Jeff Brown - cp's OEIS frontend

A336535 a(n) = (m(n)^2 + 3)(m(n)^2 + 7)/32, where m(n) = 2n - 1.

1, 6, 28, 91, 231, 496, 946, 1653, 2701, 4186, 6216, 8911, 12403, 16836, 22366, 29161, 37401, 47278, 58996, 72771, 88831, 107416, 128778, 153181, 180901, 212226, 247456, 286903, 330891, 379756, 433846, 493521, 559153, 631126, 709836, 795691, 889111, 990528, 1100386, 1219141, 1347261, 1485226, 1633528, 1792671

Offset: 1

Jeff Brown, Jul 24 2020

For m(n) = 3,5,11, and 181, the perfect numbers (A000396), 6, 28, 496, and 33550336 are produced, respectively. 3,5,11, and 181 are the numbers m(n) such that (m(n)^2+7) is a power of 2. cf A038198.

The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A000217(n). - Miguel-Ángel Pérez García-Ortega, May 16 2025

			m(2) = 2*2-1 = 3 and (3^2+3)*(3^2+7)/32 = 6, so 6 is in the sequence.

David M. Burton, Elementary Number Theory, McGraw-Hill (2011), 25.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000396, A038198.
Cf. A002061, A014206.
Cf. A000217, A058919, A383833, A383834.

Table[((n^2+3)*(n^2+7))/32, {n,1,100,2}]
a=Table[(n(n+1))/2,{n,0,40}];Apply[Join,Map[{(#+1)(2#+1)}&,a]] (* Miguel-Ángel Pérez García-Ortega, May 16 2025 *)

a(n)=(1-n+n^2)*(2-n+n^2)/2 \\ Charles R Greathouse IV, Oct 21 2022

From Stefano Spezia, Jul 25 2020: (Start)
O.g.f.: x*(1 + x + 8*x^2 + x^3 + x^4)/(1 - x)^5.
a(n) = (1 - n + n^2)*(2 - n + n^2)/2.
a(n) = A002061(n)*A014206(n-1)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)
a(n) = (A000217(n-1)+1)*(2*A000217(n-1)+1). - Miguel-Ángel Pérez García-Ortega, May 16 2025

A332333 a(n) = A335778(n)/12.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 7, 11, 8, 31, 12, 34, 14, 15, 61, 39, 29, 86, 34, 116, 44, 142, 51, 163, 79, 62, 193, 91, 68, 101, 103, 227, 76, 218, 235, 79, 91, 129, 99, 100, 143, 147, 149, 107, 310, 473, 343, 171, 125, 394, 187, 401, 191, 142, 203, 147, 418, 460, 156

Offset: 1

Jeff Brown, Jul 06 2020

nonn

Cf. A335410, A335778.

Select[IntegerQ][Table[Sqrt[2*Prime[n]^2 - 2*Prime[n-1]^2]/12, {n, 37000}]]

A335778 The square roots of the perfect squares produced by A335410 in the order in which they appear.

Original entry on oeis.org

12, 12, 24, 36, 60, 84, 84, 132, 96, 372, 144, 408, 168, 180, 732, 468, 348, 1032, 408, 1392, 528, 1704, 612, 1956, 948, 744, 2316, 1092, 816, 1212, 1236, 2724, 912, 2616, 2820, 948, 1092, 1548, 1188, 1200, 1716

Offset: 1

Jeff Brown, Jun 22 2020

nonn

			11 is in sequence A335410.  2*(11^2 - 7^2) = 144 = 12^2.  Therefore, 12 is in this sequence.

Cf. A335410.

Select[IntegerQ][Table[Sqrt[2*Prime[n]^2 - 2*Prime[n-1]^2], {n,17000}]]

lista(nn) = {my(pp=2, s); forprime (p=3, nn, if (issquare(s=2*(p^2 - pp^2)), print1(sqrtint(s), ", ")); pp = p;);} \\ Michel Marcus, Jun 25 2020

A332615 Primes prime(k) such that 2*(prime(k)^2 - prime(k-1)^2) is a fourth power.

Original entry on oeis.org

83, 2593, 194483, 388963, 31505923, 57289763, 96059603, 99574273, 169869313, 276922883, 395254163, 414720001, 3264481603, 5125781251, 6059221283, 18233242723, 35888419873, 82012500001, 135304020001, 154550410643, 159004011043, 186320859203, 206710354243, 364488705443

Offset: 1

Jeff Brown, Jun 08 2020

nonn

This is a subset of A335410.

			Prime(23)=83. Prime(22)=79. 2*(83^2 - 79^2) = 6^4.
Prime(378)=2593. Prime(377)=2591. 2*(2593^2 - 2591^2) = 12^4.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 102 terms from Chai Wah Wu)

Cf. A001248, A335410.

Select[Prime@Range[2, 500000], IntegerQ@Sqrt[Sqrt[2(#^2 - NextPrime[#, -1]^2)]]&] (* a modification of Giovanni Resta's program for A335410 *)

isok(p) = isprime(p) && ispower(2*(p^2-precprime(p-1)^2), 4); \\ Michel Marcus, Jun 08 2020

lista(nn) = {my(pp=2); forprime(p=3, nn, if (ispower(2*(p^2 - pp^2), 4), print1(p, ", ")); pp = p;);} \\ Michel Marcus, Jun 08 2020

More terms from Amiram Eldar, Jun 08 2020

More terms from Giovanni Resta, Jun 08 2020

A335410 Primes prime(k) such that 2*(prime(k)^2 - prime(k-1)^2) is a perfect square.

Original entry on oeis.org

11, 19, 73, 83, 227, 443, 883, 1091, 1153, 1931, 2593, 2609, 3529, 4051, 7451, 13691, 15139, 16649, 20809, 26921, 34849, 45377, 46819, 53147, 56171, 69193, 74507, 74531, 83233, 91811, 95483, 103067, 103969, 106937, 110459, 112339, 149059, 149771, 176419, 180001

Offset: 1

Jeff Brown, Jun 06 2020

nonn

2*(prime(n)^2 - prime(n-1)^2) represents the integer coefficient of the difference in areas between the two circles passing through the origin with centers located at (prime(n), prime(n)) and (prime(n-1), prime(n-1)).

Among the first 200000 primes 2593 and 2609 are the only consecutive primes in this sequence.

			Prime(5) = 11, prime(4) = 7, 2*(11^2 - 7^2) = 12^2, so 11 is in the sequence.
Prime(559) = 4051, prime(558) = 4049, 2*(4051^2 - 4049^2) = 180^2, so 4051 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001248, A332615.

Select[Prime@ Range[2, 17000], IntegerQ@ Sqrt[2 (#^2 - NextPrime[#, -1]^2)] &] (* Giovanni Resta, Jun 06 2020 *)

lista(nn) = {my(pp=2); forprime (p=3, nn, if (issquare(2*(p^2 - pp^2)), print1(p, ", ")); pp = p;);} \\ Michel Marcus, Jun 25 2020

More terms from Giovanni Resta, Jun 06 2020

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User: Jeff Brown

A336535 a(n) = (m(n)^2 + 3)*(m(n)^2 + 7)/32, where m(n) = 2*n - 1.

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A332333 a(n) = A335778(n)/12.

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A335778 The square roots of the perfect squares produced by A335410 in the order in which they appear.

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A332615 Primes prime(k) such that 2*(prime(k)^2 - prime(k-1)^2) is a fourth power.

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A335410 Primes prime(k) such that 2*(prime(k)^2 - prime(k-1)^2) is a perfect square.

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A336535 a(n) = (m(n)^2 + 3)(m(n)^2 + 7)/32, where m(n) = 2n - 1.