A062397 -id:A062397 - cp's OEIS frontend

A102050 a(n) = 1 if 10^(2^n)+1 is prime, otherwise smallest prime factor of 10^(2^n)+1.

1, 1, 73, 17, 353, 19841, 1265011073, 257, 10753, 1514497, 1856104284667693057, 106907803649, 458924033, 3635898263938497962802538435084289

Offset: 0

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004

The smallest known prime factors of 10^(2^15)+1 to 10^(2^18)+1 are 65537, 8257537, 175636481, 639631361. - Jeppe Stig Nielsen, Nov 04 2010
Above values for a(15)-a(18) are confirmed. a(19) = 70254593, a(20) = 167772161. - Chai Wah Wu, Oct 16 2019
a(14) <= 1702047085242613845984907230501142529. - Max Alekseyev, Feb 26 2023

			10^(2^4)+1 = 10000000000000001 = 353*449*641*1409*69857, hence a(4) = 353.

factordb.com, Status of 10^(2^n)+1.
Wilfrid Keller, Prime factors of generalized Fermat numbers Fm(10) and complete factoring status

Cf. A000533, A002275, A062397.

spf[n_]:=Module[{c=10^2^n+1},If[PrimeQ[c],1,FactorInteger[c][[1,1]]]]; Array[spf,15,0] (* Harvey P. Dale, Apr 09 2019 *)

for(k=0,8,fac=factor(10^(2^k)+1);print1(if(matsize(fac)[1]==1,1,fac[1,1]),","))

If 10^(2^n)+1 is composite, a(n) = A185121(n).

a(13) from the Keller link, added by Jeppe Stig Nielsen, Nov 04 2010

A110369 (Digit 1 repeated n times) + n.

Original entry on oeis.org

2, 13, 114, 1115, 11116, 111117, 1111118, 11111119, 111111120, 1111111121, 11111111122, 111111111123, 1111111111124, 11111111111125, 111111111111126, 1111111111111127, 11111111111111128, 111111111111111129, 1111111111111111130

Offset: 1

Amarnath Murthy, Jul 24 2005

Partial sums of A062397. - Klaus Purath, Mar 18 2021

			a(12) = 111111111111 + 12 = 111111111123.

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

Cf. A002275 (repunits).

Table[FromDigits[PadRight[{},n,1]]+n,{n,20}] (* Harvey P. Dale, May 15 2019 *)

[gaussian_binomial(n,1,10)+n for n in range(1,20)] # Zerinvary Lajos, May 29 2009

a(n) = A002275(n) + n. - Michel Marcus, Mar 04 2018

a(n) = (10^n - 1)/9 + n. - Klaus Purath, Mar 18 2021

More terms from Joshua Zucker, May 08 2006

An incorrect recurrence was deleted by N. J. A. Sloane, Sep 13 2009

A152577 a(n) = 10^(2*n - 1) + 1.

Original entry on oeis.org

11, 1001, 100001, 10000001, 1000000001, 100000000001, 10000000000001, 1000000000000001, 100000000000000001, 10000000000000000001, 1000000000000000000001, 100000000000000000000001

Offset: 1

Cino Hilliard, Dec 08 2008

These numbers are all divisible by 11. This follows from the identity a^n - b^n = (a+b)*(a^(n-1) - a^(n-2)*b + ... + b^(n-1)) for odd values of n. In this example a=10 and b=1 so a+b = 11. The sum of digits rule for divisibility by 11 also applies.

Bisection of A000533. Also, bisection of A062397. a(n) is also A084508(n+1) written in base 2. a(n) is also A087289(n-1) written in base 2. a(n) is also the concatenation of "1", 2(n-1) digits "0" and "1". - Omar E. Pol, Dec 13 2008

			From _Omar E. Pol_, Dec 14 2008: (Start)
n ....... a(n)
1 ....... 11
2 ...... 1001
3 ..... 100001
4 .... 10000001
5 ... 1000000001
(End)

Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A000533, A062397, A084508, A087289. - Omar E. Pol, Dec 13 2008

LinearRecurrence[{101,-100},{11,1001},20] (* Harvey P. Dale, Nov 05 2015 *)

g(n)=forstep(x=1,n,2,y=(10^x+1);print1(y","))

a(n) = 100*a(n-1) - 99 (with a(1)=11). - Vincenzo Librandi, Dec 14 2010
G.f.: -11*x*(-1+10*x) / ( (100*x-1)*(x-1) ). - R. J. Mathar, Sep 01 2011
a(n) = 11*A095372(n-1). - R. J. Mathar, Sep 01 2011
a(n) = 101*a(n-1)-100*a(n-2). - Wesley Ivan Hurt, Apr 24 2021
E.g.f.: (exp(100*x) + 10*exp(x) - 11)/10. - Stefano Spezia, Mar 13 2025

A168576 a(n) = (10^n+1)^4.

Original entry on oeis.org

16, 14641, 104060401, 1004006004001, 10004000600040001, 100004000060000400001, 1000004000006000004000001, 10000004000000600000040000001, 100000004000000060000000400000001

Offset: 0

Jason Earls, Nov 30 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (11111,-11222110,1122211000,-11111000000,10000000000).

Cf. A000583, A062397.

[(10^n+1)^4: n in [0..10]]; // Vincenzo Librandi, May 26 2016

Table[(10^n + 1)^4, {n, 0, 30}] (* Vincenzo Librandi, May 26 2016 *)
(10^Range[0,20]+1)^4 (* or *) LinearRecurrence[{11111,-11222110,1122211000,-11111000000,10000000000},{16,14641,104060401,1004006004001,10004000600040001},20] (* Harvey P. Dale, Sep 10 2018 *)

PARI
```
for(n=0,13,print1((10^n+1)^4","))
```

G.f. ( -16+163135*x-120938010*x^2+5861575000*x^3-14641000000*x^4 ) / ( (x-1)*(100*x-1)*(1000*x-1)*(10*x-1)*(10000*x-1) ). - R. J. Mathar, Jul 03 2012

a(n) = A000583(A062397(n)). - Michel Marcus, May 26 2016

A286315 Number of representations of 10^n as sum of 8 triangular numbers.

Original entry on oeis.org

8, 1332, 1030302, 1007141184, 1000302990372, 1000781337641904, 1000003970597090004, 1000751615026326041904, 1000203571630368710405892, 1004272191614371538730009600, 1000000970912716777250166728808, 1000834130646589459517111102258880

Offset: 0

Seiichi Manyama, May 06 2017

nonn

a(n) is nearly 10^(3*n) because a(n) is almost (10^n+1)^3.

			a(0) = Sum_{d|2, 2/d == 1 mod 2} d^3 = 2^3 = 8.
a(1) = Sum_{d|11, 11/d == 1 mod 2} d^3 = 11^3 + 1^3 = 1332.
a(2) = Sum_{d|101, 101/d == 1 mod 2} d^3 = 101^3 + 1^3 = 1030302.

Seiichi Manyama, Table of n, a(n) for n = 0..18

Cf. A007331, A062397 (10^n+1), A168575 ((10^n+1)^3), A286314.

a(n) = A007331(10^n + 1).

a(n) = Sum_{d|10^n+1, (10^n+1)/d == 1 mod 2} d^3.

A072848 Largest prime factor of 10^(6*n) + 1.

Original entry on oeis.org

9901, 99990001, 999999000001, 9999999900000001, 39526741, 3199044596370769, 4458192223320340849, 75118313082913, 59779577156334533866654838281, 100009999999899989999000000010001, 2361000305507449, 111994624258035614290513943330720125433979169

Offset: 1

Rick L. Shepherd, Jul 25 2002

nonn

According to the link, there are only 18 "unique primes" below 10^50. The first four terms above are each unique primes, of periods 12, 24, 36 and 48, respectively, according to Caldwell and the cross-referenced sequences. These are precisely the only unique primes (less than 10^50 at least) with this type of digit pattern: m 9's, m-1 0's and 1, in that order. (Also a(10) is a unique prime of period 120.)

			10^(6*4)+1 = 17 * 5882353 * 9999999900000001, so a(4) = 9999999900000001, the largest prime factor.

Max Alekseyev, Table of n, a(n) for n = 1..87 (terms n=1..51 from Ray Chandler, n=52..81 from Daniel Suteu)
C. K. Caldwell, Unique Primes
Makoto Kamada, Factorizations of 100...001.

Cf. A040017 (unique period primes), A051627 (associated periods).

Cf. A003021, A006530, A062397.

for(n=1,12,v=factor(10^(6*n)+1); print1(v[matsize(v)[1],1],","))

a(n) = A003021(6n) = A006530(A062397(6n)). - Ray Chandler, May 11 2017

A215804 Odd numbers k such that 10^k + 1 can be written in the form a^2 + 2*b^2.

Original entry on oeis.org

1, 5, 7, 13, 19, 25, 29, 31, 35, 43, 53, 61, 65, 67, 71, 95, 125, 145, 155, 175, 179, 215, 239, 263, 265, 269, 293, 305

Offset: 1

V. Raman, Aug 23 2012

nonn

These 10^k + 1 numbers have no prime factors of the form 5 or 7 (mod 8) to an odd power.

Samuel S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, for odd n's < 1200

Cf. A000533, A001562, A062397, A215805.

for(i=2, 100, a=factorint(10^i+1)~; has=0; for(j=1, #a, if(a[1, j]%8>4&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i" -\t"a[1, ])))

12 more terms from V. Raman, Aug 28 2012

A215805 Prime numbers p such that 10^p + 1 can be written in the form a^2 + 2*b^2.

Original entry on oeis.org

5, 7, 13, 19, 29, 31, 43, 53, 61, 67, 71, 179, 239, 263, 269, 293

Offset: 1

V. Raman, Aug 23 2012

nonn

These numbers have no prime factors of the form 5 or 7 (mod 8) to an odd power.

Samuel S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, for odd n's < 1200.

Cf. A000533, A001562, A062397, A215804.

forprime(i=2, 100, a=factorint(10^i+1)~; has=0; for(j=1, #a, if(a[1, j]%8>4&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i" -\t"a[1, ])))

5 more terms from V. Raman, Aug 29 2012

A234429 Numbers which are the digital sum of the square of some prime.

Original entry on oeis.org

4, 7, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178

Offset: 1

Antonio Graciá Llorente, Dec 26 2013

A123157 sorted and duplicates removed.

Cf. A067180, A067523, A123157, A056991.

Cf. A062397, A226803, A226802, A165492, A165459 A165493, A229058, A165502, A165503, A165504.

terms(nn) = {v = []; forprime (p = 1, nn, v = concat(v, sumdigits(p^2));); vecsort(v,,8);} \\ Michel Marcus, Jan 08 2014

From Robert G. Wilson v, Sep 28 2014: (Start)
Except for 3, all primes are congruent to +-1 (mod 3). Therefore, (3n +- 1)^2 = 9n^2 +- 6n + 1 which is congruent to 1 (mod 3).
2, 11, 101, ... (A062397);
5, 149, 1049, ... (A226803);
only 3;
19, 71, 179, 251, 449, 20249, 24499, 100549, ... (A226802);
7, 29, 47, 61, 79, 151, 349, 389, 461, 601, 1051, 1249, 1429, ... (A165492);
13, 23, 31, 41, 59, 103, 131, 139, 211, 229, 239, 347, 401, ... (A165459);
17, 37, 53, 73, 89, 107, 109, 127, 181, 199, 269, 271, 379, ... (A165493);
43, 97, 191, 227, 241, 317, 331, 353, 421, 439, 479, 569, 619, 641, ...;
67, 113, 157, 193, 257, 283, 311, 337, 373, 409, 419, 463, ... (A229058);
163, 197, 233, 307, 359, 397, 431, 467, 487, 523, 541, 577, 593, 631, ...;
83, 137, 173, 223, 263, 277, 281, 367, 443, 457, 547, 587, ... (A165502);
167, 293, 383, 563, 607, 617, 733, 787, 823, 859, 877, 941, 967, 977, ...;
433, 613, 683, 773, 827, 863, 1063, 1117, 1187, 1223, 1567, ... (A165503);
313, 947, 983, 1303, 1483, 1609, 1663, 1933, 1973, 1987, 2063, 2113, ...;
887, 1697, 1723, 1867, 1913, 2083, 2137, 2417, 2543, 2633, ... (A165504);
883, 937, 1367, 1637, 2213, 2447, 2683, 2791, 2917, 3313, 3583, 3833, ...;
1667, 2383, 2437, 2617, 2963, 4219, 4457, 5087, 5281, 6113, 6163, ...;
...  Also see A229058. (End)
Conjectures from Chai Wah Wu, Apr 16 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x*(2*x^4 - x^3 - x^2 - x + 4)/(x - 1)^2. (End)

a(36) from Michel Marcus, Jan 08 2014
a(37)-a(54) from Robert G. Wilson v, Sep 28 2014
More terms from Giovanni Resta, Aug 15 2019

A261173 Table read by antidiagonals: T(n,k) = smallest prime p containing only digits 0 and 1 with n 0's and k 1's, or 0 if no such p exists.

Original entry on oeis.org

11, 0, 101, 0, 0, 0, 0, 10111, 0, 0, 0, 101111, 0, 0, 0, 0, 0, 0, 1011001, 0, 0, 0, 11110111, 0, 10011101, 10010101, 0, 0, 0, 101111111, 101101111, 0, 100100111, 101001001, 0, 0, 0, 0, 1010111111, 1001110111, 0, 1000011011, 1000001011, 0, 0

Offset: 0

Felix Fröhlich, Aug 10 2015

T(n, k) = 0 if k is a term of A008585.
T(0, k) != 0 iff k is a term of A004023.
T(1, k) = A157709(k-2) for all k >= 4.
T(n, 2) != 0 iff A062397(n+1) is prime.
a(n) is in A168586 iff it is the smallest p in T with A007953(p) = k.

			Table T(n, k) starts
     k = 2        3        4        5
      -------------------------------------
n = 0 |  11       0        0        0
n = 1 |  101      0        10111    101111
n = 2 |  0        0        0        0
n = 3 |  0        0        1011001  10011101

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Cf. A020449, A036929.

a(n, k) = i=0; forprime(p=10^(n+k-1), (10^(n+k)-1)/9, if(vecmax(digits(p))==1 && sumdigits(p)==k, return(p); i++; break)); if(i==0, return(0))
table(row, col) = for(x=0, row, for(y=2, col, print1(a(x, y), " ")); print(""))
table(4, 5) \\ print 5 X 4 table

More terms from Alois P. Heinz, Aug 17 2015

cp's OEIS Frontend

A102050 a(n) = 1 if 10^(2^n)+1 is prime, otherwise smallest prime factor of 10^(2^n)+1.

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A110369 (Digit 1 repeated n times) + n.

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A152577 a(n) = 10^(2*n - 1) + 1.

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A168576 a(n) = (10^n+1)^4.

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A286315 Number of representations of 10^n as sum of 8 triangular numbers.

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A072848 Largest prime factor of 10^(6*n) + 1.

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A215804 Odd numbers k such that 10^k + 1 can be written in the form a^2 + 2*b^2.

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