A061346 -id:A061346 - cp's OEIS frontend

A061345 Powers of odd primes. Alternatively, 1 and the odd prime powers (p^k, p an odd prime, k >= 1).

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 0

N. J. A. Sloane, Jun 08 2001

Let a(n)=p^e, then tau(a(n)^2) = tau(p^(2*e)) = 2*e+1 = 2*(e+1)-1 = tau(2*a(n))-1 where tau=A000005. - Juri-Stepan Gerasimov, Jul 14 2011

For n > 0, also the allowed indices of a Cossidente-Penttila graph. - Eric W. Weisstein, Feb 24 2025

Robert Israel, Table of n, a(n) for n = 0..10000
L. J. Corwin, Irreducible polynomials over the integers which factor mod p for every p, Unpublished Bell Labs Memo, Sep 07 1967 [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Odd Prime Power.

Cf. A061346, A000961, A005408, A061344, A075109, A075110.

[1] cat [n: n in [3..300 by 2] | IsPrimePower(n)]; // Bruno Berselli, Feb 25 2016

select(t -> nops(ifactors(t)[2])<=1, [seq(2*i+1,i=0..1000)]); # Robert Israel, Jun 11 2014
# alternative:
A061345 := proc(n)
    option remember;
    local k ;
    if n = 0 then
        1;
    else
        for k from procname(n-1)+2 by 2 do
            if nops(numtheory[factorset](k)) = 1 then
                return k ;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jun 25 2016
isOddPrimepower := n -> type(n, 'primepower') and not type(n, 'even'):
A061345List := up_to -> select(isOddPrimepower, [`$`(1..up_to)]):
A061345List(240); # Peter Luschny, Feb 02 2023

t={1};k=0;Do[If[k==1,AppendTo[t,a1]];k=0;Do[c=Sqrt[a^2+b^2];If[IntegerQ[c]&&GCD[a,b,c]==1,k++;a1=a;b1=b;c1=c;],{b,4,a^2/2,2}],{a,3,260,2}];t (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
Select[2 Range@ 130 - 1, PrimeNu@# < 2 &] (* Robert G. Wilson v, Jun 12 2014 *)
Join[{1}, Select[Range[1, 200, 2], PrimePowerQ]] (* Eric W. Weisstein, Feb 23 2025 *)

is(n)=my(p); if(isprimepower(n,&p), p>2, n==1) \\ Charles R Greathouse IV, Jun 08 2016

from sympy import primepi, integer_nthroot
def A061345(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) # Chai Wah Wu, Feb 03 2025

a(n) = A061344(n)-1.

Intersection of A000961 (prime powers) and A005408 (odd numbers). - Robert Israel, Jun 11 2014

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

A024556 Odd squarefree composite numbers.

Original entry on oeis.org

15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 105, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 165, 177, 183, 185, 187, 195, 201, 203, 205, 209, 213, 215, 217, 219, 221, 231, 235, 237, 247, 249, 253, 255, 259, 265, 267, 273

Offset: 1

N. J. A. Sloane, May 22 2000

Composite numbers n such that Sum_{k=1..n-1} floor(k^3/n) = (1/4)*(n-2)*(n^2-1) (equality also holds for all primes). - Benoit Cloitre, Dec 08 2002

Zak Seidov, Table of n, a(n) for n = 1..11999.
Eric Weisstein's World of Mathematics, Lehmer's Constant
Eric Weisstein's World of Mathematics, Prime Sums

Intersection of A056911 and A071904.
Subsequence of A061346.
Cf. A010051, A046388, A078837.

a024556 n = a024556_list !! (n-1)
a024556_list = filter ((== 0) . a010051) $ tail a056911_list
-- Reinhard Zumkeller, Apr 12 2012

Complement[Select[Range[3,281,2],SquareFreeQ],Prime[Range[PrimePi[281]]]] (* Harvey P. Dale, Jan 26 2011 *)

is(n)=n>1&&n%2&&!isprime(n)&&issquarefree(n) \\ Charles R Greathouse IV, Apr 12 2012

forstep(n=3,273,2,k=omega(n);if(k>1&&bigomega(n)==k,print1(n,", "))) \\ Hugo Pfoertner, Dec 19 2018

a(n) = (Pi^2/4)*n + O(n/log n). - Charles R Greathouse IV, Mar 12 2025

More terms from James Sellers, May 22 2000

A072995 Least k > 0 such that the number of solutions to x^k == 1 (mod k) 1 <= x <= k is equal to n, or 0 if no such k exists.

Original entry on oeis.org

1, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 225, 32, 289, 54, 361, 110, 147, 242, 529, 72, 125, 338, 81, 196, 841, 0, 961, 64, 1089, 578, 1225, 108, 1369, 722, 507, 100, 1681, 0, 1849, 484, 675, 1058, 2209, 144, 343, 250, 2601, 1378, 2809

Offset: 1

Benoit Cloitre, Aug 21 2002

nonn

A072989 lists the indices for which a(n) differs from A050399(n), e.g., in n = 20, 40, 52, ... in addition to the zeros in this sequence (n = 30, 42, 66, 70, 78, 90, ...). See also A009195 vs. A072994. [Corrected and extended by M. F. Hasler, Feb 23 2014]
The sequence seems difficult to extend, as the next term a(30) is larger than 5100. However, a(32)=64, a(64)=128 and a(128)=256 can be easily calculated. It thus appears that a(2^k)=2^(k+1), for k=1,2,3,.... Is this known to be true? - John W. Layman, Aug 05 2003 -- Answer: It's true. One could have defined the sequence so that a(1)=2: then it would be true for 2^0 also. - Don Reble, Feb 23 2014
a(30), if it exists, is greater than 400000. - Ryan Propper, Sep 10 2005
a(30) doesn't exist: If N is even, and divisible by D different odd primes, but not divisible by 2^D, then a(N) doesn't exist. - Don Reble, Feb 23 2014 [This and the preceding comment refer to the former definition lacking the clause "0 if no such k exists". - Ed.]
Conjecture: a(n)=0 iff n/2 is in A061346. - Robert G. Wilson v, Feb 23 2014
[n=420 seems to be a counterexample to the above conjecture. - M. F. Hasler, Feb 24 2014]
From Robert G. Wilson v, Mar 05 2014: (Start)
Observation:
If n = 1 then a(n) = 1 by definition;
If, but not iff, n (an even number) is a member of A238367 then a(n) = 0;
If n (an even number not in A238367) is {684, 954, ...}, then a(n) = 0;
If n (an odd number) is {273, 399, 651, 741, 777, 903, ...}, then a(n) = 0;
If p is a prime [A000040] and e is its exponent, then a(p^e) = p^(e+1);
If p is a prime then a(2p^e) = 2p^(e+1);
If p is a prime then a(n) # p since the f(p)=1.
(End)
Often A072995(n) equals A050399(n). They differ at n: 20, 30, 40, 42, 52, 60, 66, 68, 70, 78, 80, 84, 90, 100, 102, 104, 110, 114, 116, 120, 126, 130, 132, ... - Robert G. Wilson v, Dec 06 2014
When A072995(n)>0 and does not equal A050399(n): 20, 40, 52, 60, 68, 80, 84, 100, 104, 116, 120, 132, 136, 140, 148, 156, 160, 164, 168, 171, 180, 200, ... - Robert G. Wilson v, Dec 06 2014
When a(n) > 1, then 2n <= a(n) <= n^2. - Robert G. Wilson v, Dec 10 2014

Robert G. Wilson v, Table of n, a(n) for n = 1..1013 (first 200 terms from Don Reble)
Robert G. Wilson v, Graph of n, a(n) for n = 1..1000

Cf. A072994.

t = Table[0, {1000}]; f[n_] := (d = If[EvenQ@ n, 2, 1]; d*Length@ Select[ Range[ n/d], PowerMod[#, n, n] == 1 &]); f[1] = 1; k = 1; While[k < 520001, If[ PrimeQ@ k, k++]; a = f@ k; If[a < 1001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++]; t (* Robert G. Wilson v, Dec 12 2014 *)

A072995(n)=(n%2||n%2^(omega(n)-1)==0)&&for(k=1,9e9,A072994(k)==n&&return(k)) \\ M. F. Hasler, Feb 23 2014

First occurrence of n in A072994.

More terms from Don Reble, Feb 23 2014

Edited, at the suggestion of Don Reble, by M. F. Hasler, Feb 23 2014

A244623 Odd prime powers that are not primes.

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641, 15625, 16129, 16807, 17161, 18769, 19321, 19683

Offset: 1

Jani Melik, Jul 02 2014

nonn

Intersection of A061345 and A014076.

A014076 set minus A061346.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A025475.

Cf. A061345 (odd prime powers), A061346 (odd neither prime nor prime power), A062739 (odd powerful), A075109 (perfect powers), A136141.

Join[{1},Select[Range[1,20001,2],PrimePowerQ[#]&&(!PrimeQ[#])&]] (* Harvey P. Dale, Dec 11 2018 *)

isok(p) = ((p%2) && !isprime(p) && isprimepower(p)) || (p==1); \\ Michel Marcus, Jul 06 2021

def isA244623(n) :
   return(n % 2 == 1 and is_prime_power(n) == 1 and is_prime(n) == 0)
[n for n in (1..20000) if isA244623(n)]

a(n) = A079290(n) at least in the range n=3..94, and perhaps beyond. - R. J. Mathar, Aug 20 2014

Sum_{n>=1} 1/a(n) = 1/2 + Sum_{p prime} 1/(p*(p-1)) = 1/2 + A136141. - Amiram Eldar, Dec 21 2020

A120890 Ordered odd leg of primitive Pythagorean triangles.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 15, 17, 19, 21, 21, 23, 25, 27, 29, 31, 33, 33, 35, 35, 37, 39, 39, 41, 43, 45, 45, 47, 49, 51, 51, 53, 55, 55, 57, 57, 59, 61, 63, 63, 65, 65, 67, 69, 69, 71, 73, 75, 75, 77, 77, 79, 81, 83, 85, 85, 87, 87, 89, 91, 91, 93, 93, 95, 95, 97, 99, 99, 101, 103

Offset: 1

Lekraj Beedassy, Jul 12 2006

nonn

Ordered union of A081874 and A081934.

Conjecture: lim_{n->oo} a(n)/n = 1/Pi. Limit is also conjectured to be equal to lim_{n->oo} A120427(n)/n, see Selle reference, chapter 2.3.10. - Lothar Selle, Jun 21 2022

Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 4th impression 2022, chapter 2.2.1., see chapter 2.3.10 for identity of lim_{n->oo} A120427(n)/n.

Ray Chandler, Table of n, a(n) for n = 1..10000
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.
F. Richman, Pythagorean Triangles

Cf. A061346, A081874, A081934, A120891.

Cf. A120427.

Corrected by T. D. Noe, Oct 25 2006

A257591 Odd numbers which are not prime powers but which have a proper divisor == 1 mod 4.

Original entry on oeis.org

15, 35, 39, 45, 51, 55, 63, 65, 75, 85, 87, 91, 95, 99, 105, 111, 115, 117, 119, 123, 135, 143, 145, 147, 153, 155, 159, 165, 171, 175, 183, 185, 187, 189, 195, 203, 205, 207, 215, 219, 221, 225, 231, 235, 245, 247, 255, 259, 261, 265, 267, 273, 275, 279, 285

Offset: 1

N. J. A. Sloane, May 14 2015

nonn

			63 = 7*9 is not a power of a prime and has a proper divisor 9 == 1 mod 4.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..3124
R. Lauterbach, Equivariant Bifurcation and Absolute Irreducibility in R^8: A Contribution to Ize Conjecture and Related Bifurcations, Journal of Dynamics and Differential Equations, Oct 2014; DOI 10.1007/s10884-014-9402-1.

Subsequence of A061346.

lista(nn) = {forstep(n=1, nn, 2, if (!isprimepower(n) && sumdiv(n, d, (d != 1) && (d != n) && ((d % 4)==1)), print1(n, ", ")););} \\ Michel Marcus, Jun 19 2015

a(7)=57 removed and a(11)-a(55) added by Hiroaki Yamanouchi, May 20 2015

A083390 m such that 2m + 1 divides lcm(1,3,5,...,2m - 1).

Original entry on oeis.org

7, 10, 16, 17, 19, 22, 25, 27, 28, 31, 32, 34, 37, 38, 42, 43, 45, 46, 47, 49, 52, 55, 57, 58, 59, 61, 64, 66, 67, 70, 71, 72, 73, 76, 77, 79, 80, 82, 85, 87, 88, 91, 92, 93, 94, 97, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 112, 115, 117, 118, 122, 123, 124, 126

Offset: 1

Lekraj Beedassy, Jun 11 2003

nonn

Also m for which A025547(m)=A025547(m+1). Query: a(n) seems to be equal to A030343(n+4) - 1. Is this true?

While any odd number>1 can be the leg of a primitive Pythagorean triangle, the m-th odd number 2m+1=A061346 forms leg common to more than one PPT. - Lekraj Beedassy, Jul 12 2006

			10 is in the sequence because we have 2*10 - 1 = 19 and lcm(1,3,5,...,19)=166966608033225=7950790858725*21 which is divisible by 2*10 + 1 = 21.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A080765.

Select[Range[150],Divisible[LCM@@Range[1,2#-1,2],2#+1]&] (* Harvey P. Dale, Jan 22 2023 *)

isok(n) = {lc = 1; for (i = 1, 2*n-1, lc = lcm(lc, i);); return (lc % (2*n+1) == 0);} \\ Michel Marcus, Jul 27 2013

a(n) = (A061346(n)-1)/2. - David Wasserman, Oct 26 2004

More terms from David Wasserman, Oct 26 2004

A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).

Original entry on oeis.org

6, 5, 14, 3, 11, 13, 2, 17, 19, 1, 23, 5, 3, 29, 62, 1, 1, 37, 1, 41, 43, 1, 47, 7, 1, 53, 1, 1, 59, 61, 2, 1, 67, 1, 71, 73, 1, 1, 79, 3, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 11, 1, 5, 254, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1

Offset: 0

Paul Barry, Jun 09 2006

The subdiagonal of A120113 is -a(n).
From Robert Israel, Dec 03 2024: (Start)
a(n) is the product of the primes p such that 2*n + 3 or 2*n + 4 is a power of p.
Thus: a(n) = 1 if and only if neither 2*n + 3 nor 2*n + 4 is in A000961.
      if n + 1 = 2^k - 1 is a Mersenne number but not a Mersenne prime, then a(n) = 2;
      if n + 1 = 2^k - 1 is a Mersenne prime, then a(n) = 2 * (2^k - 1);
      otherwise a(n) is odd. (End)
Conjectures from Davide Rotondo, Dec 02 2024: (Start)
Except for 2, if a(n) is even then a(n)/2 is a Mersenne prime.
If a(n)=1 or a(n)=2 then (n*2)+3 is in A061346, or also, or (n+1) is in A083390. (End)

Muniru A Asiru, Table of n, a(n) for n = 0..5000

Cf. A000961, A099996, A120113.

List([0..75],n->Lcm(List([1..2*n+4],i->i))/Lcm(List([1..2*n+2],i->i))); # Muniru A Asiru, Mar 04 2019

A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
[A120114(n): n in [0..100]]; // G. C. Greubel, May 05 2023

f:= proc(n) local t,x,S;
   t:= 1;
   for x from 2*n+3 to 2*n+4 do
     S:= numtheory:-factorset(x);
     if nops(S) = 1 then t:= t*S[1] fi;
   od;
   t
end proc:
map(f, [$0..100]); # Robert Israel, Dec 03 2024

Table[(LCM@@Range[2n+4])/LCM@@Range[2n+2],{n,0,100}] (* Harvey P. Dale, Dec 15 2017 *)

def A120114(n):
    return lcm(range(1,2*n+5)) // lcm(range(1,2*n+3))
[A120114(n) for n in range(101)] # G. C. Greubel, May 05 2023

a(n) = A099996(n+2)/A099996(n+1). - Michel Marcus, May 06 2023

More terms from Harvey P. Dale, Dec 15 2017

A238367 Even numbers n such that n is not divisible by 2^d where d is the number of distinct odd prime factors of n.

Original entry on oeis.org

30, 42, 66, 70, 78, 90, 102, 110, 114, 126, 130, 138, 150, 154, 170, 174, 182, 186, 190, 198, 210, 222, 230, 234, 238, 246, 258, 266, 270, 282, 286, 290, 294, 306, 310, 318, 322, 330, 342, 350, 354, 366, 370, 374, 378, 390, 402, 406, 410, 414, 418, 420, 426, 430, 434, 438, 442, 450, 462, 470, 474, 490, 494, 498

Offset: 1

Robert G. Wilson v, Feb 25 2014

nonn

Inspired by comment by Don Reble in A072995.

2*A061346 = A238367 with the exceptions of 420, 660, 780, 924, 1020, 1092, 1140, 1260, ... .

			30 is in the sequence because the odd primes that divide 30 are 3 and 5, but 2^2 does not divide 30.

Cf. A072995.

Select[2 Range[250], Mod[#, 2^(PrimeNu[#] - 1)] != 0 &]

A321644 Squarefree odd composite numbers whose factors are all twin primes (not necessarily from the same pair).

Original entry on oeis.org

15, 21, 33, 35, 39, 51, 55, 57, 65, 77, 85, 87, 91, 93, 95, 105, 119, 123, 129, 133, 143, 145, 155, 165, 177, 183, 187, 195, 203, 205, 209, 213, 215, 217, 219, 221, 231, 247, 255, 273, 285, 287, 295, 301, 303, 305, 309, 319, 321, 323, 327, 341, 355, 357, 365

Offset: 1

Dimitris Valianatos, Nov 15 2018

This sequence has infinitely many terms if and only if the twin prime conjecture is true.

			a(3) = 33 = 3 * 11; 3 and 11 are both twin primes, but not from the same pair.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A024556, and hence of A056911, A061346, and A071904.

Cf. A001097.

N:= 1000: # to get all terms <= N
P:= select(isprime, {seq(i,i=3..(N+6)/3,2)}):
TP:= P intersect map(`-`,P,2):
TP:= TP union map(`+`,TP,2):
Agenda:= map(t -> [t],TP): Res:= NULL:
while Agenda <> {} do
   Agenda:= map(proc(t) local s; seq([op(t),s], s = select(s -> s > t[-1] and s*convert(t,`*`) <= N , TP)) end proc, Agenda);
   Res:= Res, op(map(convert,Agenda,`*`));
od:
sort([Res]); # Robert Israel, Jan 27 2019

seqQ[n_] := CompositeQ[n] && SquareFreeQ[n] && Module[{f = FactorInteger[n][[;;, 1]]}, Length[Select[f, PrimeQ[# - 2] || PrimeQ[# + 2] &]] == Length[f]]; Select[ Range[1, 365, 2], seqQ] (* Amiram Eldar, Nov 15 2018 *)

{forcomposite(n=3, 1000, if(moebius(n) <> 0, v = factor(n)~; i = 0;for(k = 1, #v,p=v[1,k]; if(isprime(p-2)||isprime(p+2), i++));if(i==#v,print1(n", "))))}

cp's OEIS Frontend

A061345 Powers of odd primes. Alternatively, 1 and the odd prime powers (p^k, p an odd prime, k >= 1).

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Magma

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A024556 Odd squarefree composite numbers.

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A072995 Least k > 0 such that the number of solutions to x^k == 1 (mod k) 1 <= x <= k is equal to n, or 0 if no such k exists.

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A244623 Odd prime powers that are not primes.

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Sage

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A120890 Ordered odd leg of primitive Pythagorean triangles.

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A257591 Odd numbers which are not prime powers but which have a proper divisor == 1 mod 4.

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PARI

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A083390 m such that 2m + 1 divides lcm(1,3,5,...,2m - 1).

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