: 1

Wesley Ivan Hurt, Sep 21 2015

Numbers ending in 4, 6, 8 or 9.
Union of A017317, A017341, A017365 and A017377.
Subsequence of A118951 (numbers containing at least one composite digit).
Complement of (A197652 Union A260181).

Gerald Hillier and Didier Lachieze, Last Digit Composite.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A017317, A017341, A017365, A017377.

Cf. A118951, A197652, A260181 (last digit is prime).

[(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n) div 4) div 2) div 2: n in [1..70]]; // Vincenzo Librandi, Sep 21 2015

A262389:=n->(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2: seq(A262389(n), n=1..100);

Table[(5n+1-(-1)^n+(3+(-1)^n)*(-1)^((2n-3-(-1)^n)/4)/2)/2, {n, 100}]
LinearRecurrence[{1, 0, 0, 1, -1}, {4, 6, 8, 9, 14}, 80] (* Vincenzo Librandi, Sep 21 2015 *)
CoefficientList[Series[(4 + 2*x + 2*x^2 + x^3 + x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 80}], x] (* Wesley Ivan Hurt, Sep 21 2015 *)
Select[Range[200],CompositeQ[Mod[#,10]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 21 2019 *)

G.f.: x*(4+2*x+2*x^2+x^3+x^4)/((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(10-2*sqrt(5))*Pi - sqrt(5)*arccoth(3/sqrt(5)) - 4*log(2))/20. - Amiram Eldar, Jul 30 2024

Name edited by Jon E. Schoenfield, Feb 15 2018

A347747 Positive integers with final digit 6 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

16, 36, 56, 96, 136, 156, 176, 196, 216, 256, 276, 296, 336, 376, 396, 416, 456, 476, 496, 516, 536, 576, 616, 636, 656, 676, 696, 736, 756, 776, 816, 856, 876, 896, 936, 976, 996, 1016, 1036, 1056, 1096, 1116, 1136, 1156, 1176, 1196, 1216, 1236, 1256, 1296, 1316

Offset: 1

Stefano Spezia, Sep 12 2021

Union of A324297 and A347253.

			16 = 4*4, 36 = 6*6, 56 = 4*14, 96 = 4*24 = 6*16, 136 = 4*34, 156 = 6*26, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A017341 (supersequence), A324297, A347253, A347749.

a={}; For[n=0, n<=150, 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 && 10*n+6>Max[a] || Mod[10*n+6,10*k+6]==0 && Mod[(10*n+6)/(10*k+6),10]==6 && 10*n+6>Max[a], AppendTo[a, 10*n+6]]]]; a
tisdQ[n_]:=AnyTrue[{Mod[#,10],Mod[n/#,10]}&/@Divisors[n],#[[1]] == #[[2]]&]; Select[10 Range[150]+6,tisdQ] (* Harvey P. Dale, Dec 27 2021 *)

isok(m) = if ((m % 10) == 6, fordiv(m, d, if ((d % 10) == (m/d % 10), return(1)))); \\ Michel Marcus, Oct 06 2021

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(1317)) # Michael S. Branicky, Sep 12 2021

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

A347748 Number of positive integers with n digits 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

0, 1, 12, 159, 1859, 20704, 223525, 2370684, 24842265, 258128126, 2665475963

Offset: 1

Stefano Spezia, Sep 12 2021

a(n) is the number of n-digit numbers in A347746.

Cf. A017341, A052268, A324297, A337856, A347253, A347255, A347746, A347749.

Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Intersection[Union@Flatten@Table[a*b, {a, 4, Floor[hi/4], 10}, {b, a, Floor[hi/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[hi/6], 10}, {b, a, Floor[hi/a], 10}]], lo<#

def a(n):
  lo, hi = 10**(n-1), 10**n
  return len(set(a*b for a in range(4, hi//4+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi) & set(a*b for a in range(6, hi//6+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Oct 06 2021

a(n) < A052268(n).
a(n) = A337856(n) + A347255(n) - A347749(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

a(9)-a(10) from Michael S. Branicky, Oct 06 2021

a(11) from Frank A. Stevenson, Jan 06 2024

A347749 Number of positive integers with n digits and final digit 6 that are equal to the product of two integers ending with the same digit.

Original entry on oeis.org

0, 4, 33, 352, 3597, 36781, 374071, 3790993, 38326689, 386782889

Offset: 1

Stefano Spezia, Sep 12 2021

a(n) is the number of n-digit numbers in A347747.

Cf. A017341, A052268, A324297, A337856, A347253, A347255, A347747, A347748.

Table[{lo, hi}={10^(n-1), 10^n}; Length@Select[Union[Union@Flatten@Table[a*b, {a, 4, Floor[hi/4], 10}, {b, a, Floor[hi/a], 10}],Union@Flatten@Table[a*b, {a, 6, Floor[hi/6], 10}, {b, a, Floor[hi/a], 10}]], lo<#

def a(n):
  lo, hi = 10**(n-1), 10**n
  return len(set(a*b for a in range(4, hi//4+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi) | set(a*b for a in range(6, hi//6+1, 10) for b in range(a, hi//a+1, 10) if lo <= a*b < hi))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Oct 06 2021

a(n) < A052268(n).
a(n) = A337856(n) + A347255(n) - A347748(n).
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

a(9)-a(10) from Michael S. Branicky, Oct 06 2021

A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 21, 12, 3, 40, 31, 22, 13, 4, 50, 41, 32, 23, 14, 5, 60, 51, 42, 33, 24, 15, 6, 70, 61, 52, 43, 34, 25, 16, 7, 80, 71, 62, 53, 44, 35, 26, 17, 8, 90, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10, 110, 101, 92, 83, 74, 65, 56, 47, 38, 29, 110, 11

Offset: 0

Stefano Spezia, Jul 05 2025

			Array begins as:
   0,  1,  2,  3,  4,  5,  6,  7, ...
  10, 11, 12, 13, 14, 15, 16, 17, ...
  20, 21, 22, 23, 24, 25, 26, 27, ...
  30, 31, 32, 33, 34, 35, 36, 37, ...
  40, 41, 42, 43, 44, 45, 46, 47, ...
  50, 51, 52, 53, 54, 55, 56, 57, ...
  60, 61, 62, 63, 64, 65, 66, 67, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11475 (first 151 antidiagonals of the array, flattened)

Columns for k=0..9 give: A008592, A017281, A017293, A017305, A017317, A017329, A017341, A017353, A017365, A017377.

Cf. A001477 (1st row), A020338 (main diagonal), A055642, A385624 (antidiagonal sums).

A[n_,k_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[k]]]; Table[A[n,k],{n,0,6},{k,0,7}] (* or *)
A[n_,k_]:=If[k==0,10n,n*10^(Floor[Log10[k]]+1)+k]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

T(n, k) = fromdigits(concat(digits(n), digits(k))); \\ Michel Marcus, Jul 06 2025

A(n,0) = 10*n and A(n,k) = n*10^(floor(log_10(k)) + 1) + k for k > 0.

A017342 a(n) = (10*n + 6)^2.

Original entry on oeis.org

36, 256, 676, 1296, 2116, 3136, 4356, 5776, 7396, 9216, 11236, 13456, 15876, 18496, 21316, 24336, 27556, 30976, 34596, 38416, 42436, 46656, 51076, 55696, 60516, 65536, 70756, 76176, 81796, 87616, 93636, 99856, 106276, 112896, 119716, 126736

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A000290, A017341.

[(10*n+6)^2: n in [0..35]]; // Vincenzo Librandi, Aug 03 2011

A017342:=n->(10*n + 6)^2; seq(A017342(n), n=0..35); # Wesley Ivan Hurt, Jan 29 2014

Table[(10 n + 6)^2, {n, 0, 35}] (* Wesley Ivan Hurt, Jan 29 2014 *)
LinearRecurrence[{3,-3,1},{36,256,676},40] (* Harvey P. Dale, Dec 04 2014 *)

a(n)=(10*n+6)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A000290(A017341(n)). - Wesley Ivan Hurt, Jan 29 2014

A017352 (10*n+6)^12.

Original entry on oeis.org

2176782336, 281474976710656, 95428956661682176, 4738381338321616896, 89762301673555234816, 951166013805414055936, 6831675453247426400256, 37133262473195501387776, 163674647745587512938496, 612709757329767363772416

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A008456 (12th Powers), A017341 (10n+6).

[(10*n+6)^12: n in [0..10]]; // Vincenzo Librandi, Aug 03 2011

A017352:=n->(10*n+6)^12: seq(A017352(n), n=0..10); # Wesley Ivan Hurt, Oct 28 2014

(10 Range[0, 10] + 6)^12 (* Wesley Ivan Hurt, Oct 28 2014 *)
CoefficientList[Series[4096 (531441 + 68712568003 x + 22404773377311 x^2 + 859316242027205 x^3 + 8673413722667370 x^4 + 30946876621062078 x^5 + 44108689210889694 x^6 + 25884027384156618 x^7 + 5972410776815445 x^8 + 467792550632655 x^9 + 8736164034131 x^10 + 13841233953 x^11 + 4096 x^12)/(1 - x)^13, {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 28 2014 *)

From Wesley Ivan Hurt, Oct 28 2014: (Start)
G.f.: 4096*(531441 + 68712568003*x + 22404773377311*x^2 + 859316242027205*x^3 + 8673413722667370*x^4 + 30946876621062078*x^5 + 44108689210889694*x^6 + 25884027384156618*x^7 + 5972410776815445*x^8 + 467792550632655*x^9 + 8736164034131*x^10 + 13841233953*x^11 + 4096*x^12) / (1-x)^13.
a(n) = 13*a(n-1)-78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+1716*a(n-7)-1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13).
a(n) = (10*n+6)^12 = A008456(A017341(n)). (End)

cp's OEIS Frontend

A155151 Triangle T(n, k) = 4*n*k + 2*n + 2*k + 2, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A017343 a(n) = (10*n + 6)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A194801 Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A262389 Numbers whose last digit is composite.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A347747 Positive integers with final digit 6 that are equal to the product of two integers ending with the same digit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A347748 Number of positive integers with n digits that are equal both to the product of two integers ending with 4 and to that of two integers ending with 6.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A347749 Number of positive integers with n digits and final digit 6 that are equal to the product of two integers ending with the same digit.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

A155151 Triangle T(n, k) = 4nk + 2n + 2k + 2, read by rows.

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.