A047845 -id:A047845 - cp's OEIS frontend

A345447 Numbers of the form i+j+2ij and 2+i+j+2ij for i,j >= 1.

4, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Davide Rotondo, Jun 19 2021

nonn

Except for 1 and 2 the complement sequence c is: 3, 5, 8, 11, 20, 23, 35, 41, 50, 53, 56, 65, ...; where 2*c(i) + 1 and 2*c(i) - 3 are a pair of cousin primes. This is a consequence of the sieve of Sundaram.

			For i,j = 1, 1+1+2*1*1 = 4 and 2+1+1+2*1*1 = 6.

Wikipedia, Sieve of Sundaram.

Cf. A046132, A023200.

Union of A047845 and A153043, except for 0 and 2.

def aupto(limit):
    aset = set()
    for i in range(1, limit//3):
        for j in range(i, limit//3):
            t = i + j + 2*i*j
            if t > limit: break
            aset.update([t, t+2])
    return sorted(an for an in aset if an <= limit)
print(aupto(80)) # Michael S. Branicky, Jul 05 2021

A367647 Irregular triangle read by rows in which row n lists the positive values k such that there are no primes between kn and k(n + 1), or -1 if no such k exists.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 3, 4, 1, 2, 3, 2, 4, 2, 9, 1, 1, 6, 2, 3, 2, 7, 3, 5, 2, 6, 1, 6, 7, 10, 1, 3, 4, 2, 4, 5, 1, 2, 5, 1, 2, 3, 8, 1, 7, 1, 2, 2, 3, 5, 4, 7, 11, 3, 4, 2, 3, 1, 2, 10, 1, 4, 9, 1, 2, 6, 1, 4, 15, 4, 6, 2, 5, 8, 1, 2, 3, 4, 1, 3, 2, 3, 5, 8, 3, 5, 2, 4, 7, 2, 5, 1, 3, 12, 1, 2, 2, 4, 7

Offset: 1

Juri-Stepan Gerasimov, Nov 25 2023

			Triangle begins:
1;
1;
1;
2;
1;
4;
2;
1, 3, 4;
1;
2;
3;
2, 4;
2, 9;
1;
1, 6;
2, 3;
2, 7;
3, 5;
2, 6;
1, 6, 7, 10;
1, 3, 4;
2;
4, 5;
1, 2, 5;
1, 2, 3, 8.

Cf. A068780, A047845, A110835, A278425.

A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2k + 1) while 2k+1 is not prime.

Original entry on oeis.org

1023, 1638, 14670, 21399, 24570, 40290, 44178, 45375, 52326, 98046, 128499, 135975, 157410, 229494, 244998, 257223, 370875, 400302, 419430, 436590, 458163, 502326, 625974, 686826, 754854, 839270, 905786, 993510, 1102983, 1134546, 1142226, 1152083, 1193898, 1373238, 1374011

Offset: 1

Michel Marcus, Feb 05 2025

nonn

Intersection of A374913 and A047845.

isok(k) = (!isprime(2*k+1)) && (Mod(k, 2*k+1)^(k+1) == k+1);

cp's OEIS Frontend

A345447 Numbers of the form i+j+2ij and 2+i+j+2ij for i,j >= 1.

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A367647 Irregular triangle read by rows in which row n lists the positive values k such that there are no primes between kn and k(n + 1), or -1 if no such k exists.

Views

Author

Keywords

Examples

Crossrefs

A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2k + 1) while 2k+1 is not prime.

Views

Author

Keywords

Crossrefs

Programs

PARI

cp's OEIS Frontend

A345447 Numbers of the form i+j+2*i*j and 2+i+j+2*i*j for i,j >= 1.

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Author

Keywords

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Crossrefs

Programs

Python

A367647 Irregular triangle read by rows in which row n lists the positive values k such that there are no primes between k*n and k*(n + 1), or -1 if no such k exists.

Views

Author

Keywords

Examples

Crossrefs

A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2*k + 1) while 2*k+1 is not prime.

Views

Author

Keywords

Crossrefs

Programs

PARI

A345447 Numbers of the form i+j+2ij and 2+i+j+2ij for i,j >= 1.

A367647 Irregular triangle read by rows in which row n lists the positive values k such that there are no primes between kn and k(n + 1), or -1 if no such k exists.

A380831 Numbers k such that k^(k + 1) == k + 1 (mod 2k + 1) while 2k+1 is not prime.