A119704 -id:A119704 - cp's OEIS frontend

A104223 Retired in favor of A119704.

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, 5

Offset: 0

Parthasarathy Nambi, Apr 01 2005

dead

A119704 is a slightly newer entry which improves on this in several ways. - N. J. A. Sloane, Apr 26 2022

Cf. A119704.

a(n) = A001221(A000533(n+1)). - R. J. Mathar, Aug 24 2011

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

Original entry on oeis.org

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

A366712 Number of distinct prime divisors of 12^n + 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 3, 3, 3, 5, 3, 2, 4, 4, 4, 7, 2, 3, 5, 4, 6, 8, 6, 4, 6, 6, 5, 10, 3, 3, 5, 3, 2, 9, 4, 5, 7, 4, 5, 10, 4, 5, 6, 5, 4, 12, 3, 5, 5, 5, 8, 9, 5, 5, 10, 7, 9, 10, 7, 6, 10, 5, 6, 14, 5, 8, 11, 4, 5, 12, 9, 3, 10, 5, 8, 17, 5, 6, 9, 4, 6, 15, 3

Offset: 0

Sean A. Irvine, Oct 17 2023

nonn

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

Cf. A178248, A001221, A046799, A119704, A366713, A366714, A366707, A366686.

for(n = 0, 100, print1(omega(12^n + 1), ", "))

a(n) = omega(12^n+1) = A001221(A178248(n)).

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)).

A366686 Number of distinct prime divisors of 11^n + 1.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 16 2023

nonn

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

Cf. A034524, A001221, A366681, A102347, A366687, A366688, A366689, A366690.

Cf. A046799, A119704, A366712.

for(n = 0, 100, print1(omega(11^n + 1), ", "))

a(n) = omega(11^n+1) = A001221(A034524(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

A366605 Number of distinct prime divisors of 4^n + 1.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..583

Cf. A001221, A052539, A057940, A274903, A366604, A366606, A366607, A366608, A366609.

Cf. A046799, A366580, A366615, A366627, A366636, A366655, A366664, A119704, A366686, A366712.

PrimeNu[4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)

for(n = 0, 100, print1(omega(4^n + 1), ", "))

from sympy import primenu
def A366605(n): return primenu((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

a(n) = omega(4^n+1) = A001221(A052539(n)).

a(n) = A046799(2*n). - Max Alekseyev, Jan 08 2024

A366655 Number of distinct prime divisors of 8^n + 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 3, 3, 3, 5, 4, 4, 3, 6, 5, 3, 5, 6, 4, 4, 5, 6, 4, 5, 6, 9, 6, 5, 4, 10, 4, 3, 7, 9, 10, 6, 6, 8, 5, 6, 6, 10, 5, 7, 9, 8, 6, 7, 6, 12, 9, 5, 5, 10, 10, 8, 6, 8, 7, 8, 3, 9, 10, 4, 10, 12, 7, 8, 6, 14, 7, 8, 5, 10, 10, 8, 11, 16, 5, 7, 10

Offset: 0

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..502

Cf. A062395, A001221, A057936, A274905, A366651, A366632, A366656, A366657, A366658, A366671.

Cf. A046799, A366580, A366605, A366615, A366627, A366636, A366664, A119704, A366686, A366712.

for(n = 0, 100, print1(omega(8^n + 1), ", "))

a(n) = omega(8^n+1) = A001221(A062395(n)).

a(n) = A046799(3*n). - Max Alekseyev, Jan 09 2024

A366615 Number of distinct prime divisors of 5^n + 1.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..471

Cf. A001221, A034474, A057939, A074478, A366611, A366616, A366617, A366618.

Cf. A046799, A366580, A366605, A366627, A366636, A366655, A366664, A119704, A366686, A366712.

Table[PrimeNu[5^n+1],{n,0,90}] (* Harvey P. Dale, Apr 06 2025 *)

for(n = 0, 100, print1(omega(5^n + 1), ", "))

a(n) = omega(5^n+1) = A001221(A034474(n)).

A366627 Number of distinct prime divisors of 6^n + 1.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 0..430

Cf. A062394, A001221, A057938, A366620, A366628, A366629, A366630.

Cf. A046799, A366580, A366605, A366615, A366636, A366655, A366664, A119704, A366686, A366712.

PrimeNu[6^Range[0,84] + 1] (* Paul F. Marrero Romero, Nov 11 2023 *)

for(n = 0, 100, print1(omega(6^n + 1), ", "))

a(n) = omega(6^n+1) = A001221(A062394(n)).

cp's OEIS Frontend

A104223 Retired in favor of A119704.

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

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A366712 Number of distinct prime divisors of 12^n + 1.

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

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A366686 Number of distinct prime divisors of 11^n + 1.

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A038371 Smallest prime factor of 10^n + 1.

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A366605 Number of distinct prime divisors of 4^n + 1.

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A366655 Number of distinct prime divisors of 8^n + 1.

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