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A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

2, 5, 21, 129, 69, 1, 51, 23991, 171, 1371, 3, 322141431, 1431357020859

Offset: 0

Om S. M. Yadav, Aug 18 2025

Similar to A227547, primes are added in successive manner except that here the sequence breaks if an even-indexed term is not prime and considers preceding even-indexed prime as the last term of the sequence. For example, a(2) = 21 [21, 23, 26, 31, 38, 49] but since 49 is not prime, last two terms (38 and 49) are omitted leaving 31 as last term in the sequence.

a(12) is the last term, because b(j) is always divisible by 11 for some j in {2, 4, 6, 8, 10, 14, 16, 18, 22, 24, 26}. - Pontus von Brömssen, Aug 19 2025

			a(n) = k, b(m+1) = b(m) + prime(m); b(1) = k
For n = 0, a(0) = 2; b(m+1) = b(m) + prime(m): [2]
For n = 1, a(1) = 5; b(m+1) = b(m) + prime(m): [5, 7(5+2)]
For n = 2, a(2) = 21; b(m+1) = b(m) + prime(m): [21, 23(21+2), 26(23+3), 31(26+5)]
For n = 3, a(3) = 129; b(m+1) = b(m) + prime(m): [129, 131(129+2), 134(131+3), 139(134+5), 146(139+7), 157(146+11)]
For n = 4, a(4) = 69; b(m+1) = b(m) + prime(m): [69, 71(69+2), 74(71+3), 79(74+5), 86(79+7), 97(86+11), 110(97+13), 127(110+17)]
For n = 5, a(5) = 1; b(m+1) = b(m) + prime(m): [1, 3(1+2), 6(3+3), 11(6+5), 18(11+7), 29(18+11), 42(29+13), 59(42+17), 78(59+19), 101(78+23)]
For a(n), even-indexed term is prime. e.g. for a(3) = 129 [129, 131, 134, 139, 146, 157], even indexed terms 131, 139, 157 are primes.

Cf. A014284, A092163, A109723, A227547.

a(n) = my(vp=concat(2, vector(n+1, i, sum(k=1, 2*i+1, prime(k)))), v=concat(vector(n, i, 1), 0), k=1); while (apply(ispseudoprime, vector(n+1, i, vp[i]+k)) != v, k++); k; \\ Michel Marcus, Aug 19 2025

a(11) from Michel Marcus, Aug 19 2025

a(12) from Pontus von Brömssen, Aug 19 2025

A386980 Number of acute Heronian triangles with integer inradius n.

Original entry on oeis.org

0, 0, 1, 1, 0, 4, 0, 2, 2, 2, 0, 6, 0, 1, 4, 3, 0, 8, 0, 6, 7, 2, 0, 17, 1, 0, 2, 8, 0, 14, 0, 3, 6, 1, 4, 17, 0, 0, 4, 12, 0, 27, 0, 4, 13, 1, 0, 27, 1, 4, 2, 4, 0, 13, 5, 14, 2, 0, 0, 32, 0, 0, 14, 4, 3, 18, 0, 5, 3, 15, 0, 41, 0, 0, 10, 4, 7, 16, 0, 18, 3, 0, 0, 60, 2, 0, 2, 18, 0, 39, 9

Offset: 1

Frank M Jackson, Aug 11 2025

nonn

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is acute iff n < (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5). Also, it has been proved that other than n = 3, all acute Heronian triangles have no prime inradii. For n = 3, the Heronian triangle has sides (10, 10, 12).
Empirically, it appears that the remaining occurrences of zero counts (other than 1 and the primes excluding 3) are inradii of the form 2p where p is in the set 13, 19, 29 and all other primes > 29.
The number of right integer triangles with inradius n is given by A078644, the number of obtuse Heronian triangles with inradius n is given by A386981 and the total number of Heronian triangles with inradius n is given by A120062.

			a(6) = 4, and the 4 acute Heronian triangles with inradius 6 have sides (15, 34, 35), (17, 25, 28), (17, 25, 26), (20, 20, 24).

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
Frank M Jackson, Mathematica program

Cf. A078644, A120062, A386981.

Mathematica
```
(* See link above. *)
```

A386981 Number of obtuse Heronian triangles with integer inradius n.

Original entry on oeis.org

0, 3, 9, 14, 12, 35, 21, 39, 44, 44, 23, 124, 28, 73, 97, 81, 30, 166, 31, 130, 169, 95, 39, 283, 59, 90, 131, 208, 33, 347, 43, 160, 196, 109, 160, 466, 35, 117, 197, 304, 41, 515, 57, 267, 354, 127, 61, 550, 110, 214, 219, 258, 44, 425, 215, 484, 265, 128, 51, 977, 41, 138, 582, 269, 169, 603, 48, 325, 252, 564, 47, 1058, 65, 133, 445, 341

Offset: 1

Frank M Jackson, Aug 11 2025

nonn

If a Heronian triangle has an inradius n, and sides (x, y, z), where x <= y <= z, then the triangle is obtuse iff n > (x+y-z)/2.
The only Heronian triangle with inradius 1 is the right triangle (3, 4, 5).
The number of right integer triangles with inradius n is given by A078644, the number of acute Heronian triangles with inradius n is given by A386980 and the total number of Heronian triangles with inradius n is given by A120062.

			a(2) = 3, and the 3 obtuse Heronian triangles with inradius 2 have sides (6, 25, 29), (7, 15, 20), (9, 10, 17).

Alan F. Beardon and Paul Stephenson, The Heron parameters of a triangle, Mathematical Gazette May 8, 2014.
Frank M Jackson, Mathematica program

Cf. A078644, A120062, A386980

Mathematica
```
(* See link above. *)
```

A385974 a(n) = the number of reduced positive proper fractions p/q with phi(q) <= n.

Original entry on oeis.org

1, 7, 7, 23, 23, 47, 47, 87, 87, 107, 107, 179, 179, 179, 179, 275, 275, 347, 347, 447, 447, 491, 491, 731, 731, 731, 731, 787, 787, 847, 847, 1071, 1071, 1071, 1071, 1359, 1359, 1359, 1359, 1719, 1719, 1887, 1887, 2019, 2019, 2111, 2111, 2639, 2639, 2639, 2639, 2743, 2743, 2851, 2851

Offset: 1

Hal M. Switkay, Jul 13 2025

nonn

This sequence groups the reduced positive proper fractions p/q in a manner analogous to the Farey sequence (A005728, which adds the endpoints 0/1 and 1/1). However, instead of limiting the size of the denominator (q <= n as is done in the Farey sequence), we limit phi(q) = A000010(q), where phi is the Euler totient function. The computation requires A014197(r) = the number of natural numbers q such that phi(q) = r.

When k > 0, a(2k+1) = a(2k), because there are no natural numbers whose Euler totient equals 2k+1.

			a(4) enumerates the positive proper fractions whose denominator has Euler totient <= 4. The eligible denominators are 2; 3, 4, 6; 5, 8, 10, 12. The numerators are relatively prime to the denominators. Grouping the fractions by the order of the denominators above, we get: 1/2; 1/3, 2/3, 1/4, 3/4, 1/6, 5/6; 1/5, 2/5, 3/5, 4/5, 1/8, 3/8, 5/8, 7/8, 1/10, 3/10, 7/10, 9/10, 1/12, 5/12, 7/12, 11/12. Thus a(4) = 23.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A005728, A014197.

a(n) = sum(i=1, n, i*invphiNum(i)) - 1; \\ Michel Marcus, Jul 14 2025; using Max Alekseyev's invphi.gp

a(n) = -1 + Sum_{i=1..n} i*A014197(i).

A385722 a(n) = the n-th term of the sequence B_2 defined as the ordered union of the ranges of A_x, where the sequence A_x is defined in A384669, and x is a proper positive fraction with denominator 2, 3, 4, or 6.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 240, 256, 288, 360, 384, 420, 480, 512, 576, 720, 768, 840, 960, 1024, 1152, 1260, 1440, 1536, 1680, 1920, 2048, 2304, 2310, 2520, 2880, 3072, 3360, 3840, 4096, 4320, 4608, 4620, 5040

Offset: 1

Hal M. Switkay, Jul 07 2025

nonn

The sequences A_x were defined in A384669; please see that sequence for more details.
Because of the continuity of the functions g_k(x) defined in A384669, if k is a term of A_y, then k is a term of A_x with x rational sufficiently close to y; so it suffices to study A_x for x rational, 0 < x < 1.
Let x = p/q, with p and q natural numbers, p < q. Then B_n is the ordered union of the ranges of A_x, where q has Euler totient <= n (that is, A000010(q) <= n). B_1 is just the sequence A_(1/2), that is, A384669. The present sequence B_2 is the ordered union of the ranges of A_x, where x = 1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6.
8 is the first term in B_2 (this sequence) that is not in B_1 = A384669.
15120 appears to be the first term in B_4 that is not in B_2 (this sequence).
More discussion about B_n and B_oo appears in a linked pdf.

			8 is a term of A_(2/3) and therefore of this sequence as well as A_x for x > 2/3, even though 8 is not a term of A384669, because 3^(2/3) (corresponding to 8) > 2 = 1^(2/3) + 1^(2/3) (corresponding to 6). Thus 8 qualifies to be a term in B_2, this sequence.

Pontus von Brömssen, Table of n, a(n) for n = 1..2000 (terms 1..186 from Hal M. Switkay)
Hal M. Switkay, More information about B_n and B_oo

Cf. A000010, A000079, A002110, A002182, A025487, A095848, A384669.

s(n, q) = my(f=factor(n)); sum(k=1, #f~, f[k, 2]^q);
listaq(nn, q) = my(r=-oo, list=List()); for (n=1, nn, my(ss=s(n, q)); if (ss > r, r = ss; listput(list, n)); ); Vec(list);
putlist(list, elems) = for (i=1, #elems, listput(list, elems[i])); list;
lista(nn) = my(list=List(), vq=[1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6]); for (i=1, #vq, list = putlist(list, listaq(nn, vq[i]))); Set(Vec(list)); \\ Michel Marcus, Jul 08 2025

A385916 Positive integers m that form Gaussian integers m + i such that every Gaussian integer g with |g| <= |m + i| is a linear combination of the distinct Gaussian divisors of m + i (where i is the imaginary unit).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 12, 13, 17, 18, 21, 23, 27, 31, 32, 33, 37, 38, 41, 43, 47, 55, 57, 68, 72, 73, 75, 81, 82, 83, 89, 91, 93, 98, 99, 105

Offset: 1

Frank M Jackson, Jul 12 2025

This sequence is an example of extending the concept of a practical number to the domain of Gaussian integers. To determine if a Gaussian integer p is practical over the Gaussian integer domain it is necessary to show that the Gaussian divisors (including all their associates) of the Gaussian integer p when combined linearly and distinctly generate all Gaussian integers g where |g| <= |p|.
The Mathematica program in the link below gives a complex plot of the linear combinations of the distinct divisors of a Gaussian integer m + i to see if it is a member of this sequence.
An analogous sequence such that positive integers m that form the Gaussian integers m + i are prime is given by A005574.

			a(3) is in the sequence because the Gaussian divisors of 3 + i are 1, 1 + i, 1 + 2i, 3 + i. Each divisor has 3 other associates. In total these 16 divisors will give the complex plot below when they are combined linearly and distinctly. Note that the patten in any quadrant is a rotation by a right angle of its adjacent quadrant.
|= = = = = = = = = = + = = = = = = = = = =|
|                    * * *                |
|              * * * * * * * *            |
|        * * * * * * * * * * * * *        |
|      * * * * * * * * * * * * * * *      |
|    * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * @ @ @ * * * * * * * *  |
|* * * * * * * * @ @ @ @ @ * * * * * * *  |
|* * * * * * * @ @ @ @ @ @ @ * * * * * *  |
+*-*-*-*-*-*-*-@-@-@-@-@-@-@-*-*-*-*-*-*-*+
|  * * * * * * @ @ @ @ @ @ @ * * * * * * *|
|  * * * * * * * @ @ @ @ @ * * * * * * * *|
|    * * * * * * * @ @ @ * * * * * * * *  |
|    * * * * * * * * * * * * * * * * * *  |
|    * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * *    |
|      * * * * * * * * * * * * * * *      |
|        * * * * * * * * * * * * *        |
|            * * * * * * * *              |
|                * * *                    |
|= = = = = = = = = = + = = = = = = = = = =|

Frank M Jackson, Mathematica program that gives a complex plot

Cf. A005574, A005153, A363227, A385489.

A385533 Third prepended column of the tribonacci array of the second kind, A385436.

Original entry on oeis.org

-1, -1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24

Offset: 1

A.H.M. Smeets, Jul 02 2025

sign

The numbers that occur thrice in this sequence are given by A353084(n)-1, i.e., the first prepended column of the tribonacci array of the first kind minus one.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A353084, A385436.

A385455 First prepended column of the tribonacci array of the second kind, A385436.

Original entry on oeis.org

-1, 0, 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81

Offset: 1

A.H.M. Smeets, Jun 29 2025

sign

The numbers that do not occur in this sequence are given by A278041.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A278041, A385436.

A385532 Second prepended column of the tribonacci array of the second kind, A385436.

Original entry on oeis.org

-1, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44

Offset: 1

A.H.M. Smeets, Jul 02 2025

sign

The numbers that occur twice in this sequence are the numbers in A278039.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A278039, A385436.

A385436 Tribonacci array of the second kind, read by upward antidiagonals.

Original entry on oeis.org

0, 2, 1, 4, 5, 3, 6, 8, 10, 7, 9, 12, 16, 20, 14, 11, 18, 23, 31, 38, 27, 13, 21, 34, 44, 58, 71, 51, 15, 25, 40, 64, 82, 108, 132, 95, 17, 29, 47, 75, 119, 152, 200, 244, 176, 19, 32, 54, 88, 139, 220, 281, 369, 450, 325, 22, 36, 60, 101, 163, 257, 406, 518, 680

Offset: 1

A.H.M. Smeets, Jun 28 2025

The array is, as a sequence, a permutation of the nonnegative integers; however it does not satisfy the conditions for interspersion and dispersion as given by Eric Weisstein's World of Mathematics. However, when all terms are increased by 1, it does satisfy the conditions for interspersion and dispersion!
Rows satisfy the recurrence: T(m,k) = 2*T(m,k-1) - T(m,k-4) for all k>4.
This array belongs to a family of Wythoff like arrays, based on binary number representations like the greedy and lazy Fibonacci number representations (see A035513 and A372501 for arrays), greedy and lazy Narayana number representations (A136189 for the array related to greedy representation).
The array is related to the lazy tribonacci number representation A352103. The first column lists the even numbers, i.e., for wich 0 suffix A352103(T(m,1)). The odd numbers are represented in the columns k > 1: A352103(T(m,k)) = A352103(T(m,1)) + 1^(k-1). Here + stands for concatenation and ^ stands for repeated concatenation.

			Array including some prepended columns (p = 1..4):
  p=4 p=3 p=2 p=1 | k=1 k=2 k=3  k=4  k=5  k=6  k=7   k=8   k=9  k=10
   -2  -1  -1  -1 |   0   1   3    7   14   27   51    95   176   325
   -2  -1   0   0 |   2   5  10   20   38   71  132   244   450   829
   -2   0   0   1 |   4   8  16   31   58  108  200   369   680
   -2   0   1   2 |   6  12  23   44   82  152  281   518
   -1   0   2   4 |   9  18  34   64  119  220  406   748
   -1   1   2   5 |  11  21  40   75  139  257  474   873
   -1   1   3   6 |  13  25  47   88  163  301  555  1022
   -1   1   4   7 |  15  29  54  101  187  345  636  1171
   -1   2   4   8 |  17  32  60  112  207  382  704  1296
   -1   2   5   9 |  19  36  67  125
    0   2   6  11 |  22  42  78  145
Each row of the array satisfies the recurrence relation T(m,k) = 2*T(m,k-1) - T(m,k-4); from this, the prepended columns are obtained by rowwise backward recursion.

A.H.M. Smeets, Table of n, a(n) for n = 1..20100 (first 200 antidiagonals).
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Eric Weisstein's World of Mathematics, Interspersion.
Eric Weisstein's World of Mathematics, Sequence Dispersion.

Cf. A035513, A136189, A372501, A027084 (m=1), A351631 (k=1), A352103.

Prepended columns: A385455 (p=1), A385532 (p=2), A385533 (p=3).

def ToDual_111_Zeck(n):
    if n == 0:
        return "0"
    f0, f1, f2, sf = 1, 0, 0, 0
    while n > sf:
        f0, f1, f2 = f0+f1+f2, f0, f1
        sf += f0
    r, s = sf-n, "1"
    while f0 > 1:
        f0, f1, f2 = f1, f2, f0-f1-f2
        r, s = r%f0, s+str(1-r//f0)
    return s
def From_111_Zeck(s):
    f0, f1, f2, i, n = 1, 1, 0, len(s), 0
    while i > 0:
        i -= 1
        f0, f1, f2, n = f0+f1+f2, f0, f1, n+int(s[i])*f0
    return n
d, a, n, c1 = 0, 0, 0, []
while d < 11:
    s = ToDual_111_Zeck(a)
    if s[len(s)-1] == "0": # == even
        n, d = n+1, d+1
        print(a, end = ", ")
        i, c1, p1 = d-1, c1+[s], ""
        while i > 0:
            n, i, p1 = n+1, i-1, p1+"1"
            print(From_111_Zeck(c1[i]+p1), end = ", ")
    a += 1

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A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

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A386980 Number of acute Heronian triangles with integer inradius n.

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Mathematica

A386981 Number of obtuse Heronian triangles with integer inradius n.

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Mathematica

A385974 a(n) = the number of reduced positive proper fractions p/q with phi(q) <= n.

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A385722 a(n) = the n-th term of the sequence B_2 defined as the ordered union of the ranges of A_x, where the sequence A_x is defined in A384669, and x is a proper positive fraction with denominator 2, 3, 4, or 6.

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A385916 Positive integers m that form Gaussian integers m + i such that every Gaussian integer g with |g| <= |m + i| is a linear combination of the distinct Gaussian divisors of m + i (where i is the imaginary unit).

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A385533 Third prepended column of the tribonacci array of the second kind, A385436.

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A385455 First prepended column of the tribonacci array of the second kind, A385436.

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A385532 Second prepended column of the tribonacci array of the second kind, A385436.

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