A003021 -id:A003021 - cp's OEIS frontend

A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

1, 1, 3, 2, 2, 2, 2, 2, 5, 3, 5, 3, 3, 4, 7, 5, 4, 3, 2, 4, 8, 4, 5, 3, 5, 3, 7, 4, 3, 7, 2, 4, 9, 4, 5, 6, 4, 3, 10, 4, 3, 7, 4, 4, 12, 4, 4, 9, 4, 7, 8, 4, 2, 6, 10, 5, 6, 5, 4, 6, 3, 3, 12, 3, 6, 8, 2, 4, 10, 11, 3, 5, 4, 7, 11, 6, 12, 7, 4, 9, 11, 3, 7, 8, 8, 3, 8, 4, 4, 11, 6, 4, 8, 4, 6, 8, 4, 5, 13

Offset: 1

Patrick De Geest, Oct 15 2000

nonn

2^(a(2n)-1)-1 predicts the number of pair-solutions of even length L for AB = A^2 + B^2. For instance, with length 18 we have 10^18 + 1 = 101*9901*999999000001 or 3 divisors F which when put into the Mersenne formula 2^(F-1)-1 yields 3 pairs (see reference 'Puzzle 104' for details).

Max Alekseyev, Table of n, a(n) for n = 1..331 (first 310 terms from Ray Chandler)
Makoto Kamada, Factorizations of 100...001.
Carlos Rivera, Puzzle 104, The Prime Puzzles & Problems Connection.
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n+1): this sequence (b=10), A057935 (b=9), A057936 (b=8), A057937 (b=7), A057938 (b=6), A057939 (b=5), A057940 (b=4), A057941 (b=3), A054992 (b=2).

Cf. A003021, A001271, A046053, A001562, A057951, A119704, A344897.

PrimeOmega[10^Range[100]+1] (* Harvey P. Dale, May 02 2021 *)

a(n)=bigomega(10^n+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A057951(2n) - A057951(n). - T. D. Noe, Jun 19 2003

A119704 a(n) = number of distinct prime factors of 10^n+1 = omega(10^n+1).

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 2, 2, 2, 5, 3, 4, 3, 3, 4, 7, 5, 4, 3, 2, 4, 7, 4, 5, 3, 5, 3, 7, 4, 3, 7, 2, 4, 8, 4, 5, 6, 4, 3, 9, 4, 3, 7, 4, 4, 12, 4, 4, 9, 4, 7, 8, 4, 2, 6, 9, 5, 6, 5, 4, 6, 3, 3, 11, 3, 6, 8, 2, 4, 10, 11, 3, 5, 4, 7, 11, 6, 11, 7, 4, 9, 11, 3, 7, 8, 8, 3, 8, 4, 4, 11, 6, 4, 8, 4, 6, 8, 4

Offset: 0

Lekraj Beedassy, Jun 09 2006

nonn

			a(1) = number of distinct prime factors of 11 = 1.
a(3) = number of distinct prime factors of 1001 = 3.
a(11) = 4 because 10^11+1 = 11*11*23*4093*8779 has 4 distinct factors.

Max Alekseyev, Table of n, a(n) for n = 0..331 [First 309 terms from Ray Chandler, from the Kamada link]
Makoto Kamada, Factorizations of 100...001.

Cf. A001221, A003021, A038371, A057934, A062397.

Table[Length[FactorInteger[10^n + 1]], {n, 0, 50}] (* Stefan Steinerberger, Jun 13 2006 *)
PrimeNu[10^Range[0,100]+1] (* The program will take some time to run *) (* Harvey P. Dale, Aug 27 2019 *)

a(n) = A001221(A062397(n)). - Ray Chandler, May 02 2017

More terms from Don Reble, Jun 13 2006

A274903 Largest prime factor of 4^n + 1.

Original entry on oeis.org

2, 5, 17, 13, 257, 41, 241, 113, 65537, 109, 61681, 2113, 673, 1613, 15790321, 1321, 6700417, 26317, 38737, 525313, 4278255361, 14449, 2931542417, 30269, 22253377, 268501, 308761441, 279073, 54410972897, 536903681, 4562284561, 384773, 67280421310721

Offset: 0

Vincenzo Librandi, Jul 11 2016

nonn

			4^3 + 1 = 65 = 5*13, so a(3) = 13.

Max Alekseyev, Table of n, a(n) for n = 0..583
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Cf. largest prime factor of k^n+1: A002587 (k=2), A074476 (k=3), this sequence (k=4), A074478 (k=5), A274904 (k=6), A227575 (k=7), A274905 (k=8), A002592 (k=9), A003021 (k=10), A062308 (k=11).

Cf. A006530, A052539.

[Maximum(PrimeDivisors(4^n+1)): n in [0..35]];

Table[FactorInteger[4^n + 1][[-1, 1]], {n, 0, 30}]

a(n)=my(f=factor(4^n+1)[,1]); f[#f] \\ Charles R Greathouse IV, Jul 12 2016

a(n) = A006530(A052539(n)). - Michel Marcus, Jul 11 2016
a(2n) = A002590(n). a(2n+1) = A229747(n). - R. J. Mathar, Feb 28 2018
a(n) = A002587(2*n). - Amiram Eldar, Feb 01 2020

Terms to a(100) in b-file from  Vincenzo Librandi, Jul 12 2016
a(101)-a(531) in b-file from Amiram Eldar, Feb 01 2020
a(532)-a(583) in b-file from Max Alekseyev, Apr 25 2022, Mar 15 2025

A344897 a(n) is the number of divisors of 10^n + 1.

Original entry on oeis.org

2, 2, 2, 8, 4, 4, 4, 4, 4, 32, 8, 24, 8, 8, 16, 128, 32, 16, 8, 4, 16, 192, 16, 32, 8, 32, 8, 128, 16, 8, 128, 4, 16, 384, 16, 32, 64, 16, 8, 768, 16, 8, 128, 16, 16, 4096, 16, 16, 512, 16, 128, 256, 16, 4, 64, 768, 32, 64, 32, 16, 64, 8, 8, 3072, 8, 64, 256, 4, 16, 1024, 2048, 8, 32, 16, 128, 2048, 64, 3072, 128, 16

Offset: 0

Seiichi Manyama, Jun 01 2021

nonn

a(n) is even because 10^n + 1 is not a square number.

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000005, A000533, A003021, A038371, A046798, A057934, A070528, A119704, A344859, A087021, A087022,

a[0] = 2; a[n_] := DivisorSigma[0, 10^n + 1]; Array[a, 60, 0] (* Amiram Eldar, Jun 01 2021 *)

PARI
```
a(n) = numdiv(10^n+1);
```

a(n) = A000005(A000533(n)).

A038371 Smallest prime factor of 10^n + 1.

Original entry on oeis.org

2, 11, 101, 7, 73, 11, 101, 11, 17, 7, 101, 11, 73, 11, 29, 7, 353, 11, 101, 11, 73, 7, 89, 11, 17, 11, 101, 7, 73, 11, 61, 11, 19841, 7, 101, 11, 73, 11, 101, 7, 17, 11, 29, 11, 73, 7, 101, 11, 97, 11, 101, 7, 73, 11, 101, 11, 17, 7, 101, 11, 73, 11, 101, 7, 1265011073

Offset: 0

Miklos SZABO (mike(AT)ludens.elte.hu)

nonn

a(n) >= 7 for all n >= 1 since 10^n + 1 is then not divisible by 2, 3 or 5.
Record values are a({0, 1, 2, 16, 32, 64, ...}). - M. F. Hasler, Apr 04 2008
The record values (2, 11, 101, 353, 19841, 1265011073, ...) are also found in A185121 and A102050 (smallest prime factor of 10^2^n+1). - M. F. Hasler, Jun 28 2024

			a(12) = 73 as 10^12+1 = 1000000000001 = 73*137*99990001.

Ehrhard Behrends, Five-Minute Mathematics, translated by David Kramer. American Mathematical Society (2008) p. 7

Max Alekseyev, Table of n, a(n) for n = 0..1023 (terms 0..500 from M. F. Hasler)
Makoto Kamada, Factorizations of 100...001.

Cf. A020639 (least prime factor), A062397 (10^n + 1), A003021 (largest prime factor of 10^n + 1), A057934 (number of prime factors of 10^n + 1, with multiplicity), A119704 (as before, without multiplicity), A185121 (smallest prime factor of 10^2^n+1), A102050 (as before, but 1 if 10^2^n+1 is prime).

[Min(PrimeFactors(10^n+1)):n in[0..70]]; // Vincenzo Librandi, Nov 08 2018

Table[FactorInteger[10^n + 1][[1, 1]], {n, 0, 49}] (* Alonso del Arte, Oct 21 2011 *)

A038371(n)=A020639(10^n+1) \\ Much more efficient than the naive {factor(10^n+1)[1,1]}. - M. F. Hasler, Apr 04 2008, edited Jun 29 2024

a(n) = A020639(A062397(n)).
For odd n, a(n) <= 11 since every (base 10) palindrome of even length is divisible by 11. - M. F. Hasler, Apr 04 2008 [See below for more precise formula.]
More generally, for k >= 0 and n == 2^k (mod 2^(k+1)), a(n) <= A185121(k) = (11, 101, 73, 17, 353, ...). This follows from  x^{2q+1} + 1 = (x+1) Sum_{m=0..2q} (-x)^m, with x=10^2^k. - M. F. Hasler, Jul 30 2019
From M. F. Hasler, Jun 28 2024: (Start)
a(2k+1) = 7 iff k == 1 (mod 3), else 11. [Making the 2008 formula more precise.]
a(4k+2) = 29 iff k == 3 (mod 7), else = 61 if k == 7 (mod 15), else = 89 if k == 5 (mod 11), else 101.
a(8k+4) = 73 for all k >= 0.
a(16k+8) = 17 for all k >= 0.
a(32k+16) = 97 iff k==1 (mod 3), else 353.
a(64k+32) = 193 iff k==1 (mod 3), else 1217 if k==9 (mod 19), else 2753 if k==21 (mod 43), else 3137 if k==24 (mod 49), else 3329 if k==6 (mod 13), else 4481 if k==17 (mod 35), else 4673 if k==36 (mod 73), else 5953 if k==15 (mod 31), else 6529 if k==8 (mod 17), else 13633 if k==35 (mod 71), else 15937 if k==41 (mod 83), else 19841. (End)

More terms from Reinhard Zumkeller, Mar 12 2002

A366668 Sum of the divisors of 10^n+1.

Original entry on oeis.org

3, 12, 102, 1344, 10212, 109104, 1010004, 10909104, 105882372, 1413350400, 10102223208, 114737461440, 1021097900424, 10921790676000, 104844305394000, 1355394166984704, 10073631600468000, 110177492439680640, 1010002989998020008, 10909090909090909104

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

			a(3)=1344 because 10^3+1 has divisors {1, 7, 11, 13, 77, 91, 143, 1001}.

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000203, A003021, A057934, A062397, A102146, A119704, A344897, A366666.

a:=n->numtheory[sigma](10^n+1):
seq(a(n), n=0..100);

DivisorSigma[1, 10^Range[0,19] + 1] (* Paul F. Marrero Romero, Nov 12 2023 *)

a(n) = sigma(10^n+1) = A000203(A062397(n)).

A366669 a(n) = phi(10^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 10, 100, 720, 9792, 90900, 990000, 9090900, 94117632, 681410880, 9897840000, 86925373920, 979102080000, 9080325951840, 95255567232000, 712493107200000, 9926748531589120, 90004044661864320, 989999010000000000, 9090909090909090900, 97910150554895155200

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000010, A003021, A057934, A062397, A119704, A295503, A344897, A366668.

Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366658, A366667, A366690, A366716.

EulerPhi[10^Range[0,20] + 1] (* Paul F. Marrero Romero, Nov 10 2023 *)

PARI
```
{a(n) = eulerphi(10^n+1)}
```

a(n) = A000010(A062397(n)). - Paul F. Marrero Romero, Nov 10 2023

A366720 Largest prime factor of 12^n+1.

Original entry on oeis.org

2, 13, 29, 19, 233, 19141, 20593, 13063, 260753, 1801, 85403261, 57154490053, 2227777, 222379, 13156924369, 35671, 1200913648289, 66900193189411, 122138321401, 905265296671, 67657441, 1885339, 68368660537, 49489630860836437, 592734049, 438472201

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..306

Cf. A178248, A006530, A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590, A366712, A366713, A366714, A366715, A366716, A366717, A366718, A366719, A324941.

Table[FactorInteger[12^n + 1][[-1, 1]], {n, 0, 20}]

a(n) = A006530(A178248(n)). - Paul F. Marrero Romero, Dec 07 2023

A324941 Largest prime factor of 17^n + 1.

Original entry on oeis.org

2, 3, 29, 13, 41761, 101, 83233, 22796593, 184417, 5653, 63541, 87415373, 72337, 2001793, 100688449, 238212511, 52548582913, 45957792327018709121, 382069, 20352763, 1186844128302568601, 88109799136087, 6901823633, 1109309383381084655697725873, 48661191868691111041

Offset: 0

Vincenzo Librandi, Apr 05 2019

nonn

Max Alekseyev, Table of n, a(n) for n = 0..211

Cf. A006530, A224384.

Cf. A002587, A074476, A274903, A074478, A274904, A227575, A274905, A002592, A003021, A062308, A002590.

[Maximum(PrimeDivisors(17^n + 1)): n in [0..40]];

Table[FactorInteger[17^n + 1] [[-1,1]], {n, 0, 30}]

a(n) = vecmax(factor(17^n+1)[, 1]); \\ Jinyuan Wang, Apr 05 2019

a(n) = A006530(A224384(n)).

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

cp's OEIS Frontend

A057934 Number of prime factors of 10^n + 1 (counted with multiplicity).

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A119704 a(n) = number of distinct prime factors of 10^n+1 = omega(10^n+1).

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A274903 Largest prime factor of 4^n + 1.

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A344897 a(n) is the number of divisors of 10^n + 1.

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A038371 Smallest prime factor of 10^n + 1.

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Magma

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A366668 Sum of the divisors of 10^n+1.

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A366669 a(n) = phi(10^n+1), where phi is Euler's totient function (A000010).

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