A017317 -id:A017317 - cp's OEIS frontend

A166727 Positive integers with English names ending in "r".

4, 24, 34, 44, 54, 64, 74, 84, 94, 104, 124, 134, 144, 154, 164, 174, 184, 194, 204, 224, 234, 244, 254, 264, 274, 284, 294, 304, 324, 334, 344, 354, 364, 374, 384, 394, 404, 424, 434, 444, 454, 464, 474, 484, 494, 504, 524, 534, 544, 554, 564, 574, 584, 594

Offset: 1

Rick L. Shepherd, Oct 20 2009

			Fifty-four (54) is a term; fourteen (14) is not a term (but is a term of A060228).

Cf. A017317, A166726, A166728, A166729, A166730, A166731, A059093, A060228.

def a(n): return (n-1)//9*100 + ((n-1)%9 + 1)*((n-1)%9 > 0)*10 + 4
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Apr 24 2021

A017317 MINUS {n | n = 14 mod 100}.

a(n) = a(n-9) + 100, for n >= 10. - Michael S. Branicky, Apr 24 2021

A031914 a(n) = prime(10*n - 6).

Original entry on oeis.org

7, 43, 89, 139, 193, 251, 311, 373, 433, 491, 569, 619, 683, 757, 827, 887, 971, 1033, 1097, 1181, 1249, 1307, 1423, 1481, 1549, 1609, 1693, 1759, 1861, 1931, 2003, 2083, 2143, 2243, 2311, 2383, 2459, 2551, 2657, 2707, 2777, 2851, 2939

Offset: 1

Jeff Burch

nonn

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A000040, A017317, A031911, A031912, A031913.

[ NthPrime(10*n-6): n in [1..1000] ]; // Vincenzo Librandi, Apr 08 2011

Prime[10Range[50]-6] (* Harvey P. Dale, May 08 2011 *)

[nth_prime(10*n-6) for n in range(1,61)] # G. C. Greubel, Feb 18 2024

a(n) = A000040(A017317(n-1)). - G. C. Greubel, Feb 18 2024

A273375 Squares ending in digit 4.

Original entry on oeis.org

4, 64, 144, 324, 484, 784, 1024, 1444, 1764, 2304, 2704, 3364, 3844, 4624, 5184, 6084, 6724, 7744, 8464, 9604, 10404, 11664, 12544, 13924, 14884, 16384, 17424, 19044, 20164, 21904, 23104, 24964, 26244, 28224, 29584, 31684, 33124, 35344, 36864, 39204, 40804, 43264

Offset: 1

Vincenzo Librandi, 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, A047209.
Cf. A017317 (numbers ending in 4), A017319 (cubes ending in 4).
Cf. similar sequences listed in A273373.

/* By definition: */ [n^2: n in [0..200] | Modexp(n, 2, 10) eq 4];

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

Table[(10 n + (-1)^n - 5)^2/4, {n, 1, 50}] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {4, 64, 144, 324, 484}, 50]
Select[Range[200]^2,Mod[#,10]==4&] (* or *) LinearRecurrence[{1,1,-1},{2,8,12},40]^2(* Harvey P. Dale, Aug 06 2017 *)

G.f.: 4*x*(1 + 15*x + 18*x^2 + 15*x^3 + x^4) /((1+x)^2*(1-x)^3).
a(n) = 4*A047209(n)^2 = (10*n + (-1)^n - 5)^2/4.
Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5-sqrt(5))). - Amiram Eldar, Feb 16 2023
E.g.f.: (4 - 5*x + 25*x^2)*cosh(x) + (9 + 5*x + 25*x^2)*sinh(x) - 4. - Stefano Spezia, Feb 21 2025

Edited by Bruno Berselli, May 24 2016

A348488 Positive numbers whose square starts and ends with exactly one 4.

Original entry on oeis.org

2, 22, 68, 202, 208, 218, 222, 642, 648, 652, 658, 672, 678, 682, 692, 698, 702, 2002, 2008, 2018, 2022, 2028, 2032, 2042, 2048, 2052, 2058, 2068, 2072, 2078, 2082, 2092, 2122, 2128, 2132, 2142, 2148, 2152, 2158, 2168, 2172, 2178, 2182, 2192, 2198, 2202, 2208, 2218, 2222, 2228

Offset: 1

Bernard Schott, Oct 24 2021

When a square ends with 4 (A273375), this square may end with precisely one 4, two 4's or three 4's (A328886).
This sequence is infinite as each 2*(10^m + 1), m >= 1 or 2*(10^m + 4), m >= 2 is a term.
Numbers 2, 22, 222, ..., 2*(10^k - 1) / 9, (k >= 1), as well as numbers 2228, 22228, ..., 2*(10^k + 52) / 9, (k >= 4) are terms and have no digits 0. - Marius A. Burtea, Oct 24 2021

			22 is a term since 22^2 = 484.
638 is not a term since 638^2 = 407044.
668 is not a term since 668^2 = 446224.

Cf. A045858, A273375 (squares ending with 4), A017317, A328886 (squares ending with three 4).
Cf. A002276 \ {0} (a subsequence).
Subsequence of A305719.
Similar to: A348487 (k=1), this sequence (k=4), A348489 (k=5), A348490 (k=6).

[2] cat [n:n in [4..2300]|Intseq(n*n)[1] eq 4 and Intseq(n*n)[#Intseq(n*n)] eq 4 and Intseq(n*n)[-1+#Intseq(n*n)] ne 4 and Intseq(n*n)[2] ne 4]; // Marius A. Burtea, Oct 24 2021

Join[{2}, Select[Range[10, 2000], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 4 && d[[-2]] != 4 && d[[2]] != 4 &]] (* Amiram Eldar, Oct 24 2021 *)

isok(k) = my(d=digits(sqr(k))); (d[1]==4) && (d[#d]==4) && if (#d>2, (d[2]!=4) && (d[#d-1]!=4), 1); \\ Michel Marcus, Oct 24 2021

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [2, 8]))
  return [k for k in r if ok(k)]
print(aupto(2228)) # Michael S. Branicky, Oct 24 2021

A155156 Triangle T(n, k) = 4nk + 2n + 2k, read by rows.

Original entry on oeis.org

8, 14, 24, 20, 34, 48, 26, 44, 62, 80, 32, 54, 76, 98, 120, 38, 64, 90, 116, 142, 168, 44, 74, 104, 134, 164, 194, 224, 50, 84, 118, 152, 186, 220, 254, 288, 56, 94, 132, 170, 208, 246, 284, 322, 360, 62, 104, 146, 188, 230, 272, 314, 356, 398, 440, 68, 114, 160, 206, 252, 298, 344, 390, 436, 482, 528

Offset: 1

Vincenzo Librandi, Jan 21 2009

First column: A016933, second column: A017317, third column: A063151, fourth column: 2*A017209. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   8;
  14,  24;
  20,  34,  48;
  26,  44,  62,  80;
  32,  54,  76,  98, 120;
  38,  64,  90, 116, 142, 168;
  44,  74, 104, 134, 164, 194, 224;
  50,  84, 118, 152, 186, 220, 254, 288;
  56,  94, 132, 170, 208, 246, 284, 322, 360;
  62, 104, 146, 188, 230, 272, 314, 356, 398, 440;

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

Cf. A016933, A017317, A063151, A017209. A083487, A162254, A163832 (row sums).

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

seq(seq( 2*(2*n*k +n+k), k=1..n), n=1..15); # G. C. Greubel, Mar 20 2021

T[n_,k_]:=4*n*k +2*n +2*k; Table[T[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*(2*n*k +n+k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 20 2021

T(n, k) = 2*A083487(n, k). - R. J. Mathar, Jan 05 2011

Sum_{k=0..n} T(n,k) = n*(2*n^2 + 5*n + 1) = 2*A162254(n) = A163832(n). - G. C. Greubel, Mar 20 2021

Edited by Robert Hochberg, Jun 21 2010

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

A304158 a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).

Original entry on oeis.org

24, 84, 144, 204, 264, 324, 384, 444, 504, 564, 624, 684, 744, 804, 864, 924, 984, 1044, 1104, 1164, 1224, 1284, 1344, 1404, 1464, 1524, 1584, 1644, 1704, 1764, 1824, 1884, 1944, 2004, 2064, 2124, 2184, 2244, 2304, 2364, 2424, 2484, 2544, 2604, 2664, 2724, 2784, 2844, 2904, 2964

Offset: 1

Emeric Deutsch, May 08 2018

The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.

The M-polynomial of the linear phenylene G[n] is M(G[n];x,y) = 6*x^2*y^2 + 4*(n - 1)*x^2*y^3 + 4(n - 1)*x^3*y^3.

			a(1) = 24; indeed, G[1] is a hexagon; we have 6 edges, each with end vertices of degree 2; then the second Zagreb index is 6*2*2 =24.

Colin Barker, Table of n, a(n) for n = 1..1000
O. Bodroza-Pantic, I. Gutman, and S. J. Cyvin, Fibonacci numbers and algebraic structure count of some non-benzenoid conjugated polymers, The Fibonacci Quarterly, 35, 1, 1997, 75-83.
M. R. Darafsheh, Computation of topological indices of some graphs, Acta Appl. Math., 110, 2010, 1225-1235.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
P. Gayathri and U. Priyanka, Degree based topological indices of linear phenylene, Internat. J. of Innovative Research in Science, Engineering and Technology, 6, 8, 2017, 16986-16997.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016873, A304157.

Subsequence of A121024.

[60*n-36 for n in 1:50] |> println # Bruno Berselli, May 09 2018

Maple
```
seq(60*n - 36, n = 1 .. 40);
```

a(n) = 60*n-36; \\ Altug Alkan, May 09 2018

Vec(12*x*(2 + 3*x)/(1 - x)^2 + O(x^40)) \\ Colin Barker, May 23 2018

a(n) = 60*n - 36.
a(n) = 12 * A016873(n-1). - Alois P. Heinz, May 09 2018
From Bruno Berselli, May 09 2018: (Start)
O.g.f.: 12*x*(2 + 3*x)/(1 - x)^2.
E.g.f.: 12*(3 - 3*exp(x) + 5*x*exp(x)).
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008594(5*n-3) = A017317(6*n-4) = A072710(4*n-2) = A139245(3*n-1). (End)

A348832 Positive numbers whose square starts and ends with exactly 444.

Original entry on oeis.org

666462, 666538, 666962, 667038, 2107462, 2107538, 2107962, 2108038, 2108462, 2108538, 2108962, 2109038, 2109462, 6663462, 6663538, 6663962, 6664038, 6664462, 6664538, 6664962, 6665038, 6665462, 6665538, 6665962, 6666038, 6667462, 6667538, 6667962, 6668038, 6668462, 6668538, 6668962

Offset: 1

Bernard Schott, Nov 09 2021

The 1st problem of British Mathematical Olympiad (BMO) in 1995 (see link) asked to find all positive integers whose squares end in three 4’s (A039685); this sequence is the subsequence of these integers whose squares also start in precisely three 4's (no four or more 4's). Two such infinite subsequences are proposed below.
When a square starts and ends with digits ddd, then ddd is necessarily 444.
The first 3 digits of terms are either 210, 666 or 667, while the last 3 digits are either 038, 462, 538 or 962 (see examples).
From Marius A. Burtea, Nov 09 2021 : (Start)
The sequence is infinite because the numbers 667038, 6670038, 66700038, 667000038, ..., 667*10^k + 38, k >= 3, are terms because are square 444939693444, 44489406921444, 4448895069201444, 444889050692001444, 44488900506920001444, ...
Also, 6663462, 66633462, 666333462, 6663333462, ..., (1999*10^k + 386) / 3, k >= 4, are terms and have no digits 0, because their squares are 44401725825444, 4440018258105444, 444000282580905444, 44400012825808905444,
4440001128258088905444, ... (End)

			666462 is a term since 666462^2 = 444171597444.
21038 is not a term since 21038^2 = 442597444.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 55 and 95-96 (1995)

British Mathematical Olympiad 1975, Problem 1.

Cf. A017317, A328886.
Subsequence of A039685, A045858, A273375, A305719, A346892.
Similar to: A348488 (d=4), A348831 (dd=44), this sequence (ddd=444).

fd:=func; fs:=func; [n:n in [1..6700000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 09 2021

Select[Range[100, 7*10^6], (d = IntegerDigits[#^2])[[1 ;; 3]] == d[[-3 ;; -1]] == {4, 4, 4} && d[[-4]] != 4 && d[[4]] != 4 &] (* Amiram Eldar, Nov 09 2021 *)

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-3
def aupto(N):
  ends = [38, 462, 538, 962]
  r = takewhile(lambda x: x<=N, (1000*i+d for i in count(0) for d in ends))
  return [k for k in r if ok(k)]
print(aupto(6668962)) # Michael S. Branicky, Nov 09 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.

A017323 a(n) = (10*n + 4)^7.

Original entry on oeis.org

16384, 105413504, 4586471424, 52523350144, 319277809664, 1338925209984, 4398046511104, 12151280273024, 29509034655744, 64847759419264, 131593177923584, 250226879128704, 450766669594624, 775771085481344

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 (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A001015, A017317.

[(10*n+4)^7: n in [0..20] ]; // Vincenzo Librandi, Aug 01 2011

A017323:=n->(10*n+4)^7: seq(A017323(n), n=0..30); # Wesley Ivan Hurt, Jan 28 2017

(10*Range[0,20]+4)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{16384,105413504,4586471424,52523350144,319277809664,1338925209984,4398046511104,12151280273024},20] (* Harvey P. Dale, Aug 26 2015 *)

a(n)=(10*n+4)^7 \\ Charles R Greathouse IV, Jan 29 2017

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8); a(0)=16384, a(1)=105413504, a(2)=4586471424, a(3)=52523350144, a(4)=319277809664, a(5)=1338925209984, a(6)=4398046511104, a(7)=12151280273024. - Harvey P. Dale, Aug 26 2015

cp's OEIS Frontend

A166727 Positive integers with English names ending in "r".

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A031914 a(n) = prime(10*n - 6).

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A273375 Squares ending in digit 4.

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A348488 Positive numbers whose square starts and ends with exactly one 4.

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A155156 Triangle T(n, k) = 4*n*k + 2*n + 2*k, read by rows.

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A262389 Numbers whose last digit is composite.

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A304158 a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).

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A155156 Triangle T(n, k) = 4nk + 2n + 2k, read by rows.