A017281 -id:A017281 - cp's OEIS frontend

A208259 Numbers starting and ending with digit 1.

1, 11, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 1001, 1011, 1021, 1031, 1041, 1051, 1061, 1071, 1081, 1091, 1101, 1111, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1221, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1321, 1331

Offset: 1

Jaroslav Krizek, Feb 24 2012

A000030(a(n)) = a(n) mod 10 = 1. - Reinhard Zumkeller, Jul 16 2014

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

Intersection of A017281 and A131835. Union of A062332 and A208260.
Supersequence of A208262 (numbers with all divisors starting and ending with digit 1).
Cf. A062332 (primes starting and ending with a digit 1), A208260 (nonprime numbers starting and ending with a digit 1).
Cf. A017281, A131835, A000030.

a208259 n = a208259_list !! (n-1)
a208259_list = 1 : map ((+ 1) . (* 10)) a131835_list
-- Reinhard Zumkeller, Jul 16 2014

Select[Range[2000], First[IntegerDigits[#]] == 1 && Last[IntegerDigits[#]] == 1 &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A272914 Sixth powers ending in digit 6.

Original entry on oeis.org

4096, 46656, 7529536, 16777216, 191102976, 308915776, 1544804416, 2176782336, 7256313856, 9474296896, 24794911296, 30840979456, 68719476736, 82653950016, 164206490176, 192699928576, 351298031616, 404567235136, 689869781056, 782757789696, 1265319018496, 1418519112256, 2194972623936

Offset: 1

Bruno Berselli, May 24 2016

Other sequences of k-th powers ending in digit k are: A017281 (k=1), A017355 (k=3), A017333 (k=5), A017311 (k=7), A017385 (k=9). It is missing k=4 because the fourth powers end with 0, 1, 5 or 6.
Union of A017322 and A017346.
a(h)^(1/6) is a member of A068408 for h = 2, 4, 8, 12, 16, 20, 36, 76, ...

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Cf. A001014, A016746, A017322, A017346

Similar sequences (see comment): A017281, A017311, A017333, A017355, A017385.

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

[(10*n-3*(-1)^n-5)^6/64: n in [1..30]];

Table[(10 n - 3 (-1)^n - 5)^6/64, {n, 1, 30}]

makelist((10*n-3*(-1)^n-5)^6/64, n, 1, 30);

vector(30, n, nn; (10*n-3*(-1)^n-5)^6/64)

[(10*n-3*(-1)^n-5)^6/64 for n in (1..30)]

O.g.f.: 64*x*(64 + 665*x + 116536*x^2 + 140505*x^3 + 2023280*x^4 + 983830*x^5 + 4720240*x^6 + 983830*x^7 + 2023280*x^8 + 140505*x^9 + 116536*x^10 + 665*x^11 + 64*x^12)/((1 + x)^6*(1 - x)^7).
E.g.f.: (-8192 + 45*(91 + 182*x - 5250*x^2 + 16000*x^3 - 9375*x^4 + 1250*x^5)*exp(-x) + (4097 + 287000*x^2 + 1262500*x^3 + 1253125*x^4 + 375000*x^5 + 31250*x^6)*exp(x))/2.
a(n) = (10*n - 3*(-1)^n - 5)^6/64 = 64*A047221(n)^6.

A346509 Number of positive integers with n digits that are the product of two integers greater than 1 and ending with 1.

Original entry on oeis.org

0, 0, 12, 200, 2660, 31850, 361985, 3982799, 42914655, 455727689, 4788989458, 49930700093, 517443017072, 5336861879564

Offset: 1

Stefano Spezia, Jul 21 2021

a(n) is the number of n-digit numbers in A346507.

Cf. A017281, A052268, A087630, A337855 (ending with 5), A337856 (ending with 6), A346507.

a(n) = {my(res = 0); forstep(i = 10^(n-1) + 1, 10^n, 10, f = factor(i); if(bigomega(f) == 1, next); d = divisors(f); for(j = 2, (#d~ + 1)>>1, if(d[j]%10 == 1 && d[#d + 1 - j]%10 == 1, res++; next(2) ) ) ); res } \\ David A. Corneth, Jul 22 2021

def A346507upto(lim): return sorted(set(a*b for a in range(11, lim//11+1, 10) for b in range(a, lim//a+1, 10)))
def a(n): return len(A346507upto(10**n)) - len(A346507upto(10**(n-1)))
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jul 22 2021

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

a(6)-a(9) from Michael S. Branicky, Jul 22 2021
a(10) from David A. Corneth, Jul 22 2021
a(11) from Michael S. Branicky, Jul 23 2021
a(11) corrected and extended with a(12) by Martin Ehrenstein, Aug 03 2021
a(13)-a(14) from Martin Ehrenstein, Aug 05 2021

A132359 Numbers divisible by the square of their last decimal digit.

Original entry on oeis.org

1, 11, 12, 21, 25, 31, 32, 36, 41, 51, 52, 61, 63, 64, 71, 72, 75, 81, 91, 92, 101, 111, 112, 121, 125, 128, 131, 132, 141, 144, 147, 151, 152, 153, 161, 171, 172, 175, 181, 191, 192, 201, 211, 212, 216, 221, 224, 225, 231, 232, 241, 243, 251, 252, 261, 271, 272

Offset: 1

Jonathan Vos Post, Nov 08 2007

Subsequences are A017281 and A053742 representing last digits 1 and 5. Generators for the subsequences representing last digits 2, 3, 4, 6, 7, 8 and 9 are, in that order, the terms 12+20i, 63+90i, 64+80i, 36+180i, 147+490i, 128+320i, 729+810i, where i=0,1,2,... - R. J. Mathar, Nov 13 2007

This is a 10-automatic sequence. - Charles R Greathouse IV, Dec 28 2011

			147 belongs to the sequence because 147/7^2 = 3.

T. D. Noe and Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 values from T. D. Noe)
Index entries for 10-automatic sequences.

Cf. A034709, A225722, A221651, A225296, A225297, A034709, A034837, A005349, A007602, A034838, A225297.

isA132359 := proc(n) local ldig ; ldig := n mod 10 ; if ldig <> 0 and n mod (ldig^2) = 0 then true ; else false ; fi ; end: for n from 1 to 400 do if isA132359(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Nov 13 2007
a:=proc(n) local nn: nn:=convert(n,base,10): if 0 < nn[1] and `mod`(n,nn[1]^2) =0 then n else end if end proc: seq(a(n),n=1..250); # Emeric Deutsch, Nov 15 2007

Select[Range[250], IntegerDigits[ # ][[ -1]] > 0 && Mod[ #, IntegerDigits[ # ][[ -1]]^2] == 0 &] (* Stefan Steinerberger, Nov 12 2007 *)
dsldQ[n_]:=Module[{lidnsq=Last[IntegerDigits[n]]^2},lidnsq!=0 && Divisible[n,lidnsq]]; Select[Range[300],dsldQ] (* Harvey P. Dale, May 03 2011 *)

is(n)=n%(n%10)^2==0 \\ Charles R Greathouse IV, Dec 28 2011

def ok(n): return n%10 > 0 and n%(n%10)**2 == 0
print([k for k in range(273) if ok(k)]) # Michael S. Branicky, Jul 03 2022

which(sapply(1:500,function(x) isint(x/(x%%10)^2))) # Christian N. K. Anderson, May 04 2013

Numbers k such that fp[k / (k mod 10)] = 0.

a(n) ~ 6350400*n/1241929 = 5.113...*n. - Charles R Greathouse IV, Dec 28 2011

Corrected and extended by Stefan Steinerberger, Emeric Deutsch and R. J. Mathar, Nov 12 2007

A190816 a(n) = 5n^2 - 4n + 1.

Original entry on oeis.org

1, 2, 13, 34, 65, 106, 157, 218, 289, 370, 461, 562, 673, 794, 925, 1066, 1217, 1378, 1549, 1730, 1921, 2122, 2333, 2554, 2785, 3026, 3277, 3538, 3809, 4090, 4381, 4682, 4993, 5314, 5645, 5986, 6337, 6698, 7069, 7450, 7841, 8242, 8653, 9074

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 20 2011

For n >= 2, hypotenuses of primitive Pythagorean triangles with m = 2*n-1, where the sides of the triangle are a = m^2 - n^2, b = 2*n*m, c = m^2 + n^2; this sequence is the c values, short sides (a) are A045944(n-1), and long sides (b) are A002939(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Short sides (a) A045944(n-1), long sides (b) A002939(n).
Cf. A017281 (first differences), A051624 (a(n)-1), A202141.
Sequences of the form m*n^2 - 4*n + 1: -A131098 (m=0), A028872 (m=1), A056220 (m=2), A045944 (m=3), A016754 (m=4), this sequence (m=5), A126587 (m=6), A339623 (m=7), A080856 (m=8).

[5*n^2 - 4*n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 19 2011

Table[5*n^2 - 4*n + 1, {n, 0, 100}]
LinearRecurrence[{3,-3,1},{1,2,13},100] (* or *) CoefficientList[ Series[ (-10 x^2+x-1)/(x-1)^3,{x,0,100}],x] (* Harvey P. Dale, May 24 2011 *)

a(n)=5*n^2-4*n+1 \\ Charles R Greathouse IV, Oct 16 2015

[5*n^2-4*n+1 for n in range(41)] # G. C. Greubel, Dec 03 2023

From Harvey P. Dale, May 24 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.
G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)
E.g.f.: (1 + x + 5*x^2)*exp(x). - G. C. Greubel, Dec 03 2023

Edited by Franklin T. Adams-Watters, May 20 2011

A084854 Triangular array, read by rows: T(n,k) = concatenated decimal representations of n and k, 1<=k<=n.

Original entry on oeis.org

11, 21, 22, 31, 32, 33, 41, 42, 43, 44, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 1010, 111, 112, 113, 114, 115, 116, 117, 118, 119, 1110, 1111

Offset: 1

Reinhard Zumkeller, Jun 09 2003

Indranil Ghosh, Rows 1..100 of triangle, flattened

Cf. A017281, A020338, A055642, A066686, A067574, A084855.

def T(n, k): return int(str(n) + str(k))
def auptorow(maxrow):
    return [T(n, k) for n in range(1, maxrow+1) for k in range(1, n+1)]
print(auptorow(11)) # Michael S. Branicky, Nov 21 2021

T(n, k) = n*10^A055642(k) + k.

T(n, 1) = A017281(n); T(n, n) = A020338(n).

A130154 Triangle read by rows: T(n, k) = 1 + 2(n-k)(k-1) (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 9, 7, 1, 1, 9, 13, 13, 9, 1, 1, 11, 17, 19, 17, 11, 1, 1, 13, 21, 25, 25, 21, 13, 1, 1, 15, 25, 31, 33, 31, 25, 15, 1, 1, 17, 29, 37, 41, 41, 37, 29, 17, 1, 1, 19, 33, 43, 49, 51, 49, 43, 33, 19, 1, 1, 21, 37, 49, 57, 61, 61, 57, 49, 37, 21, 1

Offset: 1

Emeric Deutsch, May 22 2007

Column k, except for the initial k-1 0's, is an arithmetic progression with first term 1 and common difference 2(k-1). Row sums yield A116731. First column of the inverse matrix is A129779.
Studied by Paul Curtz circa 1993.
From Rogério Serôdio, Dec 19 2017: (Start)
T(n, k) gives the number of distinct sums of 2(k-1) elements in {1,1,2,2,...,n-1,n-1}. For example, T(6, 2) = the number of distinct sums of 2 elements in {1,1,2,2,3,3,4,4,5,5}, and because each sum from the smallest 1 + 1 = 2 to the largest 5 + 5 = 10 appears, T(6, 2) = 10 - 1 = 9. [In general: 2*(Sum_{j=1..(k-1)} n-j) - (2*(Sum_{j=1..k-1} j) - 1) = 2*n*(k-1) - 4*(k-1)*k/2 + 1 = 2*(k-1)*(n-k) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]
T(n, k) is the number of lattice points with abscissa x = 2*(k-1) and even ordinate in the closed region bounded by the parabola y = x*(2*(n-1) - x) and the x axis. [That is, (1/2)*y(2*(k-1)) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]
Pascal's triangle (A007318, but with apex in the middle) is formed using the rule South = West + East; the rascal triangle A077028 uses the rule South = (West*East + 1)/North; the present triangle uses a similar rule: South = (West*East + 2)/North. See the formula section for this recurrence. (End)

			The triangle T(n, k) starts:
  n\k  1  2  3  4  5  6  7  8  9 10 ...
  1:   1
  2:   1  1
  3:   1  3  1
  4:   1  5  5  1
  5:   1  7  9  7  1
  6:   1  9 13 13  9  1
  7:   1 11 17 19 17 11  1
  8:   1 13 21 25 25 21 13  1
  9:   1 15 25 31 33 31 25 15  1
 10:   1 17 29 37 41 41 37 29 17  1
 ... reformatted. - _Wolfdieter Lang_, Dec 19 2017

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A116731, A129779. A007318, A077028.

Column sequences (no leading zeros): A000012, A016813, A016921, A017077, A017281, A017533, A131877, A158057, A161705, A215145.

Flat(List([1..12], n-> List([1..n], k-> 1 + 2*(n-k)*(k-1) ))); # G. C. Greubel, Nov 25 2019

[1 + 2*(n-k)*(k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 25 2019

T:=proc(n,k) if k<=n then 2*(n-k)*(k-1)+1 else 0 fi end: for n from 1 to 14 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Flatten[Table[1+2(n-k)(k-1),{n,0,20},{k,n}]] (* Harvey P. Dale, Jul 13 2013 *)

T(n, k) = 1 + 2*(n-k)*(k-1) \\ Iain Fox, Dec 19 2017

first(n) = my(res = vector(binomial(n+1,2)), i = 1); for(r=1, n, for(k=1, r, res[i] = 1 + 2*(r-k)*(k-1); i++)); res \\ Iain Fox, Dec 19 2017

[[1 + 2*(n-k)*(k-1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 25 2019

T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).
G.f.: G(t,z) = t*z*(3*t*z^2 - z - t*z + 1)/((1-t*z)*(1-z))^2.
Equals = 2 * A077028 - A000012 as infinite lower triangular matrices. - Gary W. Adamson, Oct 23 2007
T(n, 1) = 1 and  T(n, n) = 1 for n >= 1; T(n, k) = (T(n-1, k-1)*T(n-1, k) + 2)/T(n-2, k-1), for n > 2 and 1 < k < n. See a comment above. - Rogério Serôdio, Dec 19 2017
G.f. column k (with leading zeros): (x^k/(1-x)^2)*(1 + (2*k-3)*x), k >= 1. See the g.f. of the triangle G(t,z) above: (d/dt)^k G(t,x)/k!|{t=0}. - _Wolfdieter Lang, Dec 20 2017

Edited by Wolfdieter Lang, Dec 19 2017

A346507 Positive integers k that are the product of two integers greater than 1 and ending with 1.

Original entry on oeis.org

121, 231, 341, 441, 451, 561, 651, 671, 781, 861, 891, 961, 1001, 1071, 1111, 1221, 1271, 1281, 1331, 1441, 1491, 1551, 1581, 1661, 1681, 1701, 1771, 1881, 1891, 1911, 1991, 2091, 2101, 2121, 2201, 2211, 2321, 2331, 2431, 2501, 2511, 2541, 2601, 2651, 2751, 2761

Offset: 1

Stefano Spezia, Jul 21 2021

All the terms end with 1 (A017281).

			121 = 11*11, 231 = 11*21, 341 = 11*31, 441 = 21*21, 451 = 11*41, ...

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

Cf. A017281 (supersequence), A053742 (ending with 5), A324297 (ending with 6), A346508, A346509, A346510.

a={}; For[n=1, n<=300, n++, For[k=1, kMax[a], AppendTo[a, 10n+1]]]]; a

isok(k) = fordiv(k, d, if ((d>1) && (dMichel Marcus, Jul 28 2021

def aupto(lim): return sorted(set(a*b for a in range(11, lim//11+1, 10) for b in range(a, lim//a+1, 10)))
print(aupto(2761)) # Michael S. Branicky, Jul 22 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 21 2021

A354609 Carmichael numbers ending in 1.

Original entry on oeis.org

561, 2821, 6601, 8911, 15841, 29341, 41041, 75361, 101101, 115921, 162401, 172081, 188461, 252601, 314821, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 656601, 658801, 838201, 852841, 1024651, 1152271, 1193221, 1461241, 1615681, 1857241, 1909001, 2100901, 2113921, 2433601, 2455921, 2704801, 3057601

Offset: 1

Omar E. Pol, Jul 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Carmichael numbers

Intersection of A002997 and A017281.

Cf. A352970, A355305, A355307, A355309.

Select[10*Range[0, 3*10^5] + 1, CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jul 08 2022 *)

from itertools import islice
from sympy import nextprime, factorint
def A354609_gen(): # generator of terms
    p, q = 3, 5
    while True:
        for n in range(p+2+(-p-1)%10, q, 10):
            f = factorint(n)
            if max(f.values()) == 1 and not any((n-1) % (p-1) for p in f):
                yield n
        p, q = q, nextprime(q)
A354609_list = list(islice(A354609_gen(),30)) # Chai Wah Wu, Jul 24 2022

A384714 Nonpowers of 2 whose trailing digits form a power of 2.

Original entry on oeis.org

11, 12, 14, 18, 21, 22, 24, 28, 31, 34, 38, 41, 42, 44, 48, 51, 52, 54, 58, 61, 62, 68, 71, 72, 74, 78, 81, 82, 84, 88, 91, 92, 94, 98, 101, 102, 104, 108, 111, 112, 114, 116, 118, 121, 122, 124, 131, 132, 134, 138, 141, 142, 144, 148, 151, 152, 154, 158, 161, 162

Offset: 1

Stefano Spezia, Jun 23 2025

Nonpowers of 2 that are of the form h*10^j + 2^k with j > k*log10(2) and h > 0.

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

Cf. A007524, A057716, A209229.
Complement of A000079 in A385289.
A002275 \ {0, 1} and A017281 \ {1} are subsequences.

Select[Range[162], !IntegerQ[Log2[#]] && Sum[Boole[IntegerQ[Log2[FromDigits[Drop[IntegerDigits[#],i]]]]],{i,IntegerLength[#]}] > 0 &]
(* Second program: *)
nn = 162; s = 2^Range[0, Floor@ Log2[nn]]; Complement[Union@ Reap[Map[(w = IntegerDigits[#]; i = 1; While[Set[k, FromDigits@Join[IntegerDigits[i], w]] <= nn, Sow[k]; i++]) &, s] ][[-1, 1]], s] (* Michael De Vlieger, Jun 25 2025 *)

isp2(k) = k==1<A209229
isok(k) = if (!isp2(k), for (i=1, oo, my(z=k % 10^i); if (z==k, return(0), if (z && isp2(z), return(1))))); return(0); \\ Michel Marcus, Jun 24 2025

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A208259 Numbers starting and ending with digit 1.

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A272914 Sixth powers ending in digit 6.

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A346509 Number of positive integers with n digits that are the product of two integers greater than 1 and ending with 1.

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A132359 Numbers divisible by the square of their last decimal digit.

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A190816 a(n) = 5*n^2 - 4*n + 1.

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A084854 Triangular array, read by rows: T(n,k) = concatenated decimal representations of n and k, 1<=k<=n.

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A130154 Triangle read by rows: T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).

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A190816 a(n) = 5n^2 - 4n + 1.

A130154 Triangle read by rows: T(n, k) = 1 + 2(n-k)(k-1) (1 <= k <= n).