A151800 -id:A151800 - cp's OEIS frontend

A317830 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.

1, 1, 1, 7, 1, 3, 1, 9, 11, 7, 1, 3, 1, 3, 5, 171, 1, -1, 1, -5, 5, 7, 1, -1, 11, 7, 29, 35, 1, -7, 1, -41, 5, 7, 9, 93, 1, 3, 5, 11, 1, -3, 1, -5, 3, 7, 1, -61, 11, 7, 9, 27, 1, -29, 5, -1, 9, 11, 1, -29, 1, 3, 3, 771, 9, 9, 1, -5, 5, -3, 1, -73, 1, 3, 3, 19, 9, 9, 1, -141, -45, 7, 1, -53, 5, 7, 9, 43, 1, -63, 5, 11, 9, 11, 13, 1597, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A151800, A175851, A046644 (denominators).

Cf. also A305791, A305805, A305806, A317833, A317834, A317847.

A175851[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A175851[n] - Sum[If[1 < d < n, f[d]* f[n/d], 0], {d, Divisors[n]}])];
a[n_] := Numerator[f[n]];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

A175851(n) = if(1==n,n,1 + n - precprime(n));
A317830aux(n) = if(1==n,n,(A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2);
A317830(n) = numerator(A317830aux(n));

\\ Memoized implementation:
memo317830 = Map();
A317830aux(n) = if(1==n,n,if(mapisdefined(memo317830,n),mapget(memo317830,n),my(v = (A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2); mapput(memo317830,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A175851(n) - Sum_{d|n, d>1, d 1.

A286181 Lesser of Wilson's pseudo-twin primes: primes p such that p! == 1 (mod q), where q=A151800(p) is the next prime after p, and q-p>2.

Original entry on oeis.org

7841, 594278556271608991, 4259842839142238791410741595983041626644087433

Offset: 1

Max Alekseyev and Thomas Ordowski, May 04 2017

By Wilson's theorem, p! == 1 (mod p+2) whenever p,p+2 are twin primes. This sequence and A286208 concern consecutive primes p,q satisfying p! = 1 (mod q), where d = q-p > 2.
It follows that (d-1)! == 1 (mod q), and so q divides A033312(d-1).
Listed terms correspond to d = 12, 30, 76 (cf. A286230). Further terms should have d>=140.
Also, primes p=prime(n) such that A275111(n)=1, and (prime(n),prime(n+1)) are not twin primes (i.e., p is not a term of A001359).

			For a(1)=7841, we have 7841! == 1 (mod 7853), where 7841 and 7853=7841+12 are consecutive primes. Also, 7853 | (12-1)!-1.

Cf. A275111, A286208.

A339903 Fully multiplicative with a(p) = A000265(q-1), where q = A151800(p), the next prime > p.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 3, 1, 1, 5, 3, 1, 9, 1, 11, 3, 5, 3, 7, 1, 9, 1, 1, 5, 15, 3, 9, 1, 3, 9, 15, 1, 5, 11, 1, 3, 21, 5, 23, 3, 3, 7, 13, 1, 25, 9, 9, 1, 29, 1, 9, 5, 11, 15, 15, 3, 33, 9, 5, 1, 3, 3, 35, 9, 7, 15, 9, 1, 39, 5, 9, 11, 15, 1, 41, 3, 1, 21, 11, 5, 27, 23, 15, 3, 3, 3, 5, 7, 9, 13, 33, 1, 25, 25

Offset: 1

Antti Karttunen, Dec 29 2020

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003958, A003961, A005117, A151800, A339904.

A000265(n) = (n>>valuation(n,2));
A339903(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = nextprime(1+f[i,1])-1); A000265(factorback(f)));

a(n) = A336466(A003961(n)) = A000265(A003958(A003961(n))).

For all squarefree numbers k, a(k) = A339904(k).

A249624 Numbers n such that the sum of n and the least prime>n (A151800(n)) is prime.

Original entry on oeis.org

0, 1, 2, 6, 8, 14, 18, 20, 24, 30, 34, 36, 38, 48, 50, 54, 64, 68, 78, 80, 84, 94, 96, 98, 104, 110, 114, 124, 132, 134, 138, 144, 154, 156, 164, 174, 182, 188, 198, 208, 210, 216, 220, 228, 230, 248, 252, 258, 260, 270, 284, 294, 300, 306, 308, 314, 322, 328, 330, 336, 344, 360

Offset: 1

Antonio Roldán, Dec 03 2014

nonn

			50 is in the sequence because A151800(50)=53, and 50+53=103 is prime.

Cf. A151800, A249666, A249667, A249676.

{for(i=0,10^3,k=i+nextprime(i+1);if(isprime(k),print1(i,", ")))}

A249676 Terms k of A249667 such that k-A151799(k) = A151800(k)-k.

Original entry on oeis.org

6, 30, 50, 144, 300, 560, 610, 650, 660, 714, 780, 810, 816, 870, 1120, 1176, 1190, 1806, 2130, 2470, 2490, 2550, 2922, 3030, 3240, 3330, 3390, 3480, 3600, 3620, 3840, 4266, 4368, 5796, 5850, 6270, 6786, 6954, 7074, 7710, 8280, 9400, 9990, 10146, 10350, 10380, 10530, 10660, 11064

Offset: 1

Antonio Roldán, Dec 03 2014

nonn

			610 is in A249667: the least prime>610 is 613, and 610+613=1223 is prime; the largest prime<610 is 607, and 610+607=1217 is prime. Also, 613-610=610-607=3, then 610 is in the current sequence.

Cf. A249624, A249666, A249667.

{for(i=3,2*10^4,m=nextprime(i+1);k=i+m;n=precprime(i-1);q=i+n;if(isprime(k)&&isprime(q)&&m-i==i-n,print1(i,", ")))}

A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

Original entry on oeis.org

0, 1, -1, 2, 0, 1, -2, 3, 2, -2, 0, 8, 12, -8, -7, 14, -1, 10, 2, 4, 6, -3, 20, -2, 5, -5, -27, 4, -16, 5, 5, 4, -8, 11, 13, -8, -19, 8, -36, 3, 2, -14, -5, 2, -3, -55, -19, -6, 14, -54, -13, -53, 63, -26, 38, -2, 21, 38, -30, 7, 39, 2, -23, 41, 2, -8, 5, 5, -5, -110

Offset: 2

Hugo Pfoertner, Jan 29 2021

sign

a(n) > 0 if the difference nextprime(2^n) - 2^n = A013597(n) is greater than the difference 2^n - previousprime(2^n) = A013603(n).

			a(4) = -1: 2^4 = 16, (13 + 17 - 32)/2 = -1;
a(5) = 2: 2^5 = 32, (31 + 37 - 64)/2 = 2;
a(6) = 0: 2^6 = 64, (61 + 67 - 128)/2 = 0.

Hugo Pfoertner, Table of n, a(n) for n = 2..5000 (from T. D. Noe's b-files in A013597 and A013603).

Cf. A151799, A151800, A013597, A013603, A058249, A226178.

a:= (p-> (nextprime(p)+prevprime(p))/2-p)(2^n):
seq(a(n), n=2..75);  # Alois P. Heinz, Jan 29 2021

Array[(NextPrime[2^#] + NextPrime[2^#, -1] - 2^(# + 1))/2 &, 60, 2] (* Michael De Vlieger, Aug 07 2022 *)

for(k=2,71,my(p2=2^k,pp=precprime(p2),pn=nextprime(p2));if(print1((pp+pn-2*p2)/2", ")))

a(n) = (A013597(n) - A013603(n))/2.

a(A226178(n)) = 0.

Name made more precise by Peter Luschny, Aug 08 2022

A357363 Primes p such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where q = A151800(A151800(p)).

Original entry on oeis.org

5, 19, 263, 1667

Offset: 1

Felix Fröhlich, Sep 25 2022

Cf. A151800, A357362, A357364, A357365.

is(n) = my(b=precprime(precprime(n-1)-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=5, , if(is(p), print1(p, ", ")))

A357364 Primes p such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where q = A151800(A151800(A151800(p))).

Original entry on oeis.org

11, 23, 41, 107, 389, 1987673, 35603983

Offset: 1

Felix Fröhlich, Sep 25 2022

Cf. A151800, A357362, A357363, A357365.

is(n) = my(b=precprime(precprime(precprime(n-1)-1)-1)); Mod(b, n^2)^(n-1)==1 || Mod(n, b^2)^(b-1)==1
forprime(p=7, , if(is(p), print1(p, ", ")))

A087032 a(n) = 1 if 2*A151800(n) - n is prime, otherwise 0, where A151800(n) is the smallest prime > n.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Zak Seidov, Jul 31 2003

There is no subsequence of two ones; number of zeros in each group is odd, see A087033.

			a(1)=1 because the smallest prime > 1 is 2 and 2*2-1=3 is prime.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions

Cf. A010051, A087030, A087033, A151800.

bb={}; Do[bb={bb, If[PrimeQ[2(Prime[PrimePi[n]+1])-n], 1, 0]}, {n, 1000}]; Flatten[bb]

A087032(n) = isprime((2*nextprime(1+n))-n); \\ Antti Karttunen, Oct 09 2018

a(n) = 1 if A087030(n) is prime, 0 if it is composite.

a(n) = A010051((2*A151800(n))-n). - Antti Karttunen, Oct 09 2018

Definition edited by Antti Karttunen, Oct 09 2018

A132403 Triangle read by rows: T(n,k) = nextprime( T(n-1,k) + T(n-1,k-1) ), where nextprime = A151800.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 5, 11, 11, 5, 7, 17, 23, 17, 7, 11, 29, 41, 41, 29, 11, 13, 41, 71, 83, 71, 41, 13, 17, 59, 113, 157, 157, 113, 59, 17, 19, 79, 173, 271, 317, 271, 173, 79, 19, 23, 101, 257, 449, 593, 593, 449, 257, 101, 23, 29, 127, 359, 709, 1049, 1187, 1049, 709, 359, 127, 29

Offset: 0

Jonathan Vos Post, Nov 12 2007

Each number is the smallest prime > the sum of the 2 numbers above (consider each line padded with 0 on each side).

			Triangle begins:
  1
  2....2
  3....5....3
  5...11...11....5
  7...17...23...17....7
  11..29...41...41...29...11
  13..41...71...83...71...41...13
  17..59..113..157..157..113...59...17
  19..79..173..271..317..271..173...79...19
  23.101..257..449..593..593..449..257..101...23
  29.127..359..709.1049.1187.1049..709..359..127..29
  31.157..487.1069.1759.2237.2237.1759.1069..487.157..31
  37.191..647.1559.2833.4001.4583.4001.2833.1559.647.191.37
  ...
First column is A008578.
Second column is A064337.

Cf. A008578, A151800, A007318, A064337, A199333, A362034.

cp's OEIS Frontend

A317830 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.

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A286181 Lesser of Wilson's pseudo-twin primes: primes p such that p! == 1 (mod q), where q=A151800(p) is the next prime after p, and q-p>2.

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A339903 Fully multiplicative with a(p) = A000265(q-1), where q = A151800(p), the next prime > p.

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A249624 Numbers n such that the sum of n and the least prime>n (A151800(n)) is prime.

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A249676 Terms k of A249667 such that k-A151799(k) = A151800(k)-k.

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A340707 a(n) = (prevprime(2^n) + nextprime(2^n))/2 - 2^n where prevprime(n) = A151799(n) and nextprime(n) = A151800(n).

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A357363 Primes p such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where q = A151800(A151800(p)).

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A357364 Primes p such that either p^(q-1) == 1 (mod q^2) or q^(p-1) == 1 (mod p^2), where q = A151800(A151800(A151800(p))).

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A087032 a(n) = 1 if 2*A151800(n) - n is prime, otherwise 0, where A151800(n) is the smallest prime > n.

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