A138694 -id:A138694 - cp's OEIS frontend

A144313 Prime numbers p such that p - 1 is the fourth a-figurate number, seventh b-figurate number and possibly tenth c-figurate number for some a, b and c and not a d-figurate number for any nontrivial d.

29, 71, 113, 239, 281, 449, 491, 659, 701, 827, 911, 953, 1373, 1499, 1583, 1667, 1709, 1877, 2003, 2087, 2129, 2213, 2339, 2423, 2549, 2591, 2633, 2801, 2843, 2969, 3221, 3347, 3389, 3557, 3767, 3851, 4229, 4271, 4397, 4481, 4649, 4691, 4733, 5153, 5279

Offset: 1

Reikku Kulon, Sep 17 2008

Appears to be necessarily a subset of A007528.
The 46th Mersenne prime exponent (Mpe, A000043) 43112609 is a member: 43112608 is the fourth 7185436-figurate number and the seventh 2052983-figurate number and is not a k-figurate number for any other k except 43112608 (trivially). Several other Mersenne prime exponents are members of this sequence.
It is conjectured:
- that this sequence is infinite;
- that there is a unique set {4, 7, 10, 16, ...} (A138694?) giving the possible orders in k-figurate numbers for the set S of all Mpe for which Mpe - 1 is a (4, 7) or (4, 10) k-figurate number;
- that the ratio of Mpe in this sequence to those not approaches a nonzero value;
- that a characteristic function f(n) exists which equals 1 iff n is in S.
Contribution from Reikku Kulon, Sep 18 2008: (Start)
Subset of the integers n such that n is congruent to 29 modulo 42. The case where p - 1 is a tenth c-figurate number occurs when p is also congruent to 281 modulo 630.
The first three primes where c is defined are 281, 911 and 2801, with c = 8, 22, 64; c is congruent to 8 modulo 14. All such primes are necessarily congruent to 1 modulo 10.
The first invalid values of c are 36 and 50, which correspond to the semiprimes 1541 = 23 * 67 and 2171 = 13 * 167. Both of these are members of A071331 and A098237. The next invalid value of c, 78, corresponds to 3431 = 47 * 73, once again a member of both sequences.
The first primes where a, b, c and d are all defined (which therefore excludes them from this sequence) are the consecutive 6581, 7211 and 7841, all members of A140856, A140732, A142076, A142317 and A142905. (End)

Cf. A000040, A007528, A138694, A131500, A096676, A000043, A000668
Contribution from Reikku Kulon, Sep 18 2008: (Start)
Cf. A071331, A098237 (semiprimes)
Cf. A140856, A140732, A142076, A142317, A142905 (a, b, c and d all defined) (End)

A138691 Numbers of the form 68+p^2 (where p is a prime).

Original entry on oeis.org

72, 77, 93, 117, 189, 237, 357, 429, 597, 909, 1029, 1437, 1749, 1917, 2277, 2877, 3549, 3789, 4557, 5109, 5397, 6309, 6957, 7989, 9477, 10269, 10677, 11517, 11949, 12837, 16197, 17229, 18837, 19389, 22269, 22869, 24717, 26637, 27957, 29997

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

68 + p^2 is divisible by 3 for any prime p > 3. - M. F. Hasler

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Table[2*34 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[50]]^2+68 (* Harvey P. Dale, Oct 19 2011 *)

forprime(p=1, 1e2, print1(68+p^2, ", ")) \\ Felix Fröhlich, Jul 07 2014

A138692 Numbers of the form 86+p^2 (where p is a prime).

Original entry on oeis.org

90, 95, 111, 135, 207, 255, 375, 447, 615, 927, 1047, 1455, 1767, 1935, 2295, 2895, 3567, 3807, 4575, 5127, 5415, 6327, 6975, 8007, 9495, 10287, 10695, 11535, 11967, 12855, 16215, 17247, 18855, 19407, 22287, 22887, 24735, 26655, 27975, 30015

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

86+p^2 is divisible by 3, for any prime p>3. - M. F. Hasler

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Table[2*43 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[40]]^2+86 (* Harvey P. Dale, Jan 31 2020 *)

A138693 Numbers of the form 110 + p^2. (where p is a prime).

Original entry on oeis.org

114, 119, 135, 159, 231, 279, 399, 471, 639, 951, 1071, 1479, 1791, 1959, 2319, 2919, 3591, 3831, 4599, 5151, 5439, 6351, 6999, 8031, 9519, 10311, 10719, 11559, 11991, 12879, 16239, 17271, 18879, 19431, 22311, 22911, 24759, 26679, 27999, 30039

Offset: 1

Artur Jasinski, Mar 26 2008

110+p^2 is divisible by 3, for any prime p>3. - M. F. Hasler

Cf. A000040, A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

A138693:=n->110+ithprime(n)^2: seq(A138693(n), n=1..50); # Wesley Ivan Hurt, Sep 13 2014

Table[2*55 + Prime[p + 1]^2, {p, 0, 100}]
#^2+110&/@Prime[Range[40]] (* Harvey P. Dale, Jan 21 2012 *)

a(n) = 110 + prime(n)^2; \\ Michel Marcus, Sep 14 2014

a(n) = 110 + A000040(n)^2. - Wesley Ivan Hurt, Sep 13 2014

a(n) = 110 + A001248(n). - Michel Marcus, Sep 14 2014

A144326 Prime numbers that cannot be Mersenne prime exponents, by conjecture of A144325.

Original entry on oeis.org

67, 191, 197, 211, 277, 331, 379, 397, 401, 541, 617, 631, 677, 727, 743, 751, 821, 937, 947, 971, 991, 1129, 1163, 1171, 1217, 1277, 1289, 1327, 1381, 1409, 1427, 1471, 1549, 1559, 1597, 1601, 1607, 1783, 1801, 1831, 1871, 1901, 2011, 2017, 2081, 2111

Offset: 1

Reikku Kulon, Sep 17 2008

Obviously true for the initial terms!

Conjecture: 191, 1217, 1559 and 1901 are not in fact members of this sequence, noting that they are (4, 19) k-figurate numbers; 19 is a member of A138694. Determining whether a Mersenne prime exponent one greater than a (4, 19) k-figurate number exists is sufficient to determine whether these primes are members.

Cf. A000040, A000043, A000668, A144313, A144315, A144325, A138694

A138690 Numbers of the form 56+p^2 (where p is a prime).

Original entry on oeis.org

60, 65, 81, 105, 177, 225, 345, 417, 585, 897, 1017, 1425, 1737, 1905, 2265, 2865, 3537, 3777, 4545, 5097, 5385, 6297, 6945, 7977, 9465, 10257, 10665, 11505, 11937, 12825, 16185, 17217, 18825, 19377, 22257, 22857, 24705, 26625, 27945, 29985

Offset: 1

Artur Jasinski, Mar 26 2008

nonn

56+p^2 is divisible by 3 for any prime p>3. - M. F. Hasler

Cf. A138479, A138685, A138686, A138691, A138692, A138693, A138694, A007528, A138696, A138697, A138698, A138699, A138700.

Table[2*28 + Prime[p + 1]^2, {p, 0, 100}]
Prime[Range[40]]^2+56 (* Harvey P. Dale, Dec 15 2015 *)

cp's OEIS Frontend

A144313 Prime numbers p such that p - 1 is the fourth a-figurate number, seventh b-figurate number and possibly tenth c-figurate number for some a, b and c and not a d-figurate number for any nontrivial d.

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A138691 Numbers of the form 68+p^2 (where p is a prime).

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A138692 Numbers of the form 86+p^2 (where p is a prime).

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A138693 Numbers of the form 110 + p^2. (where p is a prime).

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A144326 Prime numbers that cannot be Mersenne prime exponents, by conjecture of A144325.

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A138690 Numbers of the form 56+p^2 (where p is a prime).

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Mathematica