A055670 -id:A055670 - cp's OEIS frontend

A375703 Minimum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

2, 5, 10, 17, 26, 28, 33, 37, 50, 65, 82, 101, 122, 126, 129, 145, 170, 197, 217, 226, 244, 257, 290, 325, 344, 362, 401, 442, 485, 513, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1001, 1025, 1090, 1157, 1226, 1297, 1332, 1370, 1445, 1522, 1601, 1682, 1729

Offset: 1

Gus Wiseman, Aug 28 2024

nonn

Non-perfect-powers A007916 are numbers without a proper integer root.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n has length A375702, first a(n), last A375704, sum A375705.

For prime numbers we have A045344.
For nonsquarefree numbers we have A053806, anti-runs A373410.
For nonprime numbers we have A055670, anti-runs A005381.
For squarefree numbers we have A072284, anti-runs A373408.
The anti-run version is A216765 (same as A375703 with 2 exceptions).
For non-prime-powers we have A373673, anti-runs A120430.
For prime-powers we have A373676, anti-runs A373575.
For runs of non-perfect-powers (A007916):
- length: A375702 = A053289(n+1) - 1.
- first: A375703 (this)
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
Cf. A007674, A045542, A375708, A375713, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Min/@Split[Select[Range[100],radQ],#1+1==#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100],radQ[#]&&!radQ[#-1]&]

Numbers k > 0 such that k-1 is a perfect power (A001597) but k is not.

A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

3, 7, 15, 24, 26, 31, 35, 48, 63, 80, 99, 120, 124, 127, 143, 168, 195, 215, 224, 242, 255, 288, 323, 342, 360, 399, 440, 483, 511, 528, 575, 624, 675, 728, 783, 840, 899, 960, 999, 1023, 1088, 1155, 1224, 1295, 1330, 1368, 1443, 1520, 1599, 1680, 1727, 1763

Offset: 1

Gus Wiseman, Aug 29 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

Also numbers k > 0 such that k is a perfect power (A001597) but k+1 is not.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n begins with A375703(n), ends with a(n), adds up to A375705(n), and has length A375702(n).

For nonprime numbers: A006093, min A055670, anti-runs A068780, min A005381.
For prime numbers we have A045344.
Inserting 8 after 7 gives A045542.
For nonsquarefree numbers we have A072284(n) + 1, anti-runs A068781.
For squarefree numbers we have A373415, anti-runs A007674.
For prime-powers we have A373674 (min A373673), anti-runs A006549 (A120430).
Non-prime-powers: A373677 (min A373676), anti-runs A255346 (min A373575).
The anti-run version is A375739.
A001597 lists perfect-powers, differences A053289.
A046933 counts composite numbers between primes.
A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
For runs of non-perfect-powers (A007916):
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (this) (same as A045542 with 8 removed)
- sum: A375705
Cf. A053797, A053806, A061398, A061399, A251092, A373408, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Max/@Split[Select[Range[100],radQ],#1+1==#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100],radQ[#]&&!radQ[#+1]&]

For n > 2 we have a(n) = A045542(n+1).

A055672 Number of right-inequivalent prime Hurwitz quaternions of norm n.

Original entry on oeis.org

0, 0, 1, 4, 0, 6, 0, 8, 0, 0, 0, 12, 0, 14, 0, 0, 0, 18, 0, 20, 0, 0, 0, 24, 0, 0, 0, 0, 0, 30, 0, 32, 0, 0, 0, 0, 0, 38, 0, 0, 0, 42, 0, 44, 0, 0, 0, 48, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 60, 0, 62, 0, 0, 0, 0, 0, 68, 0, 0, 0, 72, 0, 74, 0, 0, 0, 0, 0, 80, 0, 0, 0, 84, 0, 0, 0, 0, 0, 90

Offset: 0

N. J. A. Sloane, Jun 09 2000

Two primes are considered right-equivalent if they differ by right multiplication by one of the 24 units.

The extension desired below does not exist in the following sense: Let q1 ~R q2 be the equivalence defined by q1 = q2*u (i.e. if q1 and q2 any two HQ's, a unit u exists that solves this). Let q1 ~L q2 be the equivalence defined by q1 = u*q2 (i.e. if q1 and q2 any two HQ's a unit u exists that solves this.) If we define a relation ~RL such that q1 ~RL q2 means (q1 ~R q2 or q1 ~L q2), this relation is not transitive, i.e., not an equivalence. Cause: q1 = q2*u1, q2 = u2*q3, i.e., q1 ~R q2, q2 ~L q3 does not always have a solution with either q1 = q3*u3 or q1= u3*q3. There are pairs of u1 and u2 out of the 24*24 cases where q1 ~L q3 or q1 ~L q3 cannot be solved with any u3. - R. J. Mathar, Aug 05 2025

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.

R. J. Mathar, Table of n, a(n) for n = 0..10000
L. E. Dickson, Algebras and Their Arithmetics, U. Chicago Press, 1923, Section 91.

Cf. A055669 A055670 (zeros removed), A055671, A385603 (equivalence up to left-and-right multiplication).

A055671[n_] := If[PrimeQ[n], Reduce[a^2 + b^2 + c^2 + d^2 == 4n, {a, b, c, d}, Integers] // Length, 0]; a[n_] := A055671[n]/24; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 22 2016 *)

a(n) = A055671(n)/24.

a(n) = A000593(n) * A010051(n). - R. J. Mathar, Aug 02 2025

I would also like to get the sequences of inequivalent prime Hurwitz quaternions, where two primes are considered equivalent if they differ by left or right multiplication by one of the 24 units. This will give two more sequences, analogs of A055670 and A055672.

A055669 Number of prime Hurwitz quaternions of norm prime(n).

Original entry on oeis.org

24, 96, 144, 192, 288, 336, 432, 480, 576, 720, 768, 912, 1008, 1056, 1152, 1296, 1440, 1488, 1632, 1728, 1776, 1920, 2016, 2160, 2352, 2448, 2496, 2592, 2640, 2736, 3072, 3168, 3312, 3360, 3600, 3648, 3792, 3936, 4032, 4176, 4320, 4368, 4608, 4656, 4752

Offset: 1

N. J. A. Sloane, Jun 09 2000

Number of vectors of norm p in D_4 lattice (cf. A004011).

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.

T. D. Noe, Table of n, a(n) for n=1..1000
L. E. Dickson, Algebras and Their Arithmetics, Univ. of Chicago Press, 1923, Section 91 (page 147).

Cf. A055670, A055671, A055672.

Cf. A240068 (number of prime Lipschitz quaternions having norm prime(n)).

Join[{24},24(#+1)&/@Prime[Range[2,50]]] (* Harvey P. Dale, Mar 12 2013 *)

a(n) = 24 * (prime(n)+1) = 24 * A008864(n) for n >= 2.
a(n) = 24*A055670(n).
a(n) = A004011(prime(n)). - R. J. Mathar, Aug 01 2025

More terms from David W. Wilson, May 02 2001

A061673 Even numbers k such that k+1 and k-1 are both composite.

Original entry on oeis.org

26, 34, 50, 56, 64, 76, 86, 92, 94, 116, 118, 120, 122, 124, 134, 142, 144, 146, 154, 160, 170, 176, 184, 186, 188, 202, 204, 206, 208, 214, 216, 218, 220, 236, 244, 246, 248, 254, 260, 266, 274, 286, 288, 290, 296, 298, 300, 302, 304, 320, 322, 324, 326

Offset: 1

Enoch Haga, Jun 16 2001

If a(n + 1) > a(n) + 2 then a(n) + 3 and a(n + 1) - 3 are both prime. - Joseph Wheat, Mar 16 2025

			a(3)=50 because 50 - 1 = 49 and 50 + 1 = 51 and both 49 and 51 are composite.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Twin Composites

Cf. A014574, A055670.

A025583(n-1) - 1.

Filtered([0,2..340],n->not IsPrime(n-1) and not IsPrime(n+1)); # Muniru A Asiru, Jul 01 2018;

a061673 n = a061673_list !! (n-1)
a061673_list = filter bothComp [4,6..] where
   bothComp n = (1 - a010051 (n-1)) * (1 - a010051 (n+1)) > 0
-- Reinhard Zumkeller, Feb 27 2011

fQ[n_] := !PrimeQ[n - 1] && !PrimeQ[n + 1]; Select[2 Range@ 163, fQ]
Select[Range[2,400,2],AllTrue[#+{1,-1},CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 01 2014 *)
2*SequencePosition[Table[If[CompositeQ[n],1,0],{n,1,351,2}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 04 2020 *)

{ n=0; forstep (a=2, 3986, 2, if (!isprime(a+1) && !isprime(a-1), write("b061673.txt", n++, " ", a)) ) } \\ Harry J. Smith, Jul 26 2009

from sympy import isprime
def abelow(limit):
  for k in range(2, limit, 2):
    if not isprime(k-1) and not isprime(k+1): yield k
print([an for an in abelow(327)]) # Michael S. Branicky, Jan 02 2021

A175216 The first nonprimes after the primes.

Original entry on oeis.org

4, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272

Offset: 1

Jaroslav Krizek, Mar 06 2010

Essentially the same as A135731, A055670, A028815 and A008864. [R. J. Mathar, Mar 13 2010]

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A008864, A028815, A055670, A135731.

[n eq 1 select 4 else NthPrime(n) +1: n in [1..100]]; // G. C. Greubel, Aug 06 2024

Table[Prime[n] +1 +Boole[n==1], {n,100}] (* G. C. Greubel, Aug 06 2024 *)

def A175216(n): return nth_prime(n) +1 +int(n==1)
[A175216(n) for n in range(1,101)] # G. C. Greubel, Aug 06 2024

a(1) = 4, for n >= 2, a(n) = A008864(n) = A000040(n) + 1.

A055671 Number of prime Hurwitz quaternions of norm n.

Original entry on oeis.org

0, 0, 24, 96, 0, 144, 0, 192, 0, 0, 0, 288, 0, 336, 0, 0, 0, 432, 0, 480, 0, 0, 0, 576, 0, 0, 0, 0, 0, 720, 0, 768, 0, 0, 0, 0, 0, 912, 0, 0, 0, 1008, 0, 1056, 0, 0, 0, 1152, 0, 0, 0, 0, 0, 1296, 0, 0, 0, 0, 0, 1440, 0, 1488, 0, 0, 0, 0, 0, 1632, 0, 0, 0, 1728, 0, 1776, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jun 09 2000

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.

R. J. Mathar, Table of n, a(n) for n = 0..10000

Cf. A055669 (zeros removed), A055670, A055671, A055672.

a[p_?PrimeQ] := Reduce[ a^2 + b^2 + c^2 + d^2 == 4p, {a, b, c, d}, Integers] // Length; a[] = 0; Table[ a[n], {n, 0, 78}] (* _Jean-François Alcover, Oct 03 2012 *)

a(n) = number of vectors of norm n in D_4 lattice (A004011) if n is a prime, otherwise a(n) = 0.

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jun 10 2005

A083502 Smallest k such that n*(n+k) + 1 is an n-th power.

Original entry on oeis.org

1, 2, 18, 16, 1550, 2598, 299586, 812, 29118, 348678430, 67546215506, 20345040, 61054982557998, 281241170407078, 76861433640456450, 2690404, 128583032925805678334, 211927625850, 275941052631578947368402, 174339200

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 03 2003

nonn

Sequence is obviously infinite.
If the sequence is restricted to only prime n's, the sequence increases absolutely. See comment in A083503.
[Since there is actually no comment in A083503: this probably means to say that (conjectural!) A083503(prime(n)) = A008864(n) which leads to a(p) = Sum_{s=2..p} binomial(p,s)*p^(s-1) for primes p, an increasing subsequence. - R. J. Mathar, Aug 01 2025]
a(n) = (x^n-1)/n - n, where x is the least integer > 1 with x^n == 1 (mod n). - Robert Israel, Aug 01 2025

Robert Israel, Table of n, a(n) for n = 1..388

The i's in the above Mathematica coding, except for a(1), give A055670.

Cf. A083500, A083503.

A083502 := proc(n)
    local a,b ;
    if n = 1 then
        1 ;
    else
        for b from 2 do
            a := (b^n-1)/n-n ;
            if type( a,'integer') then
                return  a;
            end if;
        end do:
    end if;
end proc:
seq(A083502(n),n=1..20) ; # R. J. Mathar, Aug 01 2025
# alternative
f:= proc(n) local X,S;
  S:= min(map(t -> subs(t,X), {msolve(X^n = 1, n)} minus {{X=1}}));
  if S = infinity then ((n+1)^n - 1)/n - n else (S^n-1)/n - n fi
end proc:
f(1):= 1:
map(f, [$1..50]); # Robert Israel, Aug 01 2025

Do[i = 2; While[k = (i^n - 1)/n - n; !IntegerQ[k], i++ ]; Print[k], {n, 1, 20}]

Edited and extended by Robert G. Wilson v, May 11 2003

A299763 a(n) = 1 + A182986(n).

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

Omar E. Pol, Mar 14 2018

Are these the indices of the rows of A299762 where there is a record?

G. C. Greubel, Table of n, a(n) for n = 1..10000

First differences are in A054541.
Cf. A008864, A028815, A182986, A299762.
Essentially the same as A008864, A028815, A055670, A135731, A175216.

A299763:= func< n | n eq 1 select 1 else NthPrime(n-1) + 1 >;
[A299763(n): n in [1..70]]; // G. C. Greubel, Aug 05 2024

A299763[n_]:= If[n==1, 1, Prime[n-1] +1];
Table[A299763[n], {n,70}] (* G. C. Greubel, Aug 05 2024 *)

def A299763(n): return 1 if n==1 else nth_prime(n-1) + 1
[A299763(n) for n in range(1,71)] # G. C. Greubel, Aug 05 2024

a(n) = A028815(n-1) - [n=1].

a(n) = A008864(n-1) for n >= 2, with a(1) = 1.

A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1

Offset: 0

Gus Wiseman, Nov 01 2023

To illustrate the definition, the sum of primes indexed by all parts greater than one of the partition (5,2,2,1) is prime(5) + prime(2) + prime(2) = 17.

			Triangle begins:
  1
  1
  1 0 0 1
  1 0 0 1 0 1
  1 0 0 1 0 1 1 1
  1 0 0 1 0 1 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 2 0 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 2 2 2 2 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 4 2 3 2 0 3 1 0 0 0 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 5 4 4 3 3 3 2 3 0 1 0 0 1 0 1
The T(8,13) = 3 partitions are: (6,1,1), (4,2,2), (3,3,2).
The T(10,17) = 4 partitions are: (7,1,1,1), (5,2,2,1), (4,4,2), (4,3,3).

Row lengths are A055670.
Columns appear to converge to A099773.
A bisected even version is A116598 (counts partitions by number of 1's).
Counting all parts (not just > 1) gives A331416, shifted A331385.
A000041 counts integer partitions, strict A000009 (also into odds).
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A330953 counts partitions with Heinz number divisible by sum of primes.
A331381 counts partitions with (product)|(sum of primes), equality A331383.
Cf. A000837, A014689, A113686, A239261, A331417, A331418, A365067, A366842.

Table[Length[Select[IntegerPartitions[n], Total[Select[Prime/@#,OddQ]]==k&]], {n,0,10}, {k,0,If[n<=1,0,Prime[n]]}]

cp's OEIS Frontend

A375703 Minimum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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Mathematica

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A055672 Number of right-inequivalent prime Hurwitz quaternions of norm n.

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Mathematica

Formula

Extensions

A055669 Number of prime Hurwitz quaternions of norm prime(n).

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Mathematica

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Extensions

A061673 Even numbers k such that k+1 and k-1 are both composite.

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GAP

Haskell

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A175216 The first nonprimes after the primes.

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Magma

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A055671 Number of prime Hurwitz quaternions of norm n.

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