Russell Easterly - cp's OEIS frontend

A173406 This sequence starts with any odd, composite number, like 15. There exists a power of two such that every 2^n + s_i is composite, where s_i is a term in the sequence less than 2^n. For example, 128+15=143, 512+15=527, 512+143=655, etc.

15, 143, 527, 655, 1039, 1167, 1551, 1679

Offset: 1

Russell Easterly, Feb 17 2010

We can easily create a k-CNF Boolean IsPrimek() function. Make a list of all composite k-bit binary numbers. Assign each bit to a variable and invert the bits to create a k-clause. Combining these clauses gives us a CNF IsPrimek() function. For example, the 4-clause for 15 (1111 base2) is (~d+~c+~b+~a). The 5-clause for 15 (01111) will be (e+~d+~c+~b+~a). We have to add a variable to the clause for each new power of two until we get to 2^7. 128+15 is composite so we can remove the variable for 128.

This sequence is the list of powers of 2 that can be removed from the CNF clause for 15. I still can't prove this is an infinite sequence. Assume this sequence is finite. Then there exists a finite width prime sieve for powers of two. For every large enough power of two we can find a prime by adding one of the numbers in this sequence (assuming the sequence is finite). The sequence can still be used as a prime sieve even if the sequence is infinite, assuming it grows slowly enough.

A173281 Let a(1) = 1. Given a(1), ..., a(2^t), find the least k such that a(1) + 2^k, a(2) + 2^k, ..., a(2^t) + 2^k are all composite and a(1) + 2^k > a(2^t). Then a(2^t+i) = a(i) + 2^k for all 1 <= i <= 2^t.

Original entry on oeis.org

1, 9, 2049, 2057, 4097, 4105, 6145, 6153, 524289, 524297, 526337, 526345, 528385, 528393, 530433, 530441, 16777217, 16777225, 16779265, 16779273, 16781313, 16781321, 16783361, 16783369, 17301505, 17301513, 17303553, 17303561, 17305601, 17305609, 17307649, 17307657

Offset: 1

Russell Easterly, Feb 14 2010

nonn

This sequence can be represented by a single clause in a CNF IsPrime() function.

step(v)=my(k=log(v[#v])\log(2));while(1, for(i=1,#v, k++; if(ispseudoprime(2^k+v[i]),next(2))); return(concat(v, vector(#v, i, 2^k+v[i])))) \\ Charles R Greathouse IV, Oct 25 2012

a(9)-a(32) from Charles R Greathouse IV, Oct 25 2012

A174088 Number of pairs (i,j) such that i*j == 0 (mod k), 0 <= i <= j < k.

Original entry on oeis.org

1, 2, 3, 5, 5, 8, 7, 11, 12, 14, 11, 21, 13, 20, 23, 26, 17, 33, 19, 37, 33, 32, 23, 51, 35, 38, 42, 53, 29, 68, 31, 58, 53, 50, 59, 87, 37, 56, 63, 91, 41, 98, 43, 85, 96, 68, 47, 122, 70, 100, 83, 101, 53, 123, 95, 131, 93, 86, 59, 181, 61, 92, 138, 132, 113, 158, 67, 133, 113

Offset: 1

Russell Easterly, Mar 06 2010

nonn

a(p) = p for p prime, since gcd(k,p) = 1 for 1 <= k < p, the product of k is also coprime to p, but multiples n*p for n >= 1 are plainly divisible by p. - Michael De Vlieger, Nov 22 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Michael De Vlieger)

Cf. A000188, A018804.

Table[If[PrimeQ@ b, b, Count[Flatten@ Array[# Range@ # &, b], ?(Mod[#, b] == 0 &)]], {b, 69}]  (* _Michael De Vlieger, Nov 22 2019 *)
f1[p_, e_] := (e*(p - 1)/p + 1)*p^e; f2[p_, e_] := p^Floor[e/2]; a[n_] := (Times @@ f1 @@@ (fct = FactorInteger[n]) + Times @@ f2 @@@ fct)/2; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n)={ my(ct=0); for(i=0,n-1,for(j=0,i, ct+=(Mod(i*j,n)==0) ) ); ct; } \\ Joerg Arndt, Aug 03 2013

a(n) = ( A018804(n) + A000188(n) ) / 2. - Max Alekseyev, Sep 05 2010

More terms from Max Alekseyev, Sep 05 2010

Better name from Joerg Arndt, Aug 03 2013

A090951 LCM of the first n numbers of the form p^q, where p and q are 1 or prime.

Original entry on oeis.org

1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360, 6126120, 116396280, 2677114440, 13385572200, 40156716600, 1164544781400, 36100888223400, 144403552893600, 5342931457063200, 219060189739591200, 9419588158802421600

Offset: 1

Russell Easterly, Feb 26 2004

Is the sum of the series 1/a(n) transcendental?

			a(11) = 6126120 = 1*2*3*2*5*7*2*3*11*13*17.

Cf. A051451.

Edited and extended by David Wasserman, Mar 13 2006

A091330 a(n) = ((p-1)!/p) - ((p-1)*(p-1)!/p!), where p is the n-th prime.

Original entry on oeis.org

0, 0, 4, 102, 329890, 36846276, 1230752346352, 336967037143578, 48869596859895986086, 10513391193507374500051862068, 8556543864909388988268015483870, 10053873697024357228864849950022572972972

Offset: 1

Russell Easterly, Mar 01 2004

Related to Wilson's Theorem. Let p be a prime number and write 1/p - (p-1)/p! = x/(p-1)!. Then x = (p-1)!/p - (p-1)*(p-1)!/p! = (p-1)!/p - (p-1)/p.
Also, a(n) = floor((p-1)!/p). [Bruno Berselli, May 31 2013]
If b(1)=1, and b(m) = ((m-1)^2 / m) *(b(m-1)+(m-3)/(m-1)) for m>1, then a(n) are the terms of b(m) for m prime. [Pedro Caceres, Dec 30 2018]

			Prime(4)=7 so a(4) = 6!/7 - 6*6!/7! = 102

Cf. A007619.

A091330[n_] := Block[{p = Prime[n]}, ((p - 1)!/p) - ((p - 1)*(p - 1)!/p!)] (* Robert G. Wilson v, Mar 02 2004 *)

More terms from Robert G. Wilson v and Ray Chandler, Mar 02 2004

A036014 a(n) is the smallest number such that the product a(1)a(2)...a(n) falls between a twin prime pair, starting with a(1)=2.

Original entry on oeis.org

2, 2, 3, 5, 3, 9, 6, 43, 18, 41, 82, 63, 47, 64, 108, 41, 192, 150, 91, 15, 5, 20, 214, 218, 46, 180, 121, 31, 80, 115, 39, 88, 2, 384, 1828, 1219, 360, 113, 2, 1111, 559, 687, 26, 1000, 368, 3130, 1198, 1731, 1752, 1240, 1237, 131, 814, 2349, 949, 64, 284, 361, 120, 3398, 47, 2068, 1001

Offset: 1

Russell Easterly

			4 between 3,5; 12 between 11,13; 60 between 59,61; etc.

Michael De Vlieger, Table of n, a(n) for n = 1..150

Cf. A014574 (average of twin primes pairs), A359948, A363274.

ms =: [: */ ,
pp =: [: *./ 1: p: <: , >:
ty =: [: pp ms
a1 =: ($: >:)`,@.ty
2 (a1~^: 30) 2x NB. Stephen Makdisi, May 06 2018

d[ 1 ]=2; d[ n_ ] := d[ n ]=Module[ {}, b=Product[ d[ i ], {i, 1, n-1} ]; i=2; While[ Not[ PrimeQ[ i b-1 ]&&PrimeQ[ i b+1 ] ], i++ ]; i ]; Table[ d[ i ], {i, 1, 30} ]
(* Second Program: *)
Nest[Append[#, Block[{k = 2}, While[! AllTrue[Times @@ #*k + {-1, 1}, PrimeQ], k++]; k]] &, {2}, 62] (* Michael De Vlieger, May 15 2018 *)

lista(nn) = {print1(a = 2, ", "); for (n=2, nn, k = 2; while (!(isprime(k*a-1) && isprime(k*a+1)), k++); print1(k, ", "); a *= k;);} \\ Michel Marcus, May 06 2018

More terms from Erich Friedman

Name edited by and more terms from Michel Marcus, May 06 2018

A036013 a(n) = smallest number > 1 such that a(1)a(2)...a(n) - 1 is prime (or 1).

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 2, 3, 6, 4, 5, 5, 5, 10, 4, 3, 5, 8, 22, 13, 14, 2, 5, 5, 2, 20, 9, 9, 24, 5, 26, 15, 14, 25, 25, 4, 9, 30, 9, 21, 12, 11, 10, 2, 40, 19, 8, 13, 11, 50, 3, 25, 25, 8, 5, 25, 46, 19, 47, 54, 9, 13, 14, 43, 4, 24, 28, 16, 33, 25, 152, 2, 11, 22, 6, 78, 87, 7, 10, 21

Offset: 1

Russell Easterly

Michael S. Branicky, Table of n, a(n) for n = 1..1600 (terms 1..300 from T. D. Noe)

Cf. A036012, A058000 (corresponding primes).

sng1[{t_,a_}]:=Module[{k=2},While[CompositeQ[t k-1],k++];{t*k,k}]; NestList[sng1,{2,2},80][[;;,2]] (* Harvey P. Dale, May 20 2023 *)

from sympy import isprime
from itertools import count
a,p = [2],2
for _ in range(100):
    q = next(filter(lambda x:isprime(x*p-1), count(2)))
    p = p*q
    a.append(q)
print(a) # Nicholas Stefan Georgescu, Mar 06 2023

More terms from Erich Friedman. More terms from Jud McCranie, Jan 26 2000.

A036012 a(n) = smallest number > 1 such that a(1)a(2)...a(n) + 1 is prime.

Original entry on oeis.org

2, 2, 3, 3, 2, 6, 3, 2, 4, 7, 7, 3, 8, 6, 2, 3, 6, 9, 6, 14, 19, 11, 4, 4, 19, 4, 13, 3, 10, 13, 15, 4, 11, 9, 2, 5, 26, 19, 52, 21, 20, 63, 4, 19, 17, 6, 29, 19, 3, 5, 51, 11, 14, 15, 7, 12, 44, 34, 7, 21, 32, 3, 22, 10, 19, 19, 7, 20, 4, 22, 4, 17, 35, 47, 40, 14, 5, 14, 36, 39, 16

Offset: 1

Russell Easterly

Except for the first term, same as A084401. - David Wasserman, Dec 22 2004

Michael S. Branicky, Table of n, a(n) for n = 1..1000 (terms 1..300 from T. D. Noe)

Equals A084716(n+1)/A084716(n).

Cf. A036013, A084716, A084717, A084718.

n := 1: while true do j := 2: while not isprime(j*n+1) do j := j+1: od: print(j): n := n*j: od:

a[1] = 2; a[n_] := a[n] = Catch[For[an = 2, True, an++, If[PrimeQ[Product[a[k], {k, 1, n - 1}]*an + 1], Throw[an]]]]; Table[a[n], {n, 1, 81}] (* Jean-François Alcover, Nov 27 2012 *)
nxt[{t_,n_}]:=Module[{k=2},While[!PrimeQ[t*k+1],k++];{t*k,k}]; NestList[ nxt,{2,2},80][[All,2]] (* Harvey P. Dale, Oct 03 2020 *)

from gmpy2 import is_prime
from itertools import count, islice
def agen(): # generator of terms
    p = 1
    while True:
        an = next(k for k in count(2) if (t:=p*k+1) == 1 or is_prime(t))
        p *= an
        yield an
print(list(islice(agen(), 81))) # Michael S. Branicky, Jan 20 2024

Conjecture: a(n) = O(n). - Thomas Ordowski, Aug 08 2017

More terms from Erich Friedman
More terms from Jud McCranie, Jan 26 2000
Description corrected by Len Smiley

cp's OEIS Frontend

User: Russell Easterly

A173406 This sequence starts with any odd, composite number, like 15. There exists a power of two such that every 2^n + s_i is composite, where s_i is a term in the sequence less than 2^n. For example, 128+15=143, 512+15=527, 512+143=655, etc.

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A173281 Let a(1) = 1. Given a(1), ..., a(2^t), find the least k such that a(1) + 2^k, a(2) + 2^k, ..., a(2^t) + 2^k are all composite and a(1) + 2^k > a(2^t). Then a(2^t+i) = a(i) + 2^k for all 1 <= i <= 2^t.

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A174088 Number of pairs (i,j) such that i*j == 0 (mod k), 0 <= i <= j < k.

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A090951 LCM of the first n numbers of the form p^q, where p and q are 1 or prime.

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A091330 a(n) = ((p-1)!/p) - ((p-1)*(p-1)!/p!), where p is the n-th prime.

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A036014 a(n) is the smallest number such that the product a(1)a(2)...a(n) falls between a twin prime pair, starting with a(1)=2.

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J

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A036013 a(n) = smallest number > 1 such that a(1)a(2)...a(n) - 1 is prime (or 1).

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A036012 a(n) = smallest number > 1 such that a(1)a(2)...a(n) + 1 is prime.

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Maple

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