A151800 -id:A151800 - cp's OEIS frontend

A218561 4-gap primes: Prime p is a term iff there is no prime between 4p and 4nextprime(p), where nextprime=A151800.

29, 71, 137, 197, 239, 269, 347, 419, 431, 641, 659, 809, 821, 1061, 1091, 1151, 1289, 1489, 1607, 1721, 1783, 1877, 1949, 1993, 2083, 2141, 2267, 2339, 2381, 2389, 2549, 2729, 2801, 2833, 2969, 2999, 3019, 3041, 3217, 3253, 3299, 3329, 3389, 3461

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Nov 02 2012

nonn

			29 is in the sequence since there are no primes in the interval(4*29,4*31)=(116,124)

M. F. Hasler, Table of n, a(n) for n = 1..4911 (all terms up to 5*10^5).

Cf. A195270, A195271, A195377, A195325, A195329, A195379.

Select[Prime[Range[500]],Count[Range[4*#,4*NextPrime[#]],?PrimeQ]==0&] (* _Harvey P. Dale, Jan 05 2013 *)

is_A218561(p)=isprime(p)&nextprime(4*p)>=4*nextprime(p+1)  \\ - M. F. Hasler, Nov 03 2012

A340148 a(n) = Product_{distinct primes p dividing n} gcd(q-1, A003961(n)-1), where q = A151800(p), the next prime larger than p.

Original entry on oeis.org

1, 2, 4, 2, 6, 4, 10, 2, 4, 4, 12, 8, 16, 4, 4, 2, 18, 4, 22, 4, 4, 4, 28, 4, 6, 4, 4, 4, 30, 16, 36, 2, 16, 4, 4, 8, 40, 4, 16, 4, 42, 16, 46, 8, 12, 4, 52, 8, 10, 4, 4, 16, 58, 4, 36, 4, 4, 4, 60, 8, 66, 4, 4, 2, 4, 8, 70, 4, 16, 40, 72, 4, 78, 4, 8, 4, 4, 8, 82, 4, 4, 4, 88, 8, 36, 4, 4, 4, 96, 16, 4, 8, 16

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Prime shifted analog of A063994.

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. A003961, A003972, A063994, A340147.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A063994(n) = { my(f=factor(n)[,1]); prod(i=1, #f, gcd(f[i]-1, n-1)); };
A340148(n) = A063994(A003961(n));

A340148(n) = { my(f=factor(n)[,1], u=A003961(n)); prod(i=1, #f, gcd(nextprime(1+f[i])-1, u-1)); };

A340148(n) = { my(u=A003961(n), f=factor(u)[,1]); prod(i=1, #f, gcd(f[i]-1, u-1)); };

a(n) = A063994(A003961(n)).

a(n) = A003972(n) / A340147(n).

A379405 a(n) = p((n+1)*p(n)), where p(x) = least prime > x; i.e., p = A151800.

Original entry on oeis.org

3, 5, 11, 23, 29, 43, 53, 89, 101, 113, 127, 157, 173, 239, 257, 277, 293, 347, 367, 461, 487, 509, 541, 701, 727, 757, 787, 821, 853, 937, 967, 1187, 1223, 1259, 1297, 1361, 1373, 1559, 1601, 1657, 1693, 1811, 1861, 2069, 2129, 2179, 2213, 2549, 2609, 2657

Offset: 0

Clark Kimberling, Jan 18 2025

nonn

			p(1) = 2, so a(1) = p(2*p(1)) = 5.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000040, A053024, A151800, A378135, A378136, A378137.

Table[NextPrime[(n+1)*NextPrime[n]], {n, 0, 80}]

p(n) = nextprime(n+1);
a(n) = p((n+1)*p(n)); \\ Michel Marcus, Jan 18 2025

from sympy import nextprime
def A379405(n): return nextprime((n+1)*nextprime(n)) # Chai Wah Wu, Jan 20 2025

a(n) = A151800(A053024(n+1)). - Alois P. Heinz, Jan 18 2025

A328915 If n = Product (p_j^k_j) then a(n) = Product (nextprime(p_j)), where nextprime = A151800.

Original entry on oeis.org

1, 3, 5, 3, 7, 15, 11, 3, 5, 21, 13, 15, 17, 33, 35, 3, 19, 15, 23, 21, 55, 39, 29, 15, 7, 51, 5, 33, 31, 105, 37, 3, 65, 57, 77, 15, 41, 69, 85, 21, 43, 165, 47, 39, 35, 87, 53, 15, 11, 21, 95, 51, 59, 15, 91, 33, 115, 93, 61, 105, 67, 111, 55, 3, 119, 195, 71, 57, 145, 231

Offset: 1

Ilya Gutkovskiy, Oct 30 2019

All terms are odd.

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2) * prime(3) = 3 * 5 = 15.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A003961, A007947, A045965, A045967, A048250, A151800, A328878, A328879.

a:= n-> mul(nextprime(i[1]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 30 2019

a[1] = 1; a[n_] := Times @@ (NextPrime[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

a(n) = my(f=factor(n)); prod(k=1, #f~, nextprime(f[k,1]+1)); \\ Michel Marcus, Oct 30 2019

If n = Product (p_j^k_j) then a(n) = Product (prime(pi(p_j) + 1)), where pi = A000720.

a(n) = A007947(A003961(n)).

A335185 a(n) = nextprime(ceiling(n/2)-1) - prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.

Original entry on oeis.org

0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0, 2, 2, 2, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0, 4, 4, 4, 4, 4, 4, 4, 0, 2, 2, 2, 0, 4, 4, 4, 4, 4, 4, 4, 0, 6, 6, 6

Offset: 4

Wesley Ivan Hurt, May 25 2020

a(n) is the difference of the smallest prime appearing among the largest parts of the partitions of n into two parts and the largest prime appearing among the smallest parts of the partitions of n into two parts.
a(n) = 0 if and only if n = 2p, where p is prime. All terms are even except a(5).
The values in the n-th run of positive integers are all equal to the n-th prime gap (A001223).
Each value specifies the run length of the block (of positive integers) in which it appears. If a(n) = 0, then it appears once. If a(n) > 0, it has a run length of 2k - 1.

			a(5) = 1; n=5 has 2 partitions into two parts: (4,1) and (3,2). Among the largest parts, the smallest prime is 3. Among the smallest parts, 2 is the largest. So a(5) = 3 - 2 = 1.
a(6) = 0; n=6 has 3 partitions into two parts: (5,1), (4,2) and (3,3). Among the largest parts, the smallest prime is 3. Among the smallest parts, the largest prime is 3. So a(6) = 3 - 3 = 0.
a(7) = 2; n=7 has 3 partitions into two parts: (6,1), (5,2) and (4,3). Among the largest parts, 5 is the smallest. Among the smallest parts, 3 is the largest. So a(7) = 5 - 3 = 2.

Index entries for sequences related to partitions

Cf. A001223 (prime gaps), A151799, A151800, A335186.

[NextPrime(Ceiling(n/2)-1) - PreviousPrime(Floor(n/2)+1) : n in [4..100]];

Table[NextPrime[Ceiling[n/2] - 1, 1] - NextPrime[Floor[n/2] + 1, -1], {n, 4, 100}]

a(n) = A151800(ceiling(n/2)-1) - A151799(floor(n/2)+1).

A335186 a(n) = nextprime(ceiling(n/2)-1) + prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.

Original entry on oeis.org

4, 5, 6, 8, 8, 8, 10, 12, 12, 12, 14, 18, 18, 18, 18, 18, 18, 18, 22, 24, 24, 24, 26, 30, 30, 30, 30, 30, 30, 30, 34, 36, 36, 36, 38, 42, 42, 42, 42, 42, 42, 42, 46, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 52, 58, 60, 60, 60, 62, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 68, 74

Offset: 4

Wesley Ivan Hurt, May 25 2020

a(n) is the sum of the smallest prime appearing among the largest parts of the partitions of n into two parts and the largest prime appearing among the smallest parts of the partitions of n into two parts.
If n = 2p where p is prime, then a(n) = n. The converse is not true since a(8) = 8, but n = 2*4 and 4 is not prime.
All terms are even except a(5).

			a(5) = 5; n=5 has 2 partitions into two parts: (4,1) and (3,2). Among the largest parts, the smallest prime is 3. Among the smallest parts, 2 is the largest. So a(5) = 3 + 2 = 5.
a(6) = 6; n=6 has 3 partitions into two parts: (5,1), (4,2) and (3,3). Among the largest parts, the smallest prime is 3. Among the smallest parts, the largest prime is 3. So a(6) = 3 + 3 = 6.
a(7) = 8; n=7 has 3 partitions into two parts: (6,1), (5,2) and (4,3). Among the largest parts, 5 is the smallest. Among the smallest parts, 3 is the largest. So a(7) = 5 + 3 = 8.

Index entries for sequences related to partitions

Cf. A001223 (prime gaps), A151799, A151800, A335185.

[NextPrime(Ceiling(n/2)-1) + PreviousPrime(Floor(n/2)+1) : n in [4..100]];

Table[NextPrime[Ceiling[n/2] - 1, 1] + NextPrime[Floor[n/2] + 1, -1], {n, 4, 100}]

a(n) = A151800(ceiling(n/2)-1) + A151799(floor(n/2)+1).

A345161 If n = Product (p_j^k_j) then a(n) = max (nextprime(p_j) - p_j), where nextprime = A151800.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 4, 4, 2, 1, 2, 2, 4, 2, 4, 2, 6, 2, 2, 4, 2, 4, 2, 2, 6, 1, 2, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 2, 6, 6, 2, 4, 2, 2, 4, 6, 2, 2, 4, 4, 2, 2, 2, 6, 6, 4, 1, 4, 2, 4, 2, 6, 4, 2, 2, 6, 4, 2, 4, 4, 4, 4, 2, 2, 2, 6, 4, 2, 4, 2, 2, 8, 2

Offset: 1

Ilya Gutkovskiy, Aug 26 2021

nonn

			a(39) = a(3 * 13) = a(prime(2) * prime(6)), prime(3) - prime(2) = 5 - 3 = 2, prime(7) - prime(6) = 17 - 13 = 4, so a(39) = max(2, 4) = 4.

Cf. A000079 (positions of 1's), A000720, A001223, A006530, A061395, A063095, A151800, A327441, A347101, A347102.

a[n_] := Max @@ (NextPrime[#[[1]]] - #[[1]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 90}]

If n = Product (p_j^k_j) then a(n) = max (prime(pi(p_j) + 1) - p_j), where pi = A000720.
a(2^j*n) = a(n).
a(n^j) = a(n), j > 0.
a(prime(n)^j) = A001223(n), j > 0.
a(n!) = A327441(n).
a(prime(n)#) = A063095(n).
2 + Sum_{k=1..n-1} a(prime(k)^j) = prime(n), j > 0.
Sum_{d|n} mu(n/d) * a(d) = 0 if n is an even number or an odd number divisible by a square > 1.

A380329 a(n) = A151800((n-1)*A151800(n)).

Original entry on oeis.org

2, 2, 5, 11, 17, 29, 37, 67, 79, 89, 101, 131, 149, 211, 223, 239, 257, 307, 331, 419, 439, 461, 487, 641, 673, 701, 727, 757, 787, 877, 907, 1117, 1151, 1187, 1223, 1259, 1297, 1481, 1523, 1559, 1601, 1721, 1777, 1979, 2027, 2069, 2129, 2441, 2503, 2549, 2609

Offset: 0

Clark Kimberling, Jan 21 2025

nonn

			p(1) = 2, so a(1) = p(0*p(1)) = 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A000040, A151800, A378135, A379405.

seq(nextprime((n-1)*nextprime(n)), n=0..100); # Robert Israel, Jan 23 2025

Table[NextPrime[(n-1)*NextPrime[n]], {n, 0, 80}]

p(n) = nextprime(n+1);
a(n) = p((n-1)*p(n)); \\ Michel Marcus, Jan 23 2025

A308022 a(n) = prevprime(2*n-1) - nextprime(n-1), where prevprime = A151799 and nextprime = A151800.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 2, 2, 6, 8, 6, 10, 6, 6, 12, 14, 12, 12, 14, 14, 18, 20, 14, 18, 18, 18, 24, 24, 22, 28, 24, 24, 24, 30, 30, 34, 32, 32, 32, 38, 36, 40, 36, 36, 42, 42, 36, 36, 44, 44, 48, 50, 44, 48, 50, 50, 54, 54, 52, 52, 46, 46, 46, 60, 60, 64, 60, 60

Offset: 2

Wesley Ivan Hurt, May 09 2019

a(n) is the difference of the largest and smallest prime(s) in the closed interval [n, 2n-2].

Also, the maximum distance between all pairs of primes (not necessarily distinct) appearing among the largest parts of the partitions of 2n into two parts < 2n-1.

			a(10) = 6; The primes in the closed interval [10, 18] are 11, 13 and 17. The difference of the largest and smallest primes is 17 - 11 = 6.

Index entries for sequences related to partitions

Cf. A151799, A151800.

Table[NextPrime[2 n - 1, -1] - NextPrime[n - 1, 1], {n, 2, 100}]

a(n) = A151799(2*n-1) - A151800(n-1).

A377403 For n >= 2, a(n) is the number of iterations needed for the map: x -> x / A085392(x) if A085392(x) > 1, otherwise x -> x + A151800(x), to (the first occurrence of) 2.

Original entry on oeis.org

0, 3, 1, 3, 1, 3, 2, 4, 1, 4, 2, 3, 1, 4, 3, 4, 2, 3, 2, 4, 1, 3, 3, 4, 1, 5, 2, 4, 2, 3, 4, 4, 1, 4, 3, 3, 1, 4, 3, 4, 2, 4, 2, 5, 1, 4, 4, 4, 2, 4, 2, 5, 3, 4, 3, 4, 1, 5, 3, 7, 1, 5, 5, 4, 2, 3, 2, 4, 2, 6, 4, 4, 1, 5, 2, 4, 2, 5, 4, 6, 1, 3, 3, 4, 1, 4, 3, 3, 3, 4, 2, 4, 1, 4, 5, 4, 2, 5, 3, 4, 2, 4, 3, 5, 1, 6, 4, 3, 2, 4, 4, 6, 2, 4, 2, 5, 1, 4, 4, 5

Offset: 2

Ctibor O. Zizka, Oct 27 2024

nonn

Also a(2*k + 1) = A001222(2*k + 1) + 2 + s, where s >= 1 for k = 5, 8, 14, 20, 21, 23, 26, 29, 30, 35, 36, 39, 48, 50, 51, ...

			n = 3: 3 -> 8 -> 4 -> 2, 3 iterations needed to reach 2, thus a(3) = 3.
n = 9: 9 -> 3 -> 8 -> 4 -> 2, 4 iterations needed to reach 2, thus a(9) = 4.
n = 11: 11 -> 24 -> 8 -> 4 - > 2, 4 iterations needed to reach 2, thus a(11) = 4.

Cf. A001222, A013634, A020639, A085392, A107286, A151800.

a[2] = 0; a[n_] := -1 + Length@ NestWhileList[If[CompositeQ[#], #/FactorInteger[#][[-1, 1]], # + NextPrime[#]] &, n, # > 2 &]; Array[a, 120, 2] (* Amiram Eldar, Oct 27 2024 *)

For n even: a(n) = A001222(n) - 1.

For n odd: a(n) = A001222(n) - 1 + A001222(A013634(A020639(n))).

cp's OEIS Frontend

A218561 4-gap primes: Prime p is a term iff there is no prime between 4*p and 4*nextprime(p), where nextprime=A151800.

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A340148 a(n) = Product_{distinct primes p dividing n} gcd(q-1, A003961(n)-1), where q = A151800(p), the next prime larger than p.

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A379405 a(n) = p((n+1)*p(n)), where p(x) = least prime > x; i.e., p = A151800.

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A328915 If n = Product (p_j^k_j) then a(n) = Product (nextprime(p_j)), where nextprime = A151800.

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A335185 a(n) = nextprime(ceiling(n/2)-1) - prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.

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Magma

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A335186 a(n) = nextprime(ceiling(n/2)-1) + prevprime(floor(n/2)+1), where nextprime = A151800 and prevprime = A151799.

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Magma

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A345161 If n = Product (p_j^k_j) then a(n) = max (nextprime(p_j) - p_j), where nextprime = A151800.

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A218561 4-gap primes: Prime p is a term iff there is no prime between 4p and 4nextprime(p), where nextprime=A151800.