A034953 -id:A034953

A085739

Partial sums of A034953(n).

+20
2

3, 9, 24, 52, 118, 209, 362, 552, 828, 1263, 1759, 2462, 3323, 4269, 5397, 6828, 8598, 10489, 12767, 15323, 18024, 21184, 24670, 28675, 33428, 38579, 43935, 49713, 55708, 62149, 70277, 78923, 88376, 98106, 109281, 120757, 133160, 146526, 160554

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A034953.

A006254

Numbers k such that 2k-1 is prime.

+10
85

2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157

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COMMENTS

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

A067076

Numbers k such that 2*k + 3 is a prime.

+10
65

0, 1, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 22, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88, 89, 94, 95, 97, 98, 104, 110, 112, 113, 115, 118, 119, 124, 127, 130, 133, 134, 137, 139, 140, 145, 152, 154, 155

(list; graph; refs; listen; history; text; internal format)

COMMENTS

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

A008837

a(n) = p*(p-1)/2 for p = prime(n).

+10
30

1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528, 19306

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COMMENTS

Whereas A034953 is the sequence of triangular numbers with prime indices, this is the sequence of triangular numbers with numbers one less than primes for indices. - Alonso del Arte, Aug 17 2014

CROSSREFS

Cf. A000010, A000040, A072230, A034953, A271780.

A054269

Length of period of continued fraction for sqrt(prime(n)).

+10
18

1, 2, 1, 4, 2, 5, 1, 6, 4, 5, 8, 1, 3, 10, 4, 5, 6, 11, 10, 8, 7, 4, 2, 5, 11, 1, 12, 6, 15, 9, 12, 6, 9, 18, 9, 20, 17, 18, 4, 5, 14, 21, 16, 13, 1, 20, 26, 4, 2, 5, 11, 12, 17, 14, 1, 12, 3, 24, 21, 13, 18, 5, 14, 16, 17, 11, 34, 19, 14, 7, 15, 4, 20, 5, 30, 8, 9, 21, 1, 21, 18, 37, 16

(list; graph; refs; listen; history; text; internal format)

COMMENTS

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

A127917

Product of three numbers: n-th prime, previous number, and following number.

+10
15

6, 24, 120, 336, 1320, 2184, 4896, 6840, 12144, 24360, 29760, 50616, 68880, 79464, 103776, 148824, 205320, 226920, 300696, 357840, 388944, 492960, 571704, 704880, 912576, 1030200, 1092624, 1224936, 1294920, 1442784, 2048256, 2247960, 2571216, 2685480, 3307800

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A036689, A034953, A084920, A117762.

A036690

Product of a prime and the following number.

+10
14

6, 12, 30, 56, 132, 182, 306, 380, 552, 870, 992, 1406, 1722, 1892, 2256, 2862, 3540, 3782, 4556, 5112, 5402, 6320, 6972, 8010, 9506, 10302, 10712, 11556, 11990, 12882, 16256, 17292, 18906, 19460, 22350, 22952, 24806, 26732, 28056, 30102

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) = 2*A034953(n). - Artur Jasinski, Feb 06 2007

CROSSREFS

Cf. A036689, A034953, A065463, A179119, A306633.

A011756

a(n) = prime(n*(n+1)/2).

+10
10

2, 5, 13, 29, 47, 73, 107, 151, 197, 257, 317, 397, 467, 571, 659, 769, 883, 1019, 1151, 1291, 1453, 1607, 1783, 1987, 2153, 2371, 2593, 2791, 3037, 3307, 3541, 3797, 4073, 4357, 4657, 4973, 5303, 5641, 5939, 6301, 6679, 7019, 7477

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A000040, A000217, A034953.

A127918

Half of product of three numbers: n-th prime, previous and following number.

+10
9

3, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A036689, A034953, A127917, A127920.

A127919

1/3 of product of three numbers: the n-th prime, the previous number and the following number.

+10
9

2, 8, 40, 112, 440, 728, 1632, 2280, 4048, 8120, 9920, 16872, 22960, 26488, 34592, 49608, 68440, 75640, 100232, 119280, 129648, 164320, 190568, 234960, 304192, 343400, 364208, 408312, 431640, 480928, 682752, 749320, 857072, 895160, 1102600

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A036689, A034953, A127917, A127918, A127920.