A061345 -id:A061345 - cp's OEIS frontend

A082916 Numbers k such that k and binomial(2*k, k) are relatively prime.

0, 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 79, 81, 83, 89, 93, 97, 101, 103, 107, 109, 111, 113, 119, 121, 125, 127, 131, 137, 139, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179, 181, 185, 191, 193, 197

Offset: 1

Benoit Cloitre, May 25 2003

nonn

Also the numbers k such that every base-p digit of k is less than p/2, for every prime divisor p of k. Contains all odd primes and their powers. - David Radcliffe, Jun 28 2025

J. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quarterly J. of Pure and Applied Math. 30 (1899), 150-156.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000984 (central binomial coefficients). Contains A061345.

Select[Range[0, 100], CoprimeQ[Binomial[2*#, #], #] &] (* Amiram Eldar, May 24 2020 *)

isok(n) = gcd(n, binomial(2*n, n)) == 1; \\ Michel Marcus, Dec 04 2013

from math import gcd
A082916_list, b = [], 1
for n in range(10**5):
    if gcd(n,b) == 1:
        A082916_list.append(n)
    b = b*(4*n+2)//(n+1) # Chai Wah Wu, Mar 25 2016

It seems that a(n) is asymptotic to c*n*log(n) with 0.7

A087688 a(n) = number of solutions to x^3 - x == 0 (mod n).

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 3, 5, 3, 6, 3, 9, 3, 6, 9, 5, 3, 6, 3, 9, 9, 6, 3, 15, 3, 6, 3, 9, 3, 18, 3, 5, 9, 6, 9, 9, 3, 6, 9, 15, 3, 18, 3, 9, 9, 6, 3, 15, 3, 6, 9, 9, 3, 6, 9, 15, 9, 6, 3, 27, 3, 6, 9, 5, 9, 18, 3, 9, 9, 18

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003

Shadow transform of A007531. - Michel Marcus, Jun 06 2013

a(n) = 3 iff n belongs to (A061345 \ {1}) Union {4}. - Bernard Schott, Sep 16 2019

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
N. J. A. Sloane, Transforms.

Cf. A034444, A224516, A060594.

A087688 := proc(n) local a,x ; a := 0 ; for x from 0 to n-1 do if (x*(x^2-1)) mod n = 0 then a := a+1 ; end if; end do; a ; end proc:
seq(A087688(n),n=1..70) ; # R. J. Mathar, Jan 07 2011

nsols[n_]:=Length[Select[Range[0,n-1],Mod[#^3-#,n]==0&]]; nsols/@Range[80]  (* Harvey P. Dale, Mar 22 2011 *)
f[2, e_] := Which[e == 1, 2, e == 2, 3, e >= 3, 5]; f[p_, e_] := 3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n)=if(n%2,3^omega(n),my(v=valuation(n,2));3^omega(n>>v)*[2,3,5][min(3,v)]) \\ Charles R Greathouse IV, Mar 22 2011

Multiplicative with a(p^e) = 3 for p an odd prime, a(2^1) = 2, a(2^2) = 3, a(2^e) = 5 for e >= 3. - Eric M. Schmidt, Apr 08 2013

A115233 Primes p which have a unique representation as p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.

Original entry on oeis.org

5, 127, 163, 179, 191, 193, 223, 239, 251, 269, 311, 337, 389, 419, 431, 457, 491, 547, 557, 569, 599, 613, 653, 659, 673, 683, 719, 739, 787, 821, 839, 853, 883, 911, 929, 953, 967, 977, 1117, 1123, 1201, 1229, 1249, 1283, 1289, 1297, 1303, 1327, 1381, 1409, 1423, 1439, 1451, 1471, 1481, 1499

Offset: 1

Reinhard Zumkeller, Jan 17 2006

nonn

			5 = 2+3 belongs to the sequence, but 23 = 2^2+19^1 = 2^4+7^1 does not.

Subsequence of A115232. Cf. A115230, A115231.

Cf. A000079, A061345.

maxp = 1500; Clear[cnt]; cnt[_] = 0;
pp = Prime[Range[PrimePi[maxp]]];
Do[p = 2^i + q^j; If[p <= maxp && PrimeQ[p], cnt[p] = cnt[p] + 1], {i, 0, Log[2, maxp] // Ceiling}, {j, 1, Log[3, maxp] // Ceiling}, {q, Rest[pp]}
];
Select[pp, cnt[#] == 1&] (* Jean-François Alcover, Aug 04 2018 *)

Recomputed (based on recomputation of A115230) by R. J. Mathar and Reinhard Zumkeller, Apr 29 2010.
Edited by N. J. A. Sloane, Apr 30 2010
Data corrected by Jean-François Alcover, Aug 04 2018

A118112 a(n) = binomial(3n,n) mod (n+1).

Original entry on oeis.org

1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 19, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 35, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Apr 13 2006

nonn

These divisibilities are analogous to those of Catalan numbers. For rather long sequences of consecutive integers, a(n)=0. For the first 10000 integers 9678 residues equals zero. See A118113.

If n+1 is in A061345, a(n)=0. This follows from Kummer's theorem. - Robert Israel, May 09 2018

			For n=9, binomial(27,7) = 4686825; 4686825 mod 10 = 5.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Kummer's theorem

Cf. A000108, A061345, A118113.

seq(binomial(3*n,n) mod (n+1), n=1..200); # Robert Israel, May 09 2018

Mathematica
```
Table[Mod[Binomial[3*k,k],k+1],{k,500}]
```

a(n) = binomial(3*n, n) % (n+1); \\ Michel Marcus, May 10 2018

a(n) = binomial(3n,n) mod (n+1).

Mathematica program corrected by Harvey P. Dale, Dec 28 2012

A132213 Number of distinct primes among the squares mod n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 0, 0, 2, 2, 4, 1, 1, 3, 3, 0, 2, 4, 3, 0, 4, 1, 4, 1, 2, 4, 2, 1, 3, 6, 2, 0, 5, 2, 6, 2, 2, 7, 5, 0, 6, 5, 3, 3, 8, 6, 3, 0, 3, 6, 8, 0, 6, 8, 3, 2, 2, 3, 7, 3, 3, 2, 7, 0, 9, 10, 3, 4, 6, 4, 9, 1, 10, 10, 11, 1, 2, 13, 3, 0, 10, 4, 5, 4, 4, 13, 4, 1, 11, 10, 4, 4

Offset: 1

T. D. Noe, Aug 13 2007, Aug 17 2007

It appears that a(n)=0 for only the 30 numbers in A065428, which appears to be related to idoneal numbers, A000926. The graph shows a(n) can be quite small even for large n. For example, a(9240)=7. Observe that the graph up to n=10000 appears to have 5 components. Why?
The logarithmic plot of the first 10^6 terms shows seven components.
From Rémy Sigrist, Nov 28 2017: (Start)
Empirically, in the logarithmic plot of the sequence:
- the set of indices of the first component (starting from the top), say S_1, is the union of A061345 and of A278568,
- the set of indices of the n-th component (for n > 1), say S_n, contains the numbers k not in a previous component and such that (omega(k) = n-1) or (omega(k) = n and val(k) = 0 or 2) or (omega(k) = n+1 and val(k) = 1) (where omega(k) = A001221(k) and val(k) = A007814(k)),
- see logarithmic scatterplot colored according to this scheme in Links section.
(End)

			For n=14, the squares (mod n) repeat 0,1,4,9,2,11,8,7,8,11,2,9,4,1,0,..., a sequence containing three distinct primes: 2, 7 and 11. Hence a(14)=3.

T. D. Noe, Table of n, a(n) for n = 1..10000
T. D. Noe, Logarithmic plot of 10^6 terms
Rémy Sigrist, Colored logarithmic plot of 2*10^6 terms

Cf. A000224 (number of squares mod n).

Cf. A000290, A001221, A007814, A010051, A061345, A278568.

import Data.List (nub, genericTake)
a132213 n = sum $ map a010051' $
            nub $ genericTake n $ map (`mod` n) $ tail a000290_list
-- Reinhard Zumkeller, Jun 23 2015, Oct 15 2011

Table[s=Union[Mod[Range[n]^2,n]]; Length[Select[s,PrimeQ]], {n,10000}]
Table[Count[Union[PowerMod[Range[n],2,n]],?PrimeQ],{n,100}] (* _Harvey P. Dale, Mar 02 2018 *)

A280152 Expansion of Product_{k>=1} (1 + floor(1/omega(2k+1))x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 6, 6, 6, 7, 7, 9, 8, 9, 10, 11, 12, 11, 14, 14, 16, 15, 18, 19, 19, 21, 22, 25, 25, 27, 28, 32, 32, 34, 36, 40, 41, 42, 47, 49, 52, 53, 57, 62, 63, 67, 71, 76, 79, 82, 88, 93, 98, 100, 108, 114, 118, 124

Offset: 0

Ilya Gutkovskiy, Dec 27 2016

nonn

Number of partitions of n into distinct odd prime powers (1 excluded).

			a(16) = 3 because we have [13, 3], [11, 5], [9, 7].

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A001221, A054685, A061345, A246655, A280151.

nmax = 78; CoefficientList[Series[Product[1 + Floor[1/PrimeNu[2 k + 1]] x^(2 k + 1), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + floor(1/omega(2*k+1))*x^(2*k+1)).

A316553 Number of elements of order 2 in the group SL(2, Z(n)).

Original entry on oeis.org

0, 3, 1, 7, 1, 7, 1, 15, 1, 7, 1, 15, 1, 7, 3, 15, 1, 7, 1, 15, 3, 7, 1, 31, 1, 7, 1, 15, 1, 15, 1, 15, 3, 7, 3, 15, 1, 7, 3, 31, 1, 15, 1, 15, 3, 7, 1, 31, 1, 7, 3, 15, 1, 7, 3, 31, 3, 7, 1, 31, 1, 7, 3, 15, 3, 15, 1, 15, 3, 15, 1, 31, 1, 7, 3, 15, 3, 15, 1

Offset: 1

Andrew Howroyd, Jul 06 2018

nonn

Equivalently, the number of cyclic subgroups of the group SL(2, Z(n)) having order 2, counting conjugates as distinct.

			Case n=2: the three 2 X 2 matrices on Z(2) having determinant 1 and order 2 are:
  [ 0 1 ]   [ 1 0 ]   [ 1 1 ]
  [ 1 0 ]   [ 1 1 ]   [ 0 1 ]

Antti Karttunen, Table of n, a(n) for n = 1..1023

Column 2 of A316564.
Cf. A174275, A316537.
Cf. A061345.

Concatenation([0], List([2..10], n->Sum(Filtered( ConjugacyClassesSubgroups( SL(2, Integers mod n)), x->Order( Representative(x))=2 and IsCyclic( Representative(x))), Size)));

a(n)={my(id=matid(2)); sum(a=0, n-1, sum(b=0, n-1, sum(c=0, n-1, sum(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, M^2==id))))) - 1}

memoA316553 = Map(); \\ Only values at 2^k are actually collected here.
A316553slow_memoized(n) = if(1==n, 0, if((n%2)&&isprimepower(n), 1, my(id=matid(2), v); if(mapisdefined(memoA316553,n,&v), v, v = (sum(a=0, n-1, sum(b=0, n-1, sum(c=0, n-1, sum(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(matdet(M)==1, M^2==id))))) - 1); mapput(memoA316553,n,v); (v))));
A316553(n) = if(1==n,0,my(f=factor(n)); -1 + prod(i=1,#f~,1+A316553slow_memoized(f[i,1]^f[i,2]))); \\ (Based on Robert Israel's multiplicativity rule) - Antti Karttunen, Dec 05 2021

Conjecture: a(n) = 2^(omega(n) + min(3, valuation(n, 2))) - 1.
From Robert Israel, Jun 15 2020: (Start)
Number of solutions mod n, other than t[1]=t[4]=1,t[2]=t[3]=0, of the equations t[2]*(t[1] + t[4])=0, t[3]*(t[1] + t[4])=0, t[1]^2 + t[2]*t[3] = 1, t[2]*t[3] + t[4]^2 = 1, t[1]*t[4] - t[2]*t[3] = 1.
If m and n are coprime, a(m*n) = a(m)*a(n)+a(m)+a(n) (i.e. a(n)+1 is multiplicative).
If n > 1 is in A061345, a(n)=1. (End)

A384233 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd noncomposite divisor is its k-th divisor, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 20, 16, 9, 12, 28, 42, 32, 11, 14, 30, 60, 84, 64, 13, 15, 40, 66, 132, 156, 128, 17, 18, 44, 78, 168, 204, 312, 256, 19, 21, 52, 88, 198, 228, 408, 684, 512, 23, 22, 56, 102, 210, 264, 456, 696, 1020, 1024, 25, 24, 68, 104, 220, 276, 468, 744, 1140, 1380

Offset: 1

Omar E. Pol, May 22 2025

This is a permutation of the positive integers.

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  20,  42,  84, 156, 312,  684, 1020, 1380, 1860, 3480, 3720,  4920, ...
      2,  5, 10,  28,  60, 132, 204, 408,  696, 1140, 1740, 2220, 3660, 4440,  5160, ...
      4,  7, 12,  30,  66, 168, 228, 456,  744, 1332, 2040, 2460, 4020, 5580,  5640, ...
      8,  9, 14,  40,  78, 198, 264, 468,  780, 1368, 2088, 2580, 4140, 6960,  6360, ...
     16, 11, 15,  44,  88, 210, 276, 510,  816, 1392, 2232, 2664, 4260, 7224,  6660, ...
     32, 13, 18,  52, 102, 220, 330, 552,  828, 1476, 2280, 2760, 4380, 7632,  7080, ...
     64, 17, 21,  56, 104, 234, 342, 570,  888, 1488, 2436, 2820, 4740, 7896,  7380, ...
    128, 19, 22,  68, 110, 252, 348, 612,  912, 1548, 2544, 2952, 4872, 8280,  7440, ...
    256, 23, 24,  70, 114, 260, 372, 624,  930, 1560, 2604, 3096, 4980, 8496,  7740, ...
    512, 25, 26,  76, 120, 272, 390, 660,  936, 1656, 2736, 3180, 5208, 8784,  8880, ...
   1024, 27, 33,  80, 126, 304, 396, 690,  984, 1692, 2790, 3384, 5220, 8904,  9912, ...
   2048, 29, 34,  90, 130, 306, 414, 792, 1032, 1710, 2832, 3420, 5256, 9030, 10248, ...
   4096, 31, 35,  92, 136, 336, 420, 870, 1044, 1776, 2928, 3540, 5328, 9324, 10440, ...
   8192, 37, 36,  99, 138, 340, 440, 920, 1104, 1908, 3060, 3612, 5340, 9648, 10512, ...
  16384, 41, 38, 100, 140, 368, 444, 966, 1110, 1932, 3108, 3816, 5520, 9660, 10836, ...
  ...
The divisors of 42 are [1, 2, 3, 6, 7, 14, 21, 42] and the largest odd noncomposite divisor is 7 and 7 is its 5th divisor, so T(1,5) = 42 because 42 the smallest number having that property.

Index entries for sequences that are permutations of the natural numbers.

Companion of A383961.
Row 1 gives A384232.
Column 1 gives A000079.
Cf. A006005, A027750, A061345, A065091, A087134.

f[n_] := FirstPosition[Divisors[n], FactorInteger[n/2^IntegerExponent[n, 2]][[-1, 1]]][[1]]; seq[m_] := Module[{t = Table[0, {m}, {m}], v = Table[0, {m}], c = 0, k = 1, i, j}, While[c < m*(m + 1)/2, i = f[k]; If[i <= m, j = v[[i]] + 1; If[j <= m - i + 1, t[[i]][[j]] = k; v[[i]]++; c++]]; k++]; Table[t[[j]][[i - j + 1]], {i, 1, m}, {j, 1, i}] // Flatten]; seq[11] (* Amiram Eldar, May 23 2025 *)

Conjecture: T(n,2) = A061345(n).

A061344 Numbers of form p^m + 1, p odd prime, m >= 1.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, 38, 42, 44, 48, 50, 54, 60, 62, 68, 72, 74, 80, 82, 84, 90, 98, 102, 104, 108, 110, 114, 122, 126, 128, 132, 138, 140, 150, 152, 158, 164, 168, 170, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240

Offset: 1

Hans Dieter Lueke (lueke(AT)ient.rwth-aachen.de), Jun 08 2001

Lengths of almost-binary sequences with perfect odd-periodic autocorrelation function.

As J. Arndt points out, each element of this sequence leads to a conference matrix (cf. link to Wikipedia and A000952). - M. F. Hasler, Mar 14 2008

H. D. Lueke, Binary odd-periodic complementary sequences. IEEE Trans. Inform. Theory, 43, pp. 365-367, 1997.

M. F. Hasler, Table of n, a(n) for n = 1..10000
Wikipedia, Conference matrix.

Equals A061345 + 1. Cf. A000952.

A061344(n)= local(m=1,p); for(c=1,n, until( isprime(m+=2) || ispower(m,[null], && p) && isprime(p),); /*print(c," ",m+1)*/); m+1 \\ - M. F. Hasler, Mar 14 2008

from sympy import primepi, integer_nthroot
def A061344(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length())))
    return bisection(f,n+1,n+1)+1 # Chai Wah Wu, Feb 03 2025

More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2001

Edited by M. F. Hasler, Mar 14 2008

A064077 Greater of odd twin prime powers (lesser = A064076).

Original entry on oeis.org

5, 7, 9, 11, 13, 19, 25, 27, 29, 31, 43, 49, 61, 73, 81, 83, 103, 109, 127, 139, 151, 169, 181, 193, 199, 229, 241, 243, 271, 283, 313, 349, 361, 421, 433, 463, 523, 571, 601, 619, 643, 661, 729, 811, 823, 829, 841, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153

Offset: 1

Reinhard Zumkeller, Sep 01 2001

nonn

A006512 is a proper subsequence of this sequence (as A001359 is of A064076).

			a(16) = 83^1 and 83 - 1 = 81 = 3^4 = A064076(16); a(20) = 139^1 and 139 - 2 = 137^1 = A064077(20).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A064076, A000961, A006512, A061345.

Select[Partition[Select[Range[1, 1200, 2], PrimePowerQ], 2, 1], Differences[#] == {2} &][[;; , 2]] (* Amiram Eldar, Mar 19 2025 *)

a(n) = A064076(n) + 2. - Amiram Eldar, Mar 19 2025

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cp's OEIS Frontend

A082916 Numbers k such that k and binomial(2*k, k) are relatively prime.

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A087688 a(n) = number of solutions to x^3 - x == 0 (mod n).

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Maple

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A115233 Primes p which have a unique representation as p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.

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Mathematica

Extensions

A118112 a(n) = binomial(3n,n) mod (n+1).

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Maple

Mathematica

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Extensions

A132213 Number of distinct primes among the squares mod n.

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Haskell

Mathematica

A280152 Expansion of Product_{k>=1} (1 + floor(1/omega(2*k+1))*x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).

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Mathematica

Formula

A316553 Number of elements of order 2 in the group SL(2, Z(n)).

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Programs

A280152 Expansion of Product_{k>=1} (1 + floor(1/omega(2k+1))x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).