Steven Lu - cp's OEIS frontend

A382863 a(2k-1) and a(2k) are a pair of prime numbers where 9a(2k-1) and 8a(2k) are neighboring integers.

17, 19, 47, 53, 79, 89, 97, 109, 113, 127, 223, 251, 239, 269, 241, 271, 337, 379, 353, 397, 383, 431, 433, 487, 463, 521, 607, 683, 673, 757, 719, 809, 863, 971, 881, 991, 1087, 1223, 1153, 1297, 1279, 1439, 1297, 1459, 1327, 1493, 1361, 1531, 1423, 1601

Offset: 1

Steven Lu, Apr 07 2025

			a(5) = 79 and a(6) = 89 are such a pair, because 79*9=711 and 89*8=712 are neighboring integers.

Steven Lu, Table of n, a(n) for n = 1..10000

Flatten[{#, FirstCase[{(9 # + 1)/8, (9 # - 1)/8}, _Integer]} & /@ Select[Prime /@ Range[225], PrimeQ[(9 # + 1)/8] || PrimeQ[(9 # - 1)/8] &]]

A382546 Positive integers whose prime factors are all in A219528.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Steven Lu, Mar 31 2025

nonn

Steven Lu, Table of n, a(n) for n = 1..10000

Cf. A219528.

Select[Range[75], And @@ Table[SubsetQ[{2, 3}, First /@ FactorInteger[p + 1]] || SubsetQ[{2, 3}, First /@ FactorInteger[p - 1]], {p, First /@ FactorInteger[#]}] &]

Sum_{n>=1} 1/a(n) = 1/Product_{p in A219528} (1-1/p) = 7.52982574262479641306... . - Amiram Eldar, Apr 14 2025

A382066 a(n) = Sum_{k=1..prime(n)-1} (-k/prime(n)) * 3^(k-1) / 2, where (p/q) is the Legendre symbol of p and q.

Original entry on oeis.org

1, 8, 151, 8083, 70568, 8910416, 39392803, 7701058213, 2325990648824, 43563061207573, 19999898090377928, 2566793589644124992, 10627327735475477203, 2179055220073884519235, 630486036620986837882904, 646895254841829205782412249, 5802709167332592724735012664

Offset: 2

Steven Lu, Mar 14 2025

nonn

All terms are integers since Sum_{k=1..p-1} (-k/p) * 3^(k-1) is even for any odd prime p.

Steven Lu, Table of n, a(n) for n = 2..316

Table[Sum[KroneckerSymbol[-i, q] 3^(i - 1), {i, q - 1}], {q, Prime /@ Range[2, 18]}]/2

A382035 a(n) is the smallest prime q such that q + prime(n) is of form 10^k or 2*10^k, or 0 if no such prime exists.

Original entry on oeis.org

0, 7, 5, 3, 89, 7, 3, 181, 977, 71, 1999969, 163, 59, 157, 53, 47, 41, 139, 1933, 29, 127, 199921, 17, 11, 3, 999999999899, 97, 999999893, 19891, 887, 73, 9999999999999999999869, 863, 61, 9851, 1999999999849, 43, 37, 9833, 827, 821, 19, 809, 7, 3, 1801, 1789

Offset: 1

Steven Lu, Mar 12 2025

nonn

a(1) is not the only term equal to 0.
For example, a(37145)=0, since prime(37145)=442609, and:
  10^k - 442609 is a multiple of 3, for k>=6,
  2*10^(2*k) - 442609 is a multiple of 11, for k>=3,
  2*10^(6*k+1) - 442609 is a multiple of 7, for k>=1,
  2*10^(6*k+3) - 442609 is a multiple of 13, for k>=1,
  2*10^(6*k+5) - 442609 is a multiple of 37, for k>=1.

			For n=11 (prime(n)=31):
For all positive integer k, 10^k-31 is multiple of 3.
200 - 31 = 169 = 13 * 13
2000 - 31 = 1969 = 11 * 179
20000 - 31 = 19969 = 19 * 1051
200000 - 31 = 199969 = 7 * 7 * 7 * 11 * 53
2000000 - 31 = 1999969 is a prime number.
thus a(11) = 1999969.

Cf. A191474 (base 2 version of this sequence).

Table[If[MissingQ[#], 0, # - Prime[i]] &@SelectFirst[Flatten[Table[{10^j, 2 10^j}, {j, 100}]], # > Prime[i] && PrimeQ[# - Prime[i]] &], {i, 1, 47}]

A381287 a(n) is the smallest nonnegative number congruent to k modulo prime(k)^(n-k+1) for k=1..n.

Original entry on oeis.org

1, 5, 353, 65153, 119966753, 3050486978753, 563678198162618753, 15413934869729743026218753, 1710386933322832904060816574218753, 14712401204424400291787297607394206774218753, 5027982881016562571248237683551040219315980699574218753, 5488604004979149030407333271782173318791620565366546226763574218753

Offset: 1

Steven Lu, Feb 19 2025

nonn

This sequence is an example demonstrating how an integer sequence (thus a rational number sequence) converges to distinct limits in all p-adic systems; that is, converges to 1 in 2-adic, to 2 in 3-adic, to k in prime(k)-adic, and so on.

Moreover, the rational number sequence a(n) / prime(n+1) ^ (primorial(n)^(n-1) * A005867(n)) converges to distinct limits in all p-adic systems as well as the real number system, with limit zero in real numbers, and limit k in prime(k)-adic, where k is any positive integer.

			For n=3, a(3)=353 since 353 is the smallest nonnegative integer x satisfying:
  x == 1 (mod 2^3),
  x == 2 (mod 3^2),
  x == 3 (mod 5^1).

Steven Lu, Table of n, a(n) for n = 1..37

Cf. A006939, A005687.

ToString[Table[ChineseRemainder[Range[n], (Prime /@ Range[n])^Range[n, 1, -1]], {n, 12}]]

A379881 Prime numbers of the form 8x^2 + 27y^2 where x and y are positive integers.

Original entry on oeis.org

59, 227, 251, 419, 443, 683, 827, 1187, 1451, 1523, 1811, 2027, 2243, 2339, 2579, 2699, 3299, 3371, 3467, 3539, 3659, 3779, 4211, 4259, 4523, 4547, 4691, 5387, 5531, 5651, 6131, 6203, 6299, 6323, 6947, 6971, 7043, 7187, 7451, 7499, 7643, 8123, 8219, 8363, 8387, 8867, 8963, 9011, 9371, 9491, 9539, 9851, 9923

Offset: 1

Steven Lu, Feb 16 2025

All terms are congruent to 11 modulo 24.

			59 = 8 * 2^2 + 27 * 1^2
227 = 8 * 5^2 + 27 * 1^2
251 = 8 * 1^2 + 27 * 3^2

Intersection of A107161 and A002145.

With[{limit = 10000}, Sort[DeleteDuplicates[Select[Flatten[Table[8 x^2 + 27 y^2, {x, Floor[Sqrt[limit/8]]}, {y, Floor[Sqrt[limit/27]]}]], PrimeQ[#] && # < limit &]]]]

A377476 Primes p such that 1..12 are quadratic residues modulo p.

Original entry on oeis.org

479, 1151, 1319, 1559, 2351, 2689, 2999, 3529, 3671, 3911, 4751, 4919, 5519, 5569, 5711, 6551, 6599, 7559, 7561, 7681, 8089, 8761, 8951, 9239, 9241, 9601, 9719, 9769, 10391, 10559, 10799, 12049, 12239, 12721, 12911, 13151, 13729, 14159, 14281, 14759, 14951, 15671, 15791, 16631, 16921

Offset: 1

Steven Lu, Feb 16 2025

nonn

An odd prime p is a term if and only if the Legendre symbol Legendre(q|p) = 1 for all q = 2,3,5,7,11; i.e., each prime q <= 12 is a quadratic residue.
Prime p is a term if and only if all the following conditions are satisfied:
  p == +-1 (mod 24)
  p == +-1 (mod 10)
  p == +-1, +-3, +-9 (mod 28)
  p == +-1, +-5, +-7, +-9, +-19 (mod 44)
Prime p is a term if and only if it is congruent to any number in the attached file modulo 9240.

			479 is a term of this sequence, since Legendre(b|479) = 1 for b = 1, 2, ..., 12.

Steven Lu, Congruence classes modulo 9240 of the terms

Select[Prime /@ Range[2000], And @@ Table[KroneckerSymbol[b, #] == 1, {b, Range[12]}] &]

isok(p)={for(i=1, 12, if(kronecker(i,p)<0, return(0))); isprime(p)} \\ Andrew Howroyd, Feb 17 2025

A380784 Prime numbers p where the cyclotomic field Q(zeta_(p-1)) has class number one.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 61, 67, 71

Offset: 1

Steven Lu, Feb 02 2025

For a prime number p, the cyclotomic field of power p-1 can take a significant part in Z/pZ or p-adic field Q_p, since 1~p-1 are all (p-1)-th unit roots in Z/pZ. It would be much better if the cyclotomic integer ring is a unique factorization domain.

A prime number p is in this sequence if and only if (p-1)/2 is in A005848 (if p equals 3 modulus 4) or p-1 is in A005848 (otherwise).

Cf. A005848.

A380832 Number of points in Z^4 of norm <= n where the sum of the four entries is even.

Original entry on oeis.org

1, 1, 49, 169, 625, 1465, 3337, 5689, 10009, 15937, 24865, 35761, 51265, 69817, 94849, 124009, 161497, 204529, 260137, 320497, 394705, 478705, 577489, 687913, 819313, 960457, 1127785, 1309153, 1517161, 1742497, 2001505, 2273473, 2585905, 2920009, 3297337, 3700153, 4144105, 4618657, 5145865, 5703073

Offset: 0

Steven Lu, Feb 05 2025

nonn

Points in Z^4 with even sum of entries forms the D_4 lattice. That is to say, the sequence is the "ball" pattern on D_4 lattice.

a(n) == 1 (mod 24).

			a(2) = 49, because in the ball with radius 2, there is 1 point (0,0,0,0), 8 points similar to (0,0,0,2), 24 points similar to (0,0,1,1), and 16 points similar to (1,1,1,1).

Steven Lu, Python program

Cf. A055410.

a(n) = sum(x=-n, n, sum(y=-n, n, sum(z=-n, n, sum(t=-n, n, (((x+y+z+t) % 2)==0) && (x^2+y^2+z^2+t^2 <=n^2))))); \\ Michel Marcus, Feb 09 2025

Python
```
# See Steven Lu's link
```

A380782 Class number of real quadratic field Q(sqrt(prime(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 3, 1

Offset: 1

Steven Lu, Feb 02 2025

nonn

A278837 contains all primes p where the class number of Q(sqrt(p)) is larger than 1.

			For n = 22, a(22) = 3 since the class number of Q(sqrt(79)) is 3 where 79 is the 22nd prime.

Cf. A003172, A076498, A278837.

Table[NumberFieldClassNumber[Sqrt[Prime[i]]], {i, 87}]

cp's OEIS Frontend

User: Steven Lu

A382863 a(2*k-1) and a(2*k) are a pair of prime numbers where 9*a(2*k-1) and 8*a(2*k) are neighboring integers.

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A382546 Positive integers whose prime factors are all in A219528.

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A382066 a(n) = Sum_{k=1..prime(n)-1} (-k/prime(n)) * 3^(k-1) / 2, where (p/q) is the Legendre symbol of p and q.

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A382035 a(n) is the smallest prime q such that q + prime(n) is of form 10^k or 2*10^k, or 0 if no such prime exists.

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A381287 a(n) is the smallest nonnegative number congruent to k modulo prime(k)^(n-k+1) for k=1..n.

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A379881 Prime numbers of the form 8*x^2 + 27*y^2 where x and y are positive integers.

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A377476 Primes p such that 1..12 are quadratic residues modulo p.

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PARI

A380784 Prime numbers p where the cyclotomic field Q(zeta_(p-1)) has class number one.

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A380832 Number of points in Z^4 of norm <= n where the sum of the four entries is even.

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A382863 a(2k-1) and a(2k) are a pair of prime numbers where 9a(2k-1) and 8a(2k) are neighboring integers.

A379881 Prime numbers of the form 8x^2 + 27y^2 where x and y are positive integers.