Angad Singh - cp's OEIS frontend

A362222 Slowest increasing sequence where a(n) + n^2 is a prime.

1, 3, 4, 7, 12, 17, 18, 19, 20, 27, 28, 29, 30, 31, 32, 37, 42, 43, 48, 49, 50, 57, 58, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 93, 96, 97, 100, 103, 104, 105, 124, 133, 138, 147, 148, 153, 154, 163, 166, 171, 184, 193, 196, 197, 198, 205

Offset: 1

Angad Singh, Apr 11 2023

nonn

			a(2) = 3, since the smallest number greater than all the previous terms which gives a prime when added to 2^2 is 3.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A053000, A107819.

R:= 1: t:= 1:
for n from 2 to 100 do
  t:= nextprime(t+n^2)-n^2;
  R:= R,t
od:
R; # Robert Israel, Apr 11 2023

a[n_] := a[n] = Module[{k = a[n - 1] + 1}, While[! PrimeQ[n^2 + k], k++]; k]; a[0] = 0; Array[a, 100] (* Amiram Eldar, Apr 12 2023 *)

seq(n)={my(a=vector(n), p=0); for(n=1, #a, p++; while(!isprime(p+n^2), p++); a[n]=p); a} \\ Andrew Howroyd, Apr 11 2023

from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    an = 1
    for n in count(2):
        yield an
        an = nextprime(an + n**2) - n**2
print(list(islice(agen(), 62))) # Michael S. Branicky, Apr 16 2023

A356364 Number of primes p of the form k^2 + 1 less than 10^n such that p+2 and 2p+1 are also primes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 10, 18, 43, 86, 185, 449, 1091, 2764, 6978, 17951, 47146, 125507, 337600, 916229, 2504458, 6898908

Offset: 1

Angad Singh, Oct 16 2022

			For n = 5, a(5) = 2 since 5 and 25601 are the only two such primes less than or equal to 10^5.

Cf. A006880, A007508, A083844, A092816, A347934.

seq[nmax_] := Module[{c = 0, pow = 10, s = {}, p}, Do[p = k^2 + 1; If[PrimeQ[p] && PrimeQ[p + 2] && PrimeQ[2*p + 1], c++]; If[p > pow, pow *= 10; AppendTo[s, c]], {k, 1, Floor[10^(nmax/2)] + 1}]; s]; seq[13] (* Amiram Eldar, Oct 16 2022 *)

A355778 Numbers k such that both k and k^2 + 2 can be written as the sum of two nonzero squares.

Original entry on oeis.org

40, 68, 72, 104, 148, 180, 320, 392, 468, 544, 612, 648, 720, 788, 832, 900, 936, 968, 1040, 1044, 1156, 1192, 1256, 1300, 1332, 1508, 1732, 1796, 1800, 1832, 1872, 1940, 2056, 2196, 2308, 2336, 2372, 2448, 2664, 2696, 2740, 2804, 2848, 2880, 3060, 3200, 3280

Offset: 1

Angad Singh, Jul 16 2022

nonn

The terms in this sequence can be considered as a solution to the "near miss" problem which occurs frequently while solving Diophantine equations. It is known that if a number k can be written as the sum of two nonzero distinct squares then so can k^2 and k^2+1. Thus, finding numbers k such that k^2+2 satisfies the same property makes it quite interesting.

			40 is a term since 40 = 2^2 + 6^2 as well as 40^2 + 2 = 1602 = 9^2 + 39^2.
320 is a term since 320 = 8^2 + 16^2 as well as 320^2 + 2 = 102402 = 201^2 + 249^2.

Cf. A000290, A000404, A000415.

is1(n)= for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2)); \\ A000404
isok(k) = is1(k) && is1(k^2+2); \\ Michel Marcus, Jul 18 2022

from itertools import count, islice
from sympy import factorint
def A355778_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue, 1)):
        c = False
        for p in (f:=factorint(n)):
            if (q:= p & 3)==3 and f[p]&1:
                break
            elif q == 1:
                c = True
        else:
            if c or f.get(2, 0)&1:
                c = False
                for p in (f:=factorint(n**2+2)):
                    if (q:= p & 3)==3 and f[p]&1:
                        break
                    elif q == 1:
                        c = True
                else:
                    if c or f.get(2, 0)&1:
                        yield n
A355778_list = list(islice(A355778_gen(),30)) # Chai Wah Wu, Sep 14 2022

A355769 Numbers k such that both k and k+1 can be written as the sum of two nonzero squares.

Original entry on oeis.org

17, 25, 40, 52, 72, 73, 89, 97, 100, 116, 136, 145, 148, 169, 180, 193, 225, 232, 233, 241, 244, 260, 288, 289, 292, 305, 313, 337, 369, 388, 400, 404, 409, 424, 449, 457, 481, 520, 521, 544, 548, 577, 584, 585, 592, 612, 625, 628, 640, 656, 673, 676, 697, 724

Offset: 1

Angad Singh, Jul 16 2022

nonn

The numbers in the sequence are useful in solving various second-degree Diophantine equations.

The identity (3n-12)^2 + (4n-12)^2 + 1 = (3n-8)^2 + (4n-15)^2 proves that there are infinitely many such numbers in this sequence.

			17 is a term since 17 = 4^2 + 1^2 and 17 + 1 = 18 = 3^2 + 3^2.
169 is a term since 169 = 5^2 + 12^2 and 169 + 1 = 170 = 1^2 + 13^2.

Cf. A000290, A000404, A000415.

is1(n)= for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2)); \\ A000404
isok(k) = is1(k) && is1(k+1); \\ Michel Marcus, Jul 18 2022

A354626 Numbers that can't be written as the sum of a Fibonacci number and the square of a Fibonacci number.

Original entry on oeis.org

15, 16, 18, 19, 20, 23, 24, 29, 31, 32, 36, 37, 39, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 60, 61, 62, 63, 68, 70, 71, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115

Offset: 1

Angad Singh, Jul 09 2022

nonn

			16 is a term since there does not exist any pair of integers i,j >= 0 such that Fibonacci(i) + Fibonacci(j)^2 = 16.

Cf. A000045, A007598, A059727.

Numbers k such that the coefficient of x^k in the product (Sum_{i>=0} x^Fibonacci(i)) * (Sum_{j>=0} x^(Fibonacci(j)^2)) is 0.

A347934 Primes p of the form k^2 + 1 such that p+2 and 2p+1 are also prime.

Original entry on oeis.org

5, 25601, 193601, 1144901, 1464101, 4326401, 6100901, 12390401, 23522501, 72931601, 127012901, 245862401, 256960901, 351937601, 441840401, 732784901, 802588901, 951722501, 1621672901, 2024100101, 2070250001, 2217468101, 2219352101, 2428518401, 2930056901, 2963713601

Offset: 1

Angad Singh, Sep 20 2021

nonn

All primes (except 5) in the sequence are of the form 100*k^2 + 1.

Intersection of A002496 and A045536.

Select[Range[50000]^2 + 1, AllTrue[{#, # + 2, 2*# + 1}, PrimeQ] &] (* Amiram Eldar, Sep 20 2021 *)

A347932 Least m such that the first n digits (after the decimal point) of e and Sum_{k=0..m} 1/k! are the same.

Original entry on oeis.org

4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 16, 16, 16, 17, 18, 19, 20, 20, 22, 22, 22, 23, 24, 24, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47, 48, 48, 49, 50, 50

Offset: 1

Angad Singh, Sep 20 2021

Cf. A001113.

a[n_] := Module[{e = Floor[10^n*E], s = 0, k = 0}, While[Floor[10^n*s] != e, s += 1/k!; k++]; k - 1]; Array[a, 65] (* Amiram Eldar, Sep 20 2021 *)

A347038 Primes p such that there are no solutions to d(k+p) = sigma(k).

Original entry on oeis.org

29, 37, 41, 53, 67, 89, 101, 109, 113, 127, 137, 151, 157, 173, 181, 197, 227, 229, 233, 257, 269, 277, 281, 293, 313, 349, 373, 389, 401, 409, 421, 439, 461, 557, 587, 593, 601, 613, 617, 641, 643, 653, 661, 673, 677, 701, 709, 739, 761, 773, 787, 821, 829

Offset: 1

Angad Singh, Aug 12 2021

nonn

If p is not in the sequence and d(k+p) = sigma(k), then k <= 1+2*sqrt(p). Proof: We have d(m) <= 2*sqrt(m) (see formula in A000005), so 2*sqrt(k+p) >= d(k+p) = sigma(k) >= k+1 (if k > 1). After squaring and simplifying, we get k <= 1+2*sqrt(p). - Pontus von Brömssen, Aug 20 2021

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A247485.

filter:= proc(p) isprime(p) and not ormap(k -> numtheory:-tau(k+p) = numtheory:-sigma(k), [$1 .. 1 + 2*isqrt(p)]) end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Aug 06 2025

from sympy import divisor_count as d, divisor_sigma as sigma, primerange
from math import isqrt
def A347038_list(pmax):
    a = []
    for p in primerange(2, pmax + 1):
        if not any(d(k + p) == sigma(k) for k in range(1, 2 + isqrt(4 * p))):
            a.append(p)
    return a  # Pontus von Brömssen, Aug 20 2021

A345746 Decimal expansion of Pi^(-1/9).

Original entry on oeis.org

8, 8, 0, 5, 6, 4, 4, 0, 3, 4, 5, 2, 2, 1, 5, 0, 9, 5, 3, 2, 6, 6, 6, 4, 7, 9, 1, 9, 4, 5, 0, 0, 8, 6, 3, 5, 4, 7, 5, 2, 4, 9, 8, 8, 6, 7, 4, 0, 3, 8, 1, 1, 8, 6, 3, 5, 7, 7, 5, 1, 4, 0, 8, 3, 9, 2, 6, 7, 6, 7, 4, 8, 7, 5, 9, 4, 6, 3, 1, 5, 1, 3, 1, 0, 0, 2, 3, 7, 9, 9, 6, 3

Offset: 0

Angad Singh, Jun 25 2021

			0.880564403452215095326664791945008635475249886...

Index entries for transcendental numbers

Cf. A000796 (Pi), A049541 (1/Pi), A093209 (Pi^(-1/8)).

RealDigits[Pi^(-1/9), 10, 100][[1]] (* Amiram Eldar, Jun 29 2021 *)

A344209 Decimal expansion of zeta(sqrt(2)).

Original entry on oeis.org

3, 0, 2, 0, 7, 3, 7, 6, 7, 9, 4, 8, 6, 0, 3, 2, 6, 6, 8, 2, 7, 0, 9, 9, 2, 2, 8, 3, 7, 0, 0, 4, 0, 9, 0, 1, 8, 6, 5, 3, 6, 7, 7, 5, 9, 6, 4, 5, 8, 3, 7, 2, 6, 9, 9, 7, 7, 9, 7, 9, 1, 9, 1, 6, 3, 9, 3, 0, 4, 1, 0, 5, 8, 0, 7, 5, 2, 6, 8, 9, 9, 8, 8, 6, 1, 9, 1, 8, 9, 3, 0, 6, 0, 9, 7, 8, 7, 2, 7, 2, 1, 7, 6, 5, 8, 5, 7, 9

Offset: 1

Angad Singh, May 12 2021

			3.0207376794860326682709922837004090186536775964583726997797919163...

Cf. A002193.

RealDigits[Zeta[Sqrt[2]], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

zeta(sqrt(2)) \\ Michel Marcus, Jun 15 2021

Equals Sum_{k>=1} 1/k^(sqrt(2)).

cp's OEIS Frontend

User: Angad Singh

A362222 Slowest increasing sequence where a(n) + n^2 is a prime.

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A356364 Number of primes p of the form k^2 + 1 less than 10^n such that p+2 and 2p+1 are also primes.

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A355778 Numbers k such that both k and k^2 + 2 can be written as the sum of two nonzero squares.

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A355769 Numbers k such that both k and k+1 can be written as the sum of two nonzero squares.

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A354626 Numbers that can't be written as the sum of a Fibonacci number and the square of a Fibonacci number.

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Formula

A347934 Primes p of the form k^2 + 1 such that p+2 and 2p+1 are also prime.

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A347932 Least m such that the first n digits (after the decimal point) of e and Sum_{k=0..m} 1/k! are the same.

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Mathematica

A347038 Primes p such that there are no solutions to d(k+p) = sigma(k).

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Maple

Python

A345746 Decimal expansion of Pi^(-1/9).

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