Ryan Propper - cp's OEIS frontend

A133801 Number of distinct prime divisors of 3^n - 1.

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 5, 2, 3, 4, 5, 3, 6, 3, 5, 4, 5, 3, 7, 4, 3, 6, 6, 4, 8, 4, 6, 5, 6, 5, 9, 3, 5, 6, 7, 4, 8, 3, 8, 8, 4, 5, 12, 7, 7, 6, 6, 4, 11, 6, 9, 7, 7, 3, 12, 3, 6, 6, 7, 6, 10, 4, 9, 6, 8, 2, 12, 5, 6, 9, 8, 7, 12, 4, 11, 9, 6, 5, 14, 6, 4, 8, 12, 4, 16, 5, 7, 7, 8, 6, 15, 4, 10, 8

Offset: 1

Ryan Propper, Jan 06 2008

nonn

			a(4) = omega(3^4 - 1) = omega(80) = omega(2^4 * 5) = 2.

Table of n, a(n) for n = 1..690
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001221, A024023, A113913.

Table[PrimeNu[3^n - 1], {n, 1, 50}] (* G. C. Greubel, May 21 2017 *)

for(n = 1, 100, print1(omega(3^n - 1), ", "))

a(n) = omega(3^n - 1) = A001221(3^n - 1).

Terms to a(660) in b-file from Amiram Eldar, Feb 03 2020

a(661)-a(690) in b-file from Max Alekseyev, May 22 2022

A133927 Numbers k such that the sum of the first k Catalan numbers, C_1 + C_2 + ... + C_k, is divisible by k.

Original entry on oeis.org

1, 30, 494, 6331, 36851, 95450, 122614, 940745, 5126032, 7519204

Offset: 1

Ryan Propper, Jan 07 2008

The sum of the first 30 Catalan numbers, 1 + 2 + 5 + ... + 3814986502092304 = 5175497420902740, is divisible by 30, so 30 is a term.

No more terms < 5*10^7. - Lars Blomberg, Nov 07 2011

Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A014138.

Module[{nn=40000,lst},lst=Thread[{Range[nn],Accumulate[CatalanNumber[Range[nn]]]}]; Position[ lst,?(Mod[#[[2]],#[[1]]]==0&),1,Heads->False]]//Flatten (* The program generates the first 5 terms of the sequence. *) (* _Harvey P. Dale, Jun 18 2024 *)

A134656 Corresponding GCD values in A116894.

Original entry on oeis.org

2, 10453, 129341, 157519, 555097, 595411, 624363411431

Offset: 1

Ryan Propper, Feb 01 2008

Cf. A116894.

C
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// See Links section in A116893.
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for(n = 1, 10^9, my(x = gcd(n! + 1, n^n + 1)); if(x != 1 && x != 2*n + 1, print1(x, ", ")))

a(n) = gcd(A116894(n)! + 1, A116894(n)^A116894(n) + 1)

a(6)-a(7) from Nick Hobson, Feb 20 2024

A125284 Lower indices of duplicate terms in A125204, i.e., k such that A125204(k) = A125204(k + 1).

Original entry on oeis.org

3, 18, 35, 37, 148, 741, 752, 6814, 13976, 150095, 517213, 11874105

Offset: 1

Ryan Propper, Jan 25 2007

			18 is in the sequence because A125204(18) = A125204(19) = 266.

Cf. A125204.

l = {0, 1}; Do[x = l[[n]] + l[[Mod[l[[n]], n] + 1]]; If[x == l[[n]], Print[n - 1]]; AppendTo[l, x], {n, 2, 10^9}]

A125567 Numbers k such that Sum_{x=2..k} (x-1)3^(x-2) = ((2k-3)*3^(k-1)+1)/4 is prime.

Original entry on oeis.org

3, 6, 10, 135, 186, 542, 1162, 1726, 4179, 13910, 14514, 78915, 170026

Offset: 1

Ryan Propper, Jan 01 2007

a(14) > 2*10^5. - Robert Price, Jan 25 2015

			For k = 3, 6, 10 the corresponding primes are 7, 547, 83653.

Cf. A119529.

s = 0; Do[s += (x-1)*3^(x-2); If[PrimeQ[s], Print[x]], {x, 10^4}]

a(10) from Max Alekseyev, Dec 10 2011

a(11)-a(13) from Robert Price, Jan 25 2015

A125570 Numbers n such that Sum_(x=1..n) (x-1)*6^(x-1)/6 is prime.

Original entry on oeis.org

3, 7, 8, 19, 69, 77, 104, 107, 162, 163, 399, 4787, 4818

Offset: 1

Ryan Propper, Jan 01 2007

No more terms through 10^4.

Note that Sum(x=1,n,(x-1)*6^(x-1))/6 = (6^(n-1)*(5*n-6)+1)/25. Therefore this sequence consists of n such that (6^(n-1)*(5*n-6)+1)/25 is prime. - Max Alekseyev, Oct 18 2008

Cf. A119529.

s = 0; Do[s += (x-1)*6^(x-1)/6; If[PrimeQ[s], Print[x]], {x, 10^4}]

A125569 Numbers k such that Sum_{j=1..k-1} j*5^(j-1) is prime.

Original entry on oeis.org

3, 10, 11, 38, 735, 1083, 2063

Offset: 1

Ryan Propper, Jan 01 2007

No more terms through 10^4.

No more terms through 5*10^4. - Michael S. Branicky, Jun 30 2024

Cf. A119529.

s = 0; Do[s += (x-1)*5^(x-1)/5; If[PrimeQ[s], Print[x]], {x, 10^4}]
Select[Range[2100],PrimeQ[Sum[x*5^(x-1),{x,#-1}]]&] (* Harvey P. Dale, Feb 09 2023 *)

Name simplified by Jon E. Schoenfield, Sep 23 2018

A118710 Smallest positive integer k such that k^k + F(n) is prime, where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 2, 444

Offset: 1

Ryan Propper, May 20 2006

Next term is not known. Sequence continues: 1, 1, 1, 2, 444, ?, 2, 4, 3, 2, ?, ?, 6, ?, 1059, 2, 2, ?, ?, 14, 3, 66, 2, ?, 2, 46, 15, 8, 78, 273, 2, 2. All unknown terms are >= 2000. All known terms except a(15) = 1059 correspond to certified primes.

a(6) = A087037(8) > 30300.

Do[k = 1; While[ !PrimeQ[k^k + Fibonacci[n]], k++ ]; Print[k], {n, 32}]

a(n) = A087037(A000045(n)).

A119734 a(n) = least k such that 2^k + n is a perfect power, or -1 if no such k exists.

Original entry on oeis.org

0, 3, 1, 0, 2, 2, 1, 0, 0, 4, -1, 4, 2

Offset: 0

Ryan Propper, Jun 15 2006

Comments from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007: (Start) The first term for which k does not exist is a(10) since neither for k=0 nor 1 do we get a perfect power and for k>1 2^k+10=2*(2^(k-1)+5)=2*odd which can't be a perfect power since the exponent of 2 is not greater than 1.
For n=2*n' where n' is odd: if n+1 is a perfect power, a(n)=0; else if n+2 is a perfect power, a(n)=1. Otherwise a(n)=-1 because if we assume, for some k>1, that 2^k + n = 2*(2^(k-1) + n') is a perfect power m^e then, since 2^(k-1)+n' is odd, m must have its factor 2 raised to a multiple of e equal to 1 and so e=1, a contradiction. For example:
a(2*1) = 1 since n+2=2*1+2=4, a perfect power.
a(2*3) = 1 since n+2=2*3+2=8, a perfect power.
a(2*5) = -1 since n+1=11 and n+2=12 are not perfect powers.
a(2*7) = 1 since n+2=2*7+2=16, a perfect power.
a(2*9) = -1 since n+1=19 and n+2=20 are not perfect powers.
a(2*11) = -1 since n+1=23 and n+2=24 are not perfect powers.
a(2*13) = 0 since n+1=2*13+1=27, a perfect power.
a(2*15) = 1 since n+2=2*15+2=32, a perfect power.
a(2*17) = 1 since n+2=2*17+2=36, a perfect power. (End)

			The least k such that 2^k + 5 is a perfect power is 2, since 2^2 + 5 = 9 = 3^2, so a(5) = 2.

Cf. A001597.

a(10)-a(12) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

A120484 a(n) = Sum(Sum p_i, {Sum p_i=prime(n)}, p_i is prime).

Original entry on oeis.org

2, 3, 10, 21, 66, 117, 289, 437, 920, 2523, 3441, 8103, 13776, 17759, 28858, 57399, 109150, 134078, 243210, 355497, 427853, 733278, 1036504, 1711648, 3243583, 4414609, 5136713, 6919797, 8012154, 10692625, 28100655, 36616596, 54001290

Offset: 1

Ryan Propper, Jul 21 2006

nonn

			a(5) = (2+2+2+2+3) + (2+2+2+5) + (2+2+7) + (2+3+3+3) + (3+3+5) + 11 = 66.

Cf. A103275.

More terms from Ryan Propper, Sep 27 2006

cp's OEIS Frontend

User: Ryan Propper

A133801 Number of distinct prime divisors of 3^n - 1.

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A133927 Numbers k such that the sum of the first k Catalan numbers, C_1 + C_2 + ... + C_k, is divisible by k.

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A134656 Corresponding GCD values in A116894.

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C

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A125284 Lower indices of duplicate terms in A125204, i.e., k such that A125204(k) = A125204(k + 1).

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A125567 Numbers k such that Sum_{x=2..k} (x-1)*3^(x-2) = ((2*k-3)*3^(k-1)+1)/4 is prime.

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A125570 Numbers n such that Sum_(x=1..n) (x-1)*6^(x-1)/6 is prime.

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A125569 Numbers k such that Sum_{j=1..k-1} j*5^(j-1) is prime.

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A118710 Smallest positive integer k such that k^k + F(n) is prime, where F(n) is the n-th Fibonacci number.

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A119734 a(n) = least k such that 2^k + n is a perfect power, or -1 if no such k exists.

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A125567 Numbers k such that Sum_{x=2..k} (x-1)3^(x-2) = ((2k-3)*3^(k-1)+1)/4 is prime.