A049002 -id:A049002 - cp's OEIS frontend

A286256 Compound filter: a(n) = P(A046523(n), A046523(2+n)), where P(n,k) is sequence A000027 used as a pairing function.

2, 12, 5, 40, 5, 84, 12, 86, 14, 142, 5, 148, 23, 216, 27, 367, 5, 265, 23, 148, 27, 412, 12, 430, 59, 142, 44, 832, 5, 1860, 23, 698, 61, 826, 27, 856, 23, 412, 27, 1402, 5, 850, 80, 148, 90, 1384, 12, 1759, 40, 265, 27, 607, 23, 1105, 61, 430, 27, 2086, 5, 2140, 80, 2352, 148, 4342, 27, 850, 23, 832, 27, 5080, 5, 2998, 80, 142, 148, 832, 27, 2956, 138, 1426

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A286255, A286257, A286258.

Cf. A001359 (gives the positions of 5's), A049002 (of 12's), A115093 (of 23's).

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286256(n) = (2 + ((A046523(n)+A046523(2+n))^2) - A046523(n) - 3*A046523(2+n))/2;
for(n=1, 10000, write("b286256.txt", n, " ", A286256(n)));

from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), a046523(n + 2)) # Indranil Ghosh, May 07 2017

(define (A286256 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ 2 n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ 2 n)))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A046523(2+n))^2) - A046523(n) - 3*A046523(2+n)).

A237367 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that 2*k - 1, prime(k)^2 - 2 and prime(m)^2 - 2 are all prime.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 3, 3, 2, 3, 2, 4, 3, 5, 2, 6, 3, 6, 2, 4, 3, 4, 2, 4, 3, 4, 4, 4, 3, 8, 3, 4, 5, 6, 6, 5, 6, 5, 5, 3, 4, 7, 5, 6, 3, 7, 3, 3, 5, 4, 5, 6, 5, 8, 10, 4, 5, 11, 6, 3, 6, 5, 5, 5, 6, 5, 8, 4, 3, 5, 6, 5, 1, 7, 6, 3, 3, 5, 6, 4

Offset: 1

Zhi-Wei Sun, Feb 07 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 73, 81, 534.
(ii) Any integer n > 2 can be written as k + m with k > 0 and m > 0 such that 2*k - 1, prime(k) + k*(k-1) and prime(m) + m*(m-1) are all prime.
(iii) Every n = 9, 10, ... can be written as k + m with k > 0 and m > 0 such that 6*k - 1, prime(k) + 2*k and prime(m) + 2*m are all prime.
Clearly, part (i) of this conjecture implies that there are infinitely many primes p with p^2 - 2 also prime. Similar comments apply to parts (ii) and (iii).

			a(3) = 1 since 3 = 2 + 1 with 2*2 - 1 = 3, prime(2)^2 - 2 = 3^2 - 2 = 7 and prime(1)^2 - 2 = 2^2 - 2 = 2 all prime.
a(73) = 1 since 73 = 55 + 18 with 2*55 - 1 = 109, prime(55)^2 - 2 = 257^2 - 2 = 66047 and prime(18)^2 - 2 = 61^2 - 2 = 3719 all prime.
a(81) = 1 since 81 = 34 + 47 with 2*34 - 1 = 67, prime(34)^2 - 2 = 139^2 - 2 = 19319 and prime(47)^2 - 2 = 211^2 - 2 = 44519 all prime.
a(534) = 1 since 534 = 100 + 434 with 2*100 - 1 = 199, prime(100)^2 - 2 = 541^2 - 2 = 292679 and prime(434)^2 - 2 = 3023^2 - 2 = 9138527 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A049002, A062326, A076297, A218829, A230502, A233204, A237348.

pq[k_]:=PrimeQ[Prime[k]^2-2]
a[n_]:=Sum[If[PrimeQ[2k-1]&&pq[k]&&pq[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A237414 Primes p with p^2 - 2 and prime(p)^2 - 2 both prime.

Original entry on oeis.org

2, 3, 43, 47, 107, 139, 191, 211, 223, 239, 293, 313, 337, 541, 743, 757, 863, 1013, 1153, 1231, 1619, 2113, 2137, 2287, 2297, 2423, 2543, 2729, 2749, 2897, 3079, 3089, 3313, 3863, 3947, 4241, 4271, 4583, 4649, 4993, 5581, 6571, 6637, 6911, 7547, 8629, 8849, 8867, 9049, 9661

Offset: 1

Zhi-Wei Sun, Feb 07 2014

nonn

According to the conjecture in A237413, this sequence should have infinitely many terms.

			a(1) = 2 since 2^2 - 2 = 2 and prime(2)^2 - 2 = 3^2 - 2 = 7 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A049002, A062326, A230502, A237413.

p[n_]:=PrimeQ[n^2-2]
n=0;Do[If[p[Prime[k]]&&p[Prime[Prime[k]]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[1200]],AllTrue[{#^2-2,Prime[#]^2-2},PrimeQ]&] (* Harvey P. Dale, Apr 06 2022 *)

A253257 Least positive integer k such that prime(k*n) has the form p^2 - 2 with p prime, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 3, 1, 3, 12, 47, 9, 1, 100, 502, 6, 3, 1817, 1, 362, 3141, 4, 104, 50, 14157, 251, 222, 3, 27, 76, 25, 5423, 416, 73, 28764, 181, 488, 3860, 1249, 2, 138, 52, 1, 25, 8734, 65719, 7089, 214, 15, 111, 7, 990, 6254, 20, 1047, 38, 367, 880, 435, 3712, 3287, 208, 5194, 598

Offset: 1

Zhi-Wei Sun, Jul 05 2015

nonn

Conjecture: a(n) > 0 for all n > 0.
This is stronger than the conjecture that there are infinitely many primes of the form p^2-2 with p prime.
I also conjecture that for any positive integer n there is a positive integer k such that prime(k*n) has the form 2*p^2-1 (or 4*p^2+1, or p^2+p+1) with p prime.

			a(1) = 1 since prime(1*1) = 2 = 2^2-2 with 2 prime.
a(6) = 12 since prime(12*6) = 359 = 19^2-2 with 19 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..800
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A049002, A060429, A060800, A062326, A179262, A237367, A237413, A259731.

SQ[n_]:=IntegerQ[Sqrt[n]]&&PrimeQ[Sqrt[n]]
Do[k=0;Label[bb];k=k+1;If[SQ[Prime[k*n]+2],Goto[aa],Goto[bb]];Label[aa];Print[n, " ", k];Continue,{n,1,60}]

use ntheory ":all"; use Math::Prime::Util::PrimeArray qw/$probj/; my %v; forprimes { undef $v{$*$-2} } 4e7; for my $n (1..800) { my $k=1; $k++ until exists $v{$probj->FETCH($k*$n-1)}; say "$n $k"; } # Dana Jacobsen, Dec 15 2015

A267944 Primes that are a prime power minus two.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 23, 29, 41, 47, 59, 71, 79, 101, 107, 137, 149, 167, 179, 191, 197, 227, 239, 241, 269, 281, 311, 347, 359, 419, 431, 461, 521, 569, 599, 617, 641, 659, 727, 809, 821, 827, 839, 857, 881

Offset: 1

Robert C. Lyons, Jan 22 2016

nonn

The sequence is probably infinite, since it includes all the terms of A001359 (Lesser of twin primes).

Also includes A049002. The generalized Bunyakovsky conjecture implies that for every k there are infinitely many terms of the form p^k - 2. - Robert Israel, Jan 22 2016

			2 is in the sequence because 2 = 2^2 - 2.
3 is in the sequence because 3 = 5^1 - 2.
5 is in the sequence because 5 = 7^1 - 2.
7 is in the sequence because 7 = 3^2 - 2.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Generalized Bunyakovsky conjecture

Cf. A000961, A049002, A246655, A267945.

select(t -> isprime(t) and nops(numtheory:-factorset(t+2))=1, [2, seq(i,i=3..1000, 2)]); # Robert Israel, Jan 22 2016

A267944Q = PrimeQ@# && Length@FactorInteger[# + 2] == 1 & (* JungHwan Min, Jan 24 2016 *)
Select[Array[Prime, 100], Length@FactorInteger[# + 2] == 1 &] (* JungHwan Min, Jan 24 2016 *)
Select[Prime[Range[300]],PrimePowerQ[#+2]&] (* Harvey P. Dale, Nov 28 2016 *)

lista(nn) = {forprime(p=2, nn, if (isprimepower(p+2), print1(p, ", ")););} \\ Michel Marcus, Jan 22 2016

[n - 2 for n in prime_powers(1, 1000) if is_prime(n - 2)]

A347194 Numbers such that the two adjacent integers are a prime and the square of another prime.

Original entry on oeis.org

8, 10, 24, 48, 168, 360, 840, 1368, 1848, 2208, 3720, 5040, 7920, 10608, 11448, 16128, 17160, 19320, 29928, 36480, 44520, 49728, 54288, 57120, 66048, 85848, 97968, 113568, 128880, 177240, 196248, 201600, 218088, 241080, 273528, 292680, 323760, 344568, 368448, 426408, 458328, 516960, 528528, 537288, 552048, 564000, 573048, 579120

Offset: 1

Bernard Schott, Sep 23 2021

nonn

-> Equivalently, numbers k such that tau(k^2-1) = A347191(k) = 6 (see example; used for Maple code).
Proof: tau(k^2-1) = 6 <==> k^2-1 = p^5 or k^2-1 = p*q^2 with p <> q primes; but k^2-p^5 = 1 is impossible, as a consequence of the Catalan-Mihăilescu theorem; now, (k-1)*(k+1) = p*q^2 ==> (k-1 = p and k+1 = q^2) or (k-1 = q^2 and k+1 = p), because k-1 = q and k+1 = p*q is not possible, otherwise 2 = q*(p-1), which would contradict p <> q.
-> There are two possible configurations with p, q primes: (q^2 < a(n) < p) or (p < a(n) < q^2).
The unique configuration q^2 < a(n) < p is for q = 3, a(2) = 10 and  p = 11.
All the other configurations, for n = 1 or n >= 3, are of the form p < a(n) < q^2 with p = A049002(n) and q = A062326(n).
-> Note that there is only one integer such that the two adjacent integers are a prime and the square of that prime: it is 3, which lies between 2 and 2^2; in this case, tau(3^2-1) = 4.

			8 is a term since 8 lies between 7 (prime) and 9 = 3^2 (square of prime); also tau(8^2-1) = tau(63) = 6.
10 is a term since 10 lies between 9 = 3^2 (square of prime) and 11 (prime); also tau(10^2-1) = tau(99) = 6.
24 is a term since 24 lies between 23 (prime) and 25 = 5^2 (square of prime); also tau(24^2-1) = tau(575) = 6.

Diophante, A1885, Cachés derrière leurs diviseurs (in French).
Adrian Dudek, On the Number of Divisors of n^2-1, arXiv:1507.08893 [math.NT], 2015.
Wikipedia, Catalan's conjecture.

Cf. A049002, A062326, A146981.
Cf. A347191, A347192, A347193.
Subsequence of A163492 (between prime and a perfect square).

with(numtheory):
filter := q-> tau(q^2-1) = 6 : select(filter, [$2..580000]);

q[n_] := Module[{e1 = FactorInteger[n - 1][[;; , 2]], e2 = FactorInteger[n + 1][[;; , 2]]}, (e1 == {1} && e2 == {2}) || (e1 == {2} && e2 == {1})]; Select[Range[4, 600000], q] (* Amiram Eldar, Sep 23 2021 *)

isok(m) = my(pa, pb); (isprimepower(m-1, &pa)*isprimepower(m+1, &pb) == 2) && (pa != pb); \\ Michel Marcus, Sep 23 2021

upto(n) = { my(res = List()); forprime(i = 3, sqrtint(n-1), if(isprime(i^2 - 2), listput(res, i^2-1); ); if(isprime(i^2 + 2), listput(res, i^2 + 1); ) ); res } \\ David A. Corneth, Sep 23 2021

For n >= 3: a(n) = A049002(n) + 1 = a(n) = A146981(n) - 1 = (A049002(n) + A146981(n))/2 = A062326(n)^2 - 1.

A065017 Primes of the form p*q + p + q, where (p, q=p+2) are twin primes.

Original entry on oeis.org

23, 47, 167, 359, 1847, 3719, 10607, 19319, 97967, 177239, 273527, 657719, 1042439, 1104599, 1329407, 1515359, 1745039, 2042039, 4464767, 5013119, 5148359, 9740639, 11095559, 11377127, 12538679, 16024007, 16410599, 16752647

Offset: 1

Stephan Wagler (stephanwagler(AT)aol.com), Nov 01 2001

nonn

The resulting prime can never be a twin prime since the odd number preceding it is divisible by three and the following odd number is a perfect square.

			(3*5) + (3+5) = 23.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A049001, A049002.

NextPrim[n_] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; k = 1; Do[k = NextPrim[k]; If[ PrimeQ[k + 2], p = k*(k + 2) + 2k + 2; If[ PrimeQ[p], Print[p]]], {n, 1, 700} ]
f[n_]:=Module[{x=Total[n]+Times@@n},If[PrimeQ[x],x,0]]; Select[f/@ (Select[Partition[Prime[Range[700]],2,1],Last[#]-First[#]==2&]), #!=0&] (* Harvey P. Dale, May 11 2011 *)

{ n=p=0; for (m=1, 10^9, p=nextprime(p + 1); if (isprime(q=p + 2) && isprime(a=p*q + p + q), write("b065017.txt", n++, " ", a); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 03 2009

p^2 + 4*p + 2.

Offset changed from 0 to 1 by Harry J. Smith, Oct 03 2009

A146981 Numbers k of the form q^2, q = prime, such that k-2 is a prime.

Original entry on oeis.org

4, 9, 25, 49, 169, 361, 841, 1369, 1849, 2209, 3721, 5041, 7921, 10609, 11449, 16129, 17161, 19321, 29929, 36481, 44521, 49729, 54289, 57121, 66049, 85849, 97969, 113569, 128881, 177241, 196249, 201601, 218089, 241081, 273529, 292681, 323761

Offset: 1

Giovanni Teofilatto, Nov 04 2008

nonn

Except for initial term, a(n) - 1 is not a squarefree number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049002, A062326.

Select[Prime[Range[150]]^2,PrimeQ[#-2]&] (* Harvey P. Dale, Jan 15 2020 *)

a(n) = A049002(n) + 2.

a(n) = A062326(n)^2. - Amiram Eldar, Oct 21 2019

Corrected (3721 inserted, 10321 replaced by 19321, 49729 and 218089 inserted) by R. J. Mathar, Apr 22 2010

A226977 Places n where A225867(n) <= 2.

Original entry on oeis.org

7, 8, 11, 17, 23, 29, 41, 47, 59, 71, 239, 359, 419, 839

Offset: 1

Vladimir Shevelev, Jun 27 2013

Except for a(2)=8, all terms are prime.
Is this sequence finite?
There are no other terms up to 86000. - Peter J. C. Moses, Jun 28 2013
There are no more terms up to 10^9. - Charles R Greathouse IV, Nov 25 2014

Cf. A225867, A049002.

is(n)=for(k=2,n\2-1,if(sumdiv(n+k,d,(n+d)%k==0 && d>1)>2, return(0))); n>6 \\ Charles R Greathouse IV, Nov 25 2014

A381330 Numbers that are the sum of a prime and the square of a prime in more than one way.

Original entry on oeis.org

11, 27, 28, 32, 38, 51, 52, 54, 56, 62, 66, 68, 72, 78, 80, 86, 92, 96, 98, 108, 110, 116, 122, 126, 128, 132, 134, 138, 140, 146, 150, 152, 156, 158, 162, 164, 171, 172, 174, 176, 180, 182, 186, 188, 192, 198, 200, 204, 206, 210, 212, 216, 218, 222, 224, 228

Offset: 1

Chai Wah Wu, Feb 20 2025

nonn

Subsequence of A081053. Most terms are even. The odd terms are 11, 27, 51, 171, 363, 843, 1371, 1851 and must be of the form 2+p^2=4+q for primes p, q. In particular, the odd terms are exactly A049002(n)+4 for n>1.

			11 is a term since 11 = 2^2+7  = 3^2+2.
27 is a term since 27 = 2^2+23 = 5^2+2.
28 is a term since 28 = 3^2+19 = 5^2+3.
32 is a term since 32 = 3^2+23 = 5^2+7.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A081053, A049002.

isok(k) = my(nb=0); forprime(p=2, sqrtint(k), if (isprime(k-p^2), nb++);); nb > 1; \\ Michel Marcus, Feb 21 2025

from math import isqrt
from itertools import count, islice
from sympy import isprime, primerange
def A381330_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        c = 0
        for p in primerange(isqrt(n)+1):
            if isprime(n-p**2):
                c += 1
            if c>1:
                yield n
                break
A381330_list = list(islice(A381330_gen(),30))

cp's OEIS Frontend

A286256 Compound filter: a(n) = P(A046523(n), A046523(2+n)), where P(n,k) is sequence A000027 used as a pairing function.

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A237367 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that 2*k - 1, prime(k)^2 - 2 and prime(m)^2 - 2 are all prime.

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Mathematica

A237414 Primes p with p^2 - 2 and prime(p)^2 - 2 both prime.

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A253257 Least positive integer k such that prime(k*n) has the form p^2 - 2 with p prime, or 0 if no such k exists.

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A267944 Primes that are a prime power minus two.

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A347194 Numbers such that the two adjacent integers are a prime and the square of another prime.

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A065017 Primes of the form p*q + p + q, where (p, q=p+2) are twin primes.

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