A017341 -id:A017341 - cp's OEIS frontend

A017353 a(n) = 10*n + 7.

7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, 147, 157, 167, 177, 187, 197, 207, 217, 227, 237, 247, 257, 267, 277, 287, 297, 307, 317, 327, 337, 347, 357, 367, 377, 387, 397, 407, 417, 427, 437, 447, 457, 467, 477, 487, 497, 507, 517, 527, 537

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008592, A016873, A017341.

[10*n+7: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017353:=n->10*n+7: seq(A017353(n), n=0..70); # Wesley Ivan Hurt, Mar 26 2015

Range[7, 497, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

vector(100,n,10*n-3) \\ Derek Orr, Mar 29 2015

From Vincenzo Librandi, May 28 2011: (Start)
a(n) = 10*n + 7.
a(n) = 2*a(n-1) - a(n-2). (End)
G.f.: (7+3*x)/(x-1)^2. - Wesley Ivan Hurt, Mar 26 2015
From Elmo R. Oliveira, Apr 05 2025: (Start)
E.g.f.: exp(x)*(7 + 10*x).
a(n) = A016873(2*n+1). (End)

A324297 Positive integers k that are the product of two integers ending with 6.

Original entry on oeis.org

36, 96, 156, 216, 256, 276, 336, 396, 416, 456, 516, 576, 636, 676, 696, 736, 756, 816, 876, 896, 936, 996, 1056, 1116, 1176, 1196, 1216, 1236, 1296, 1356, 1376, 1416, 1456, 1476, 1536, 1596, 1656, 1696, 1716, 1776, 1836, 1856, 1896, 1956, 1976, 2016, 2076, 2116

Offset: 1

Stefano Spezia, Mar 16 2019

All the terms end with 6 (A017341).

			36 = 6*6, 96 = 6*16, 216 = 6*36, 256 = 16*16, 276 = 6*46, ...

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000400, A017341 (supersequence), A324298, A053742 (ending with 5).

a={}; For[n=0,n<=250,n++,For[k=0,k<=n,k++,If[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

isok6(n) = (n%10) == 6; \\ A017341
isok(n) = {if (isok6(n), my(d=divisors(n)); fordiv(n, d, if (isok6(d) && isok6(n/d), return(1)));); return (0);} \\ Michel Marcus, Apr 14 2019

def aupto(lim): return sorted(set(a*b for a in range(6, lim//6+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2117)) # Michael S. Branicky, Aug 18 2021

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

The conjecture is true since it can be proved that a(n) = (sqrt(a(n-1)) + g(n-1))^2 where [g(n): n > 1] is a bounded sequence of positive real numbers. - Stefano Spezia, Aug 18 2021

A337856 Number of positive integers with n digits that are the product of two integers ending with 6.

Original entry on oeis.org

0, 2, 20, 230, 2515, 26889, 282211, 2930013, 30196730, 309564822, 3161099901, 32182595954, 326874672928, 3313770788984

Offset: 1

Stefano Spezia, Sep 27 2020

a(n) is the number of n-digit numbers in A324297.

Index entries for sequences related to digits.

Cf. A017341, A324297, A337855, A346509.

def A337856(n):
    k, n1, n2, pset = 0, 10**(n-1)//2-18, 10**n//2-18, set()
    while 50*k**2+60*k < n2:
        a, b = divmod(n1-30*k,50*k+30)
        m = max(k,a+int(b>0))
        r = 50*k*m+30*(k+m)
        while r < n2:
            pset.add(r)
            m += 1
            r += 50*k+30
        k += 1
    return len(pset) # Chai Wah Wu, Sep 26 2021

Conjecture: Lim_{n->infinity} a(n)/a(n-1) = 10.

a(5) corrected by and a(6)-a(9) from Jinyuan Wang, Oct 01 2020
a(10)-a(13) from Bert Dobbelaere, Oct 20 2020
a(14) from Martin Ehrenstein, Aug 06 2021

A347253 Positive integers that are the product of two integers ending with 4.

Original entry on oeis.org

16, 56, 96, 136, 176, 196, 216, 256, 296, 336, 376, 416, 456, 476, 496, 536, 576, 616, 656, 696, 736, 756, 776, 816, 856, 896, 936, 976, 1016, 1036, 1056, 1096, 1136, 1156, 1176, 1216, 1256, 1296, 1316, 1336, 1376, 1416, 1456, 1496, 1536, 1576, 1596, 1616, 1656

Offset: 1

Stefano Spezia, Aug 24 2021

			16 = 4*4, 56 = 4*14, 96 = 4*24, 136 = 4*34, 176 = 4*44, 196 = 14*14, 216 = 4*54, ...

Cf. A017341 (supersequence), A053742 (ending with 5), A139245 (ending with 2), A324297 (ending with 6), A346950 (ending with 3), A347254, A347255.

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 && 10*n+6>Max[a], AppendTo[a, 10*n+6]]]]; a

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)))
print(aupto(1660)) # Michael S. Branicky, Aug 24 2021

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

A324298 Positive integers k such that 10*k+6 is equal to the product of two integers ending with 6 (A324297).

Original entry on oeis.org

3, 9, 15, 21, 25, 27, 33, 39, 41, 45, 51, 57, 63, 67, 69, 73, 75, 81, 87, 89, 93, 99, 105, 111, 117, 119, 121, 123, 129, 135, 137, 141, 145, 147, 153, 159, 165, 169, 171, 177, 183, 185, 189, 195, 197, 201, 207, 211, 213, 217, 219, 223, 225, 231, 233, 237, 243, 249

Offset: 1

Stefano Spezia, Mar 16 2019

All the terms of this sequence are odd.

Why? If an integer 10*k+6 = (10*a+6) * (10*b+6), then k = 10*a*b + 6*(a+b) + 3, so k is odd. - Bernard Schott, May 13 2019

			145 is a term because 26*56 = 1456 = 145*10 + 6. - _Bernard Schott_, May 13 2019

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A017341, A053742 (ending with 5), A324297, A337856, A346389.

a={}; For[n=0,n<=250,n++,For[k=0,k<=n,k++,If[Mod[10*n+6,10*k+6]==0 && Mod[(10*n+6)/(10*k+6),10]==6 && 10*n+6>Max[10*a+6],AppendTo[a,n]]]]; a

isok6(n) = (n%10) == 6; \\ A017341
isok(k) = {my(n=10*k+6, d=divisors(n)); fordiv(n, d, if (isok6(d) && isok6(n/d), return(1))); return (0);} \\ Michel Marcus, Apr 14 2019

def aupto(lim): return sorted(set(a*b//10 for a in range(6, 10*lim//6+2, 10) for b in range(a, 10*lim//a+2, 10) if a*b//10 <= lim))
print(aupto(249)) # Michael S. Branicky, Aug 21 2021

a(n) = (A324297(n) - 6)/10.
Conjecture: lim_{n->infinity} a(n)/a(n-1) = 1.
The conjecture is true since a(n) = (A324297(n) - 6)/10 and lim_{n->infinity} A324297(n)/A324297(n-1) = 1. - Stefano Spezia, Aug 21 2021

A166728 Positive integers with English names ending in "x".

Original entry on oeis.org

6, 26, 36, 46, 56, 66, 76, 86, 96, 106, 126, 136, 146, 156, 166, 176, 186, 196, 206, 226, 236, 246, 256, 266, 276, 286, 296, 306, 326, 336, 346, 356, 366, 376, 386, 396, 406, 426, 436, 446, 456, 466, 476, 486, 496, 506, 526, 536, 546, 556, 566, 576, 586, 596

Offset: 1

Rick L. Shepherd, Oct 20 2009

			Fifty-six (56) is a term; sixteen (16) is not a term (but is a term of A060228).

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

Cf. A017341, A166726, A166727, A166729, A166730, A166731, A059093, A060228.

seq(seq(6+10*i+100*j,i=[0,$2..9]),j=0..10); # Robert Israel, Jul 01 2018

Rest@ CoefficientList[Series[x (6 + 20 x + 10 x^2 + 10 x^3 + 10 x^4 + 10 x^5 + 10 x^6 + 10 x^7 + 10 x^8 + 4 x^9)/(1 - x - x^9 + x^10), {x, 0, 54}], x] (* Michael De Vlieger, Jul 01 2018 *)

def agen(lim): yield from (k for k in range(6, lim+1, 10) if k%100 != 16)
print([an for an in agen(600)]) # Michael S. Branicky, Jun 26 2021

A017341 MINUS {n | n = 16 mod 100}.
From Robert Israel, Jul 01 2018: (Start)
  a(n+9) = a(n)+100.
  G.f.: x*(6+20*x+10*x^2+10*x^3+10*x^4+10*x^5+10*x^6+10*x^7+10*x^8+4*x^9)/(1-x-x^9+x^10). (End)

A202803 a(n) = n(5n+1).

Original entry on oeis.org

0, 6, 22, 48, 84, 130, 186, 252, 328, 414, 510, 616, 732, 858, 994, 1140, 1296, 1462, 1638, 1824, 2020, 2226, 2442, 2668, 2904, 3150, 3406, 3672, 3948, 4234, 4530, 4836, 5152, 5478, 5814, 6160, 6516, 6882, 7258, 7644, 8040, 8446, 8862, 9288, 9724, 10170

Offset: 0

Jeremy Gardiner, Dec 24 2011

First bisection of A219190. - Bruno Berselli, Nov 15 2012

a(n)*Pi is the total length of 5 points circle center spiral after n rotations. The spiral length at each rotation (L(n)) is A017341. The spiral length ratio rounded down [floor(L(n)/L(1))] is A032793. See illustration in links. - Kival Ngaokrajang, Dec 27 2013

			G.f. = 6*x + 22*x^2 + 48*x^3 + 84*x^4 + 130*x^5 +186*x^6 + 252*x^7 + 328*x^8 + ...

Jeremy Gardiner, Table of n, a(n) for n = 0..3000
Kival Ngaokrajang, Illustration of 5 points circle center spiral.
Leo Tavares, Illustration: Twin Trapezoids.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033429, A168668, A126264, A135705, A124080.
Cf. sequences listed in A254963.
Cf. A001622, A005475.

List([0..50], n-> n*(5*n+1)); # G. C. Greubel, Jul 04 2019

[n*(5*n+1):n in [0..50]]; // Vincenzo Librandi, Aug 11 2014

CoefficientList[Series[2x(3+2x)/(1-x)^3, {x, 0, 50}] ,x] (* Vincenzo Librandi, Aug 11 2014 *)
Table[5*n^2+n, {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)

a(n)=n*(5*n+1) \\ Charles R Greathouse IV, Jun 17 2017

[n*(5*n+1) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = 5*n^2 + n.
a(n) = A033429(n) + n. - Omar E. Pol, Dec 24 2011
G.f.: 2*x*(3+2*x)/(1-x)^3. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 6, a(2) = 22. - Philippe Deléham, Mar 27 2013
a(n) = A131242(10n+5). - Philippe Deléham, Mar 27 2013
a(n) = 2*A005475(n). - Philippe Deléham, Mar 27 2013
a(n) = A168668(n) - n. - Philippe Deléham, Mar 27 2013
a(n) = (n+1)^3 - (1 + n + n*(n-1) + n*(n-1)*(n-2)). - Michael Somos, Aug 10 2014
E.g.f.: x*(6+5*x)*exp(x). - G. C. Greubel, Aug 22 2017
Sum_{n>=1} 1/a(n) = 5*(1-log(5)/4) - sqrt(1+2/sqrt(5))*Pi/2 -sqrt(5)*log(phi)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Jul 19 2022

A273373 Squares ending in digit 6.

Original entry on oeis.org

16, 36, 196, 256, 576, 676, 1156, 1296, 1936, 2116, 2916, 3136, 4096, 4356, 5476, 5776, 7056, 7396, 8836, 9216, 10816, 11236, 12996, 13456, 15376, 15876, 17956, 18496, 20736, 21316, 23716, 24336, 26896, 27556, 30276, 30976, 33856, 34596, 37636, 38416, 41616

Offset: 1

Vincenzo Librandi, May 21 2016

These are the only squares whose second last digit is odd. This implies that the only squares whose last two digits are the same are those ending with 0 or 4; those ending with 1, 5, and 9 are paired with even second last digits. - Waldemar Puszkarz, May 24 2016

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A047221, A090773.
Cf. A017341 (numbers ending in 6), A017343 (cubes ending in 6).
Cf. squares with last digit k: A017270 (k=0), A273372 (k=1), A273375 (k=4), A017330 (k=5), this sequence (k=6), A273374 (k=9).

/* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 6];

[(10*n - 3*(-1)^n - 5)^2/4: n in [1..50]];

seq(seq((10*i+j)^2,j=[4,6]),i=0..20); # Robert Israel, May 24 2016

Table[(10 n - 3 (-1)^n - 5)^2/4, {n, 1, 50}]
CoefficientList[Series[4 (4 + 5 x + 32 x^2 + 5 x^3 + 4 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x]
Select[Range[250]^2,Mod[#,10]==6&] (* Harvey P. Dale, May 31 2020 *)

G.f.: 4*x*(4 + 5*x + 32*x^2 + 5*x^3 + 4*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = 4*A047221(n)^2 = (10*n - 3*(-1)^n - 5)^2/4.
a(n) = A090773(n)^2. - Michel Marcus, May 25 2016
Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5+sqrt(5))). - Amiram Eldar, Feb 16 2023

Corrected and extended by Bruno Berselli, May 23 2016

A346389 a(n) is the number of proper divisors of A324297(n) ending with 6.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 4, 2, 1, 2, 2, 3, 3, 2, 2, 4, 2, 5, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 4, 3, 4, 2, 5, 3, 3, 2, 2, 2, 2, 7, 2, 1, 2, 2, 3, 2, 3, 2, 2, 5, 3, 6, 3, 3, 2, 2, 2, 5, 2, 2, 3, 4, 3, 5, 2, 5, 4, 3, 2, 3, 6, 2, 2, 2, 6, 2, 2, 3, 2, 2, 3, 7

Offset: 1

Stefano Spezia, Jul 15 2021

			a(12) = 4 since there are 4 proper divisors of A324297(12) = 576 ending with 6: 6, 16, 36 and 96.

Cf. A017341, A032741, A324297, A324298, A337856, A346388 (ending with 5), A346392.

b={}; For[n=0, n<=450, n++, For[k=0, k<=n, k++, If[Mod[10*n+6, 10*k+6]==0 && Mod[(10*n+6)/(10*k+6), 10]==6 && 10*n+6>Max[b], AppendTo[b, 10*n+6]]]]; (* A324297 *) a={}; For[i =1, i<=Length[b], i++, AppendTo[a, Length[Drop[Select[Divisors[Part[b, i]], (Mod[#,10]==6&)], -1]]]]; a

a(n) = A346392(A324297(n)).

A347255 Number of positive integers with n digits that are the product of two integers ending with 4.

Original entry on oeis.org

0, 3, 25, 281, 2941, 30596, 315385, 3231664, 32972224, 335346193, 3402373313, 34454358909, 348373701706, 3518101287286, 35491654274101

Offset: 1

Stefano Spezia, Aug 24 2021

a(n) is the number of n-digit numbers in A347253.

Cf. A017341, A052268, A347253.

Cf. A346509 (ending with 1), A346952 (ending with 3), A337855 (ending with 5), A337856 (ending with 6).

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))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Aug 24 2021

a(n) < A052268(n).

Conjecture: lim_{n->infinity} a(n)/a(n-1) = 10.

a(9)-a(11) from Michael S. Branicky, Aug 25 2021

a(12)-a(15) from Martin Ehrenstein, Sep 29 2021

cp's OEIS Frontend

A017353 a(n) = 10*n + 7.

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A324297 Positive integers k that are the product of two integers ending with 6.

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A337856 Number of positive integers with n digits that are the product of two integers ending with 6.

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A347253 Positive integers that are the product of two integers ending with 4.

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A324298 Positive integers k such that 10*k+6 is equal to the product of two integers ending with 6 (A324297).

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A166728 Positive integers with English names ending in "x".

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A202803 a(n) = n*(5*n+1).

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A202803 a(n) = n(5n+1).