A057934 -id:A057934 - cp's OEIS frontend

A115439 Numbers m such that the square of m is the concatenation of two numbers k and k+5.

4, 7, 45, 56, 38163, 61838, 83618, 346980, 653021, 950051, 8647555, 9534265, 8167822283, 9007920992, 9209900792, 9950000501, 4737445289221, 4990568257187, 5009431742814, 5262554710780, 8373808925585, 8626931893551, 34323166122692, 34532758615690, 49625657225895, 49835249718893

Offset: 1

Giovanni Resta, Jan 24 2006

All numbers of the form f(n)=9(n).5.0(2n).5.0(n-1).1 where n>0 are in the sequence because if k(n)=9(n).0(n).25.0(n-1).9(n).6 then f(n)^2=k(n).(k(n)+5). For example f(2)=9950000501; k(2)=9900250996 and f(2)^2=9950000501^2=9900250996.9900251001 =k(2).(k(2)+5). - Farideh Firoozbakht, Nov 26 2006

m^2 = (k)|(k+5) = (k)|(k) + 5 = (10^q + 1)*k + 5 where | denotes concatenation and q is the number of digits of k gives a nonlinear equation that can be solved using the solver below. - David A. Corneth, Jan 02 2021

			38163^2 = 14564_14569.

David A. Corneth, Table of n, a(n) for n = 1..3733
Dario A. Alpern, Generic two integer variable equation solver

Cf. A030467, A057934, A106497, A115428, A115427, A115438, A115440, A115441, A115442, A115443, A115444, A115445, A115446, A115447.

More terms from David A. Corneth, Jan 02 2021

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

A366713 Number of prime factors of 12^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 4, 2, 4, 5, 4, 7, 2, 3, 5, 4, 6, 9, 6, 4, 6, 6, 5, 10, 3, 3, 6, 3, 2, 9, 4, 5, 7, 4, 5, 11, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 10, 9, 5, 5, 10, 7, 9, 11, 8, 6, 10, 5, 6, 15, 5, 9, 11, 4, 5, 12, 10, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

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

Cf. A178248, A001222, A054992, A057934, A057935, A057936, A057937, A057938, A057939, A057940, A057941, A366712, A366714, A366715, A366716, A366719, A366720, A366708, A366687.

Mathematica
```
PrimeOmega[12^Range[70]+1]
```
PARI
```
a(n)=bigomega(12^n+1)
```

a(n) = bigomega(12^n+1) = A001222(A178248(n)).

A366687 Number of prime factors of 11^n + 1 (counted with multiplicity).

Original entry on oeis.org

1, 3, 2, 5, 2, 4, 4, 4, 3, 7, 3, 7, 4, 6, 5, 8, 3, 6, 5, 7, 4, 7, 4, 7, 7, 6, 3, 10, 6, 6, 6, 7, 4, 13, 6, 11, 7, 5, 4, 11, 5, 6, 9, 5, 6, 13, 6, 7, 5, 8, 6, 11, 3, 7, 9, 13, 7, 12, 6, 7, 8, 6, 4, 13, 3, 10, 8, 9, 7, 14, 8, 6, 10, 8, 8, 13, 6, 12, 12, 7, 10

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..325

Cf. A034524, A001222, A057934, A062308, A366686, A366688, A366689, A366690.

Cf. A054992, A057941, A057940, A057939, A057938, A057937, A057936, A057935, A057934, A366713.

Mathematica
```
PrimeOmega[11^Range[70]+1]
```
PARI
```
a(n)=bigomega(11^n+1)
```

a(n) = bigomega(11^n+1) = A001222(A034524(n)).

A001271 Irregular table read by rows: row n lists prime factors of 10^n +1, with multiplicity.

Original entry on oeis.org

2, 11, 101, 7, 11, 13, 73, 137, 11, 9091, 101, 9901, 11, 909091, 17, 5882353, 7, 11, 13, 19, 52579, 101, 3541, 27961, 11, 11, 23, 4093, 8779, 73, 137, 99990001, 11, 859, 1058313049, 29, 101, 281, 121499449, 7, 11, 13, 211, 241, 2161, 9091, 353, 449, 641, 1409, 69857

Offset: 0

N. J. A. Sloane, revised Jul 13 2009

Except for 2, all these primes are in A028416, and all the primes in A028416 appear here. - Davide Rotondo, Aug 12 2023

			Table begins:
    2;
   11;
  101,
    7,      11, 13;
   73,     137;
   11,    9091;
  101,    9901;
   11,  909091;
   17, 5882353;
  ...

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Ray Chandler, Rows n=0..310, flattened (from Kamada link)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Makoto Kamada, Factorizations of 100...001.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A028416, A057934, A062397.

Term ordering corrected by Sean A. Irvine, Apr 11 2012

Minor edits to description by Ray Chandler, May 02 2017

A309358 Numbers k such that 10^k + 1 is a semiprime.

Original entry on oeis.org

4, 5, 6, 7, 8, 19, 31, 53, 67, 293, 586, 641, 922, 2137, 3011

Offset: 1

Hugo Pfoertner, Jul 29 2019

a(16) > 12000.
10^k + 1 is composite unless k is a power of 2, and it can be conjectured that it is composite for all k > 2, cf. A038371 and A185121. - M. F. Hasler, Jul 30 2019
Suppose k is odd. Then k is a term if and only if (10^k+1)/11 is prime. - Chai Wah Wu, Jul 31 2019

			a(1) = 4 because 10^4 + 1 = 10001 = 73*137.

Cf. A003021, A038371, A046413, A057934, A062397, A092559.

Odd terms in sequence: A001562.

IsSemiprime:=func; [n: n in [2..200] | IsSemiprime(s) where s is 10^n+1]; // Vincenzo Librandi, Jul 31 2019

Mathematica

Select[Range[200], Plus@@Last/@FactorInteger[10^# + 1] == 2 &] (* Vincenzo Librandi, Jul 31 2019 *)

cp's OEIS Frontend

A115439 Numbers m such that the square of m is the concatenation of two numbers k and k+5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A038371 Smallest prime factor of 10^n + 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366713 Number of prime factors of 12^n + 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366687 Number of prime factors of 11^n + 1 (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A001271 Irregular table read by rows: row n lists prime factors of 10^n +1, with multiplicity.

Views

Author