A081053 -id:A081053 - cp's OEIS frontend

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

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))

A381333 Smallest integer that is the sum of a prime and the square of a prime in n or more ways.

Original entry on oeis.org

6, 11, 56, 176, 188, 362, 398, 668, 1448, 1448, 1592, 2390, 3372, 3632, 4532, 6342, 6342, 6368, 6368, 10632, 12920, 12920, 12942, 19502, 23168, 25038, 25038, 25038, 25472, 32238, 32238, 39800, 39800, 39800, 53360, 64998, 72740, 72740, 72740, 81542, 82880, 82880

Offset: 1

Chai Wah Wu, Feb 20 2025

nonn

Subsequence of A081053. All terms are even except for a(2) = 11.

			a(1) = 6 as 6 = 2 + 2^2.
a(2) = 11 as 11 = 7 + 2^2 = 2 + 3^2.
a(3) = 56 as 56 = 47 + 3^2 = 31 + 5^2 = 7 + 7^2.
a(4) = 176 as 176 = 167 + 3^2 = 151 + 5^2 = 127 + 7^2 = 7 + 13^2.
a(5) = 188 as 188 = 179 + 3^2 = 163 + 5^2 = 139 + 7^2 = 67 + 11^2 = 19 + 13^2.
a(6) = 362 as 362 = 353 + 3^2 = 337 + 5^2 = 313 + 7^2 = 241 + 11^2 = 193 + 13^2 = 73 + 17^2.

Cf. A001172, A081053, A381330.

f(k) = my(nb=0); forprime(p=2, sqrtint(k), if (isprime(k-p^2), nb++);); nb;
a(n) = my(k=1); while (f(k) < n, k++); k; \\ Michel Marcus, Feb 21 2025

from itertools import count
from math import isqrt
from sympy import isprime, primerange
def A381333(n):
    for m in count(1):
        c = 0
        for p in primerange(isqrt(m)+1):
            if isprime(m-p**2):
                c += 1
            if c>=n:
                return m

A080713 Numbers not of the form p + q^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 10, 13, 18, 19, 24, 25, 29, 31, 34, 37, 39, 43, 49, 53, 55, 58, 59, 61, 64, 67, 69, 73, 74, 79, 81, 85, 89, 91, 94, 95, 97, 99, 100, 103, 109, 115, 119, 121, 125, 127, 129, 130, 133, 137, 139, 142, 145, 147, 149, 151, 154, 157, 159, 163, 165, 169, 170

Offset: 1

Robert G. Wilson v, Mar 05 2003

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A081053.

Complement[ Table[k, {k, 1, 200}], Take[ Union[ Flatten[ Table[ Prime[i] + Prime[j]^2, {i, 1, 80}, {j, 1, 10}]]], 250]]

is(n)=if(n%2, !isprime(n-4) && isprimepower(n-2)!=2, forprime(q=3, sqrtint(n), if(isprime(n-q^2), return(0))); n!=6) \\ Charles R Greathouse IV, Oct 25 2017

A381334 Smallest integer that is the sum of a prime and the square of a prime in exactly n ways.

Original entry on oeis.org

6, 11, 56, 176, 188, 362, 398, 668, 1568, 1448, 1592, 2390, 3372, 3632, 4532, 6888, 6342, 8582, 6368, 10632, 13002, 12920, 12942, 19502, 23168, 26990, 26292, 25038, 25472, 33648, 32238, 41048, 40640, 39800, 53360, 64998, 77348, 74718, 72740, 81542, 89682, 82880

Offset: 1

Chai Wah Wu, Feb 20 2025

nonn

Subsequence of A081053. All terms are even except for a(2) = 11.

a(n) >= A381333(n).

			a(1) = 6 as 6 = 2 + 2^2.
a(2) = 11 as 11 = 7 + 2^2 = 2 + 3^2.
a(3) = 56 as 56 = 47 + 3^2 = 31 + 5^2 = 7 + 7^2.
a(4) = 176 as 176 = 167 + 3^2 = 151 + 5^2 = 127 + 7^2 = 7 + 13^2.
a(5) = 188 as 188 = 179 + 3^2 = 163 + 5^2 = 139 + 7^2 = 67 + 11^2 = 19 + 13^2.
a(6) = 362 as 362 = 353 + 3^2 = 337 + 5^2 = 313 + 7^2 = 241 + 11^2 = 193 + 13^2 = 73 + 17^2.
a(9) = 1568 as 1568 = 1559 + 3^2 = 1543 + 5^2 = 1447 + 11^2 = 1399 + 13^2 = 1279 + 17^2 = 1039 + 23^2 = 727 + 29^2 = 607 + 31^2 = 199 + 37^2.
Note that a(9) > A381333(9) = 1448 as 1448 has 10 decompositions: 1448 = 1439 + 3^2 = 1423 + 5^2 = 1399 + 7^2 = 1327 + 11^2 = 1279 + 13^2 = 1087 + 19^2 = 919 + 23^2 = 607 + 29^2 = 487 + 31^2 = 79 + 37^2.

Robert Israel, Table of n, a(n) for n = 1..564

Cf. A081053, A381333.

N:= 10^6: # for a(1)..a(k) where a(k+1) is the first term > N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
W:= Vector(1..N/2,datatype=integer[4]):
for i from 2 while P[i]^2 < N do
m:= ListTools:-BinaryPlace(P, N - P[i]^2);
J:= (P[2..m] +~ P[i]^2)/~ 2;
  W[J]:= W[J] +~ 1;
od:
imax:= max[index](W):
R:= Vector(W[imax]):
R[1]:= 6: R[2]:= 11:
for i from 1 to imax do
r:= W[i];
if r > 0 and R[r] = 0 then R[r]:= 2*i fi;
od:
if member(0,R,'i') then convert(R[1..i-1],list) else convert(R,list) fi; # Robert Israel, Feb 24 2025

from itertools import count
from math import isqrt
from sympy import isprime, primerange
def A381334(n): return next(filter(lambda m:sum(1 for p in primerange(isqrt(m)+1) if isprime(m-p**2))==n,count(1)))

A080693 Numbers of the form p^2q + rs where p,q,r,s are (not necessarily distinct) primes.

Original entry on oeis.org

12, 14, 16, 17, 18, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Mario Maqueda Garcia [Garci'a] (israelmira(AT)terra.es), Mar 03 2003

nonn

A conjecture of Goldbach type says every number >= 26 is of this form.

			12=2^2*2 + 2*2

Cf. A081053.

H := proc(n::posint) local p,q,r,s; p := 2; while p<=floor(sqrt((n-4)/2)) do q := 2; while q<=floor((n-4)/p^2) do s := 2; while s<=floor((n-p^2*q)/2) do r := (n-p^2*q)/s; if type(r,posint) then if isprime(r) then return(true,p,q,s,r); end if; end if; s := nextprime(s); end do; q := nextprime(q); end do; p := nextprime(p); end do; return(false); end:

Take[ Union[ Flatten[ Table[ Prime[p]^2*Prime[q] + Prime[r]*Prime[s], {p, 1, 6}, {q, 1, 15}, {r, 1, 15}, {s, 1, 15}]]], 70]

Edited and extended by Robert G. Wilson v, Mar 05 2003

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

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A381333 Smallest integer that is the sum of a prime and the square of a prime in n or more ways.

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A080713 Numbers not of the form p + q^2.

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A381334 Smallest integer that is the sum of a prime and the square of a prime in exactly n ways.

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A080693 Numbers of the form p^2*q + r*s where p,q,r,s are (not necessarily distinct) primes.

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A080693 Numbers of the form p^2q + rs where p,q,r,s are (not necessarily distinct) primes.