,11, - cp's OEIS frontend

A386969 A variant of Recamán's sequence (A005132): a(0) = 0; for n > 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n + 3.

0, 4, 2, 8, 15, 10, 19, 12, 23, 14, 27, 16, 31, 18, 35, 20, 39, 22, 43, 24, 47, 26, 51, 28, 55, 30, 59, 32, 63, 34, 67, 36, 71, 38, 75, 40, 79, 42, 83, 44, 87, 46, 91, 48, 95, 50, 99, 52, 103, 54, 107, 56, 111, 58, 115, 60, 119, 62, 123, 64, 127, 66, 131, 68, 135, 70, 139, 72, 143, 74, 147, 76, 151

Offset: 0

Jules Beauchamp, Aug 11 2025

When visualized in connecting semicircular lines after a(n) = 8 it produces one increasingly large conical spiral.

Cf. A005132, A386947.

a[n_] := a[n] = If[a[n-1] >= n && FreeQ[Table[a[k], {k, 0, n-1}], a[n-1] - n], a[n-1] - n, a[n-1] + n + 3]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Aug 22 2025 *)

A386962 Number of equivalence classes of connected 3-regular graphs on 2n unlabeled nodes up to local complementation.

Original entry on oeis.org

0, 1, 2, 4, 15, 60

Offset: 1

Tristan Cam, Aug 11 2025

Number of equivalences classes of 3-regular graphs on 2n nodes up to a sequence of local complementation or isomorphism, also called orbits for the local equivalence relation.
a(n) is necessarily less than:
  A005638(n) (number of non-isomorphic, not necessarily connected 3-regular graphs);
  A002851(n) (number of non-isomophic connected 3-regular graphs);
  A090899(n) (number of local equivalence classes of connected graphs); and
  A156800(n) (number of equivalence classes for connected graphs up to pivots and isomorphism).
This is relevant in the study of optimal quantum circuit synthesis for graph state preparation.

			There are only two 3-regular graphs with 6 nodes and they are not equivalent up to a sequence of local complementation, thus a(3) = 2.

Niels Bohr Institute Center for Hybrid Quantum Networks, graph_table (github)
Tristan Cam, Cyril Gavoille, Yvan Le Borgne, and Simon Martiel, Universal Graph Theory Operations for Graph State Preparation

Cf. A002851, A005638, A090899, A156800.

A386963 Gaps between positions of odd terms in A065090.

Original entry on oeis.org

5, 4, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 2, 4, 2, 3, 4, 2, 3, 3, 2, 3, 2, 2, 3, 4, 4, 3, 2, 2, 2, 2, 2, 3, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 3, 2, 3, 2, 4, 2, 2, 2, 4, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 4, 3, 2, 4, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 4, 3, 2, 2, 2, 2, 2, 3, 2

Offset: 1

Aied Sulaiman, Aug 11 2025

For n >= 2 we have a(n) in {2,3,4}:
  a(n) = 2 if no prime lies between the two successive odd terms,
  a(n) = 3 if a single prime lies between them,
  a(n) = 4 if two primes lie between them.
The initial 5 comes from 3, 5, 7 between 1 and 9.
Conjecture: a(n) tends to 2 in frequency (i.e., {n : a(n) = 2} has natural density 1).
Conjecture is true because the primes have natural density 0. - Robert Israel, Aug 29 2025

			A065090: 1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ...
Odd terms occur at positions: 1, 6, 10, 14, 17, 19, 23, 25, ...
Hence a(n): 5, 4, 4, 3, 2, 4, 2, ...

Cf. A065090, A014076.

a065090=Select[Range[335],#==2||!PrimeQ[#]&];l=Length[a065090];p={};Do[If[OddQ[a065090[[i]] ],AppendTo[p,i]],{i,l}];Differences[p] (* James C. McMahon, Aug 29 2025 *)

lista(nn) = my(vio = select(x->(x % 2), select(m->(!isprime(m) || m==2), [1..nn]), 1)); vector(#vio-1, k, vio[k+1] - vio[k]); \\ Michel Marcus, Aug 16 2025

from sympy import isprime
def gaps_generator():
    pos = 0
    last = None
    k = 1
    while True:
        if not (k % 2 == 1 and isprime(k)):  # in A065090
            pos += 1
            if k % 2 == 1:  # odd term (A014076)
                if last is None:
                    last = pos
                else:
                    yield pos - last
                    last = pos
        k += 1
def a(n: int) -> int:
    g = gaps_generator()
    for _ in range(n - 1):
        next(g)
    return next(g)

A385822 Numbers k such that phi(k) is not a perfect square.

Original entry on oeis.org

3, 4, 6, 7, 9, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89

Offset: 1

Aidan Chen, Aug 11 2025

			Since phi(35) = 24 and there is no integer n such that n^2 = 24.

Cf. A000010. Complement of A039770.

Select[Range[100], !IntegerQ[Sqrt[EulerPhi[#]]] &] (* Amiram Eldar, Aug 18 2025 *)

isok(k) = !issquare(eulerphi(k)); \\ Michel Marcus, Aug 18 2025

from math import isqrt
from sympy import totient as phi
def ok(n): return isqrt(p:=phi(n))**2 != p
print([k for k in range(1, 110) if ok(k)]) # Michael S. Branicky, Aug 17 2025

A386985 Smallest k > 0 such that the base-n number formed by concatenating k, k - 1, ..., 2, 1 (each written in base n) is prime, or -1 if no such k exists for the given n.

Original entry on oeis.org

2, 2, 4, 2, 2, 373, 2, 2, 82, 2, 3

Offset: 2

Marco Ripà, Aug 11 2025

If a(13) != -1, then the corresponding prime must have more than 4178 decimal digits.
Sequence continues n=13..36: ?, 2, 2, 28, 362, 2, ?, 2, 2, 4, 2, 3, ?, 2, 5, 4, 2, 2, 37, 3, 2, 4, 2, 2.
The increasing-order analog begins 15, 2, ?. See A048436.

			a(3) = 2 since 21_(base-3) = 7_(base-10), which is prime.

Mathematics Stack Exchange, Concatenation of consecutive positive integers: Does a prime also exist in base 4 (as in bases 2, 3, 5, 6, 7, 8, 9,...)?.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.

Cf. A000422, A048436, A281571.

f(n, b) = my(p=1, L=1); sum(k=1, n, k*p*=L+(k==L&&!L*=b)); \\ adapted from A000422
a(n) = my(k=1); while (!ispseudoprime(f(k, n)), k++); k; \\ Michel Marcus, Aug 16 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. *)
```

A386972 Numbers that are the product of a semiprime and the square of another semiprime.

Original entry on oeis.org

96, 144, 160, 224, 240, 324, 336, 352, 360, 400, 416, 486, 504, 528, 540, 544, 560, 600, 608, 624, 736, 756, 784, 792, 810, 816, 880, 900, 912, 928, 936, 992, 1040, 1104, 1134, 1176, 1184, 1188, 1215, 1224, 1232, 1260, 1312, 1350, 1360, 1368, 1376, 1392, 1400, 1404, 1456, 1488, 1500

Offset: 1

Ian Hahus, Aug 11 2025

nonn

Numbers with prime signature [5, 1], [4, 2], [4, 1, 1], [3, 2, 1], [2, 2, 2] or [2, 2, 1, 1]. So, necessarily but not sufficiently, terms t have bigomega(t) = 6. - David A. Corneth, Aug 11 2025

			96 = 6 * 4^2;
144 = 9 * 4^2 or 4 * 6^2.

David A. Corneth, First 1000000 terms colorcoded by prime signature

Cf. A001358 (semiprimes), A046306, A054753, A386977.

M:= 2000: # for terms <= M
P:= select(isprime, [2,seq(i,i=3..M/8,2)]): nP:= nops(P):
S:= {}:
for i1 from 1 to nP do
  p1:= P[i1];
  if p1^2*4^2 > M then break fi;
  for i2 from i1 to nP do
    p2:= P[i2];
    if p1*p2*4^2 > M then break fi;
    for i3 from 1 to nP do
      p3:= P[i3];
      if p1*p2*p3^4 > M then break fi;
      for i4 from i3 to nP do
        p4:= P[i4];
        v:= p1*p2*(p3*p4)^2;
        if v > M then break fi;
        if p1*p2 = p3*p4 then next fi;
        S:= S union {v}
od od od od:
sort(convert(S,list)); # Robert Israel, Aug 11 2025

Select[Range@ 1500, MemberQ[{{1,5}, {2,4}, {1,1,4}, {1,2,3}, {2,2,2}, {1,1,2,2}}, Sort[ Last /@ FactorInteger[#]]] &] (* Giovanni Resta, Aug 12 2025 *)

is(n) = {my(f = factor(n), b = bigomega(f)); if(b != 6, return(0)); f = vecsort(f[,2]~); #setminus(Set([f]), Set([[1, 5], [2, 4], [1, 1, 4], [1, 2, 3], [2, 2, 2], [1, 1, 2, 2]])) == 0} \\ David A. Corneth, Aug 12 2025

A386216 Values of v in the quartets (2, u, v, w) of type 2; i.e., values of v for solutions to 2(2 + u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see A385884.

Original entry on oeis.org

6, 10, 12, 14, 16, 6, 18, 5, 20, 22, 8, 24, 26, 7, 28, 6, 10, 30, 8, 32, 34, 9, 12, 36, 38, 8, 10, 40, 7, 14, 42, 11, 44, 46, 12, 16, 48, 10, 50, 13, 52, 9, 18, 54, 8, 14, 56, 58, 10, 12, 15, 20, 60, 62, 16, 64, 11, 22, 66, 17, 68, 10, 14, 70, 9, 12, 18, 24

Offset: 1

Clark Kimberling, Aug 11 2025

nonn

Cf. A385884.

A386984 Number of 2-dense sublists of divisors of the n-th hexagonal number.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 3, 5, 1, 5, 1, 5, 1, 5, 1, 3, 1, 7, 3, 3, 1, 5, 3, 7, 1, 5, 1, 3, 1, 7, 3, 5, 1, 3, 1, 5, 1, 7, 1, 7, 1, 5, 3, 5, 1, 5, 1, 7, 3, 5, 1, 7, 1, 7, 5, 7, 1, 5, 3, 3, 1, 7, 1, 7, 1, 7, 5, 5, 1, 7, 1, 7, 3, 5, 1, 3, 1, 9, 3, 7, 1, 7, 1, 7, 1, 5, 1, 5, 1, 9, 3, 3, 1, 3, 1, 9, 1

Offset: 0

Omar E. Pol, Aug 11 2025

In a sublist of divisors of k the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of k.
The 2-dense sublists of divisors of k are the maximal sublists whose terms increase by a factor of at most 2.
Conjecture: all terms are odd.

			For n = 3 the third positive hexagonal number is 15. The list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15], so a(3) = 3.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Bisection of A384928.

Cf. A000384, A174973 (2-dense numbers), A237271, A379288, A384149, A384222, A384225, A384226, A384930, A384931, A386989.

A386984[n_] := Length[Split[Divisors[PolygonalNumber[6, n]], #2 <= 2*# &]];
Array[A386984, 100, 0] (* Paolo Xausa, Aug 29 2025 *)

a(n) = A237271(A000384(n)) for n >= 1 (conjectured).

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A386969 A variant of Recamán's sequence (A005132): a(0) = 0; for n > 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n + 3.

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A386962 Number of equivalence classes of connected 3-regular graphs on 2n unlabeled nodes up to local complementation.

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A386963 Gaps between positions of odd terms in A065090.

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A385822 Numbers k such that phi(k) is not a perfect square.

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A386985 Smallest k > 0 such that the base-n number formed by concatenating k, k - 1, ..., 2, 1 (each written in base n) is prime, or -1 if no such k exists for the given n.

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A386980 Number of acute Heronian triangles with integer inradius n.

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A386981 Number of obtuse Heronian triangles with integer inradius n.

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A386972 Numbers that are the product of a semiprime and the square of another semiprime.

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