A263951 -id:A263951 - cp's OEIS frontend

A247687 Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one.

9, 25, 49, 50, 98, 121, 169, 242, 289, 338, 361, 484, 529, 578, 676, 722, 841, 961, 1058, 1156, 1369, 1444, 1681, 1682, 1849, 1922, 2116, 2209, 2312, 2738, 2809, 2888, 3362, 3364, 3481, 3698, 3721, 3844, 4232, 4418, 4489, 5041, 5329, 5476, 5618, 6241, 6724, 6728, 6889, 6962, 7396, 7442, 7688, 7921, 8836, 8978, 9409

Offset: 1

Hartmut F. W. Hoft, Sep 22 2014

nonn

The symmetric representation of sigma(m) has 3 regions of width 1 where the two extremal regions each have 2^k - 1 legs and the central region starts with the p-th leg of the associated Dyck path for sigma(m) precisely when m = 2^(k - 1) * p^2 where 2^k < p <= row(m), k >= 1, p >= 3 is prime and row(m) = floor((sqrt(8*m + 1) - 1)/2). Furthermore, the areas of the two outer regions are (2^k - 1)*(p^2 + 1)/2 each so that the area of the central region is (2^k - 1)*p; for a proof see the link.
Since the sequence is defined by a two-parameter expression it can be written naturally as a triangle as shown in the Example section.
A263951 is a subsequence of this sequence containing the squares of all those primes p for which the areas of the 3 regions in the symmetric representation of p^2 (p once and (p^2 + 1)/2 twice) are primes; i.e., p^2 and p^2 + 1 are semiprimes (see A070552). - Hartmut F. W. Hoft, Aug 06 2020

			We show portions of the first eight columns, powers of two 0 <= k <= 7, and 55 rows of the triangle through prime(56) = 263.
p/k     0       1       2       3       4       5       6       7
3       9
5       25      50
7       49      98
11      121     242     484
13      169     338     676
17      289     578     1156    2312
19      361     722     1444    2888
23      529     1058    2116    4232
29      841     1682    3364    6728
31      961     1922    3844    7688
37      1369    2738    5476    10952   21904
41      1681    3362    6724    13448   26896
43      1849    3698    7396    14792   29584
47      2209    4418    8836    17672   35344
53      2809    5618    11236   22472   44944
59      3481    6962    13924   27848   55696
61      3721    7442    14884   29768   59536
67      4489    8978    17956   35912   71824   143648
71      5041    10082   20164   40328   80656   161312
.       .       .       .       .       .       .
.       .       .       .       .       .       .
131     17161   34322   68644   137288  274567  549152  1098304
137     18769   37538   75076   150152  300304  600608  1201216
.       .       .       .       .       .       .       .
.       .       .       .       .       .       .       .
257     66049   132098  264196  528392  1056784 2113568 4227136 8454272
263     69169   138338  276676  553352  1106704 2213408 4426816 8853632
Number 4 is not in this sequence since the symmetric representation of sigma(4) consists of a single region. Column k=0 contains the squares of primes (A001248(n), n>=2), column k=1 contains double the squares of primes (A079704(n), n>=2) and column k=2 contains four times the squares of primes (A069262(n), n>=5).

Hartmut F. W. Hoft, Three regions width one - triangle formula proof

Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A250068, A250070, A250071.

Cf. A070552, A263951.

(* path[n] and a237270[n] are defined in A237270 *)
atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
(* data *)
Select[Range[10000], atmostOneDiagonalsQ[#] && Length[a237270[#]]==3 &]
(* expression for the triangle in the Example section *)
TableForm[Table[2^k Prime[n]^2, {n, 2, 57}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth -> 2, TableHeadings -> {Map[Prime, Range[2, 57]], Range[0, Floor[Log[2, Prime[57] - 1]]]}]

As an irregular triangle, T(n, k) = 2^k * prime(n)^2 where n >= 2 and 0 <= k <= floor(log_2(prime(n)) - 1).

A263990 Nonsquare numbers k such that k and k+1 are semiprimes.

Original entry on oeis.org

14, 21, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697, 698, 706, 717, 745, 766, 778, 793, 802, 817, 842, 865, 878

Offset: 1

Zak Seidov, Oct 31 2015

nonn

If k and k+1 are semiprimes then k+1 is always nonsquare while k can be a square (see A263951). The sequence gives the nonsquare terms of A070552. Each of the numbers k and k+1 is a product of two distinct primes.
Numbers that are terms in A070552 but not in A263951.
The subsequence of triples of consecutive squarefree semiprimes is A039833. - R. J. Mathar, Aug 13 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Subsequence of A070552, A086263.

Cf. A006881, A039833, A109288, A263951.

Select[Range[1000], ! IntegerQ[Sqrt[#]] && 2 == PrimeOmega[#] == PrimeOmega[# + 1] &]

is(n)=if(n%2, isprime((n+1)/2) && bigomega(n)==2 && !isprimepower(n), isprime(n/2) && bigomega(n+1)==2) \\ Charles R Greathouse IV, Apr 25 2016

a(n) = A109288(n) - 1. - Amiram Eldar, Aug 08 2025

A292989 Triangular numbers having exactly 6 divisors.

Original entry on oeis.org

28, 45, 153, 171, 325, 4753, 7381, 29161, 56953, 65341, 166753, 354061, 5649841, 6060421, 6835753, 6924781, 12708361, 19478161, 24231241, 52035301, 56791153, 147258541, 186660181, 282304441, 326081953, 520273153, 536657941, 704531953, 784139401, 1215121753

Offset: 1

Jon E. Schoenfield, Dec 08 2017

nonn

Intersection of A000217 (triangular numbers) and A030515 (numbers with exactly 6 divisors).
This sequence is also the union of
(1) numbers of the form p*(2p-1) where p is prime and 2p-1 is the square of a prime (this sequence begins 45, 325, 7381, 65341, 354061, ...),
(2) numbers of the form p^2*(2p^2 - 1) where both p and 2p^2 - 1 are prime (this sequence begins 28, 153, 4753, 29161, ...), and
(3) numbers of the form p^2*(2p^2 + 1) where both p and 2p^2 + 1 are prime (the only such number is 171).

			28 = 2^2 * 7, so it has 6 divisors: {1, 2, 4, 7, 14, 28};
45 = 3^2 * 5, so it has 6 divisors: {1, 3, 5, 9, 15, 45};
171 = 3^2 * 19, so it has 6 divisors: {1, 3, 9, 19, 57, 171}.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000217 (triangular numbers), A030515 (numbers with exactly 6 divisors).
Cf. A067756 (primes p such that 2p-1 is the square of a prime), A106483 (primes p such that 2p^2 - 1 is prime).
Cf. A263951.

Select[Array[PolygonalNumber, 10^5], DivisorSigma[0, #] == 6 &] (* Michael De Vlieger, Dec 09 2017 *)

A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.

Original entry on oeis.org

9, 21, 25, 33, 57, 85, 93, 121, 133, 145, 177, 205, 213, 217, 253, 361, 393, 445, 553, 565, 633, 697, 793, 817, 841, 865, 913, 933, 973, 1137, 1285, 1345, 1417, 1437, 1465, 1477, 1513, 1537, 1717, 1765, 1837, 1857, 1893, 2101, 2173, 2245, 2305, 2517, 2577, 2581, 2605, 2641, 2653

Offset: 1

Hartmut F. W. Hoft, Jan 09 2021

nonn

For the squares p^2 in this sequence the area of the central region of the three regions in the symmetric representation of sigma(p^2) is equal to p.

p^2 is a term iff p is in A048161, and this subsequence of p^2 is A263951. - Bernard Schott, Jan 10 2021

			a(1) = 9 = 3*3 is the first number for which SRS(a(1)) consists of three regions ( 5, 3, 5 ).
a(6) = 85 = 5*17, both (1+85)/2 = 43 and (5+17)/2 = 11 are primes, and SRS(a(6)) consists of the 4 regions ( 43, 11, 11, 43 ).

Union of A128283 and A263951.
Subsequence of A046315 (all odd semiprimes).
Cf. A048161, A070552, A128281, A237591, A237593, A249223, A262045, A340482.

dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[-#]])/2&, Range[Length[d]/2]], PrimeQ]]
a340482[n_] := Select[Range[n], PrimeOmega[#]==2&&dQ[#]&]
a340482[2700]

isok(m) = if ((m % 2) && (bigomega(m)==2), if (issquare(m), isprime((m+1)/2), my(p=factor(m)[1,1], q=factor(m)[2,1]); isprime((p*q+1)/2) && isprime((p+q)/2))); \\ Michel Marcus, Jan 10 2021

A068485 One-sixtieth of the even leg of Pythagorean triangles whose other sides are both primes (other than 3, 5 or 13).

Original entry on oeis.org

1, 3, 7, 29, 31, 42, 52, 85, 143, 161, 273, 330, 612, 1015, 1197, 1394, 1680, 1771, 2262, 2698, 2717, 3318, 3424, 3641, 4551, 4700, 5617, 6468, 7192, 8184, 8858, 8996, 9205, 9523, 9919, 10622, 11040, 11427, 11623, 15436, 17256, 17739, 18476, 18725, 19533

Offset: 1

Lekraj Beedassy, Mar 11 2002

nonn

The (primitive) Pythagorean triple is {A048161(n), A067755(n), A067756(n)}.

Cf. A048161, A067755, A067756, A263951.

a068485[n_] := (Select[Map[Prime[#]^2&, Range[4, n]], PrimeQ[(#+1)/2]&]-1)/120
a068485[250] (* data - Hartmut F. W. Hoft, Aug 06 2020 *)

From Hartmut F. W. Hoft, Aug 06 2020: (Start)
a(n) = A067755(n+2)/60, n>=1.
a(n) = (A263951(n+2) - 1)/120, n>=1. (End)

More terms from Sascha Kurz, Mar 26 2002

a(34)-a(45) from Ray Chandler, Apr 12 2010

cp's OEIS Frontend

A247687 Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one.

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A263990 Nonsquare numbers k such that k and k+1 are semiprimes.

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A292989 Triangular numbers having exactly 6 divisors.

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A340482 Numbers that are the product of two not necessarily distinct odd primes p*q with the property that (p*q+1)/2 and (p+q)/2 are primes.

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A068485 One-sixtieth of the even leg of Pythagorean triangles whose other sides are both primes (other than 3, 5 or 13).

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A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.