A089581 -id:A089581 - cp's OEIS frontend

A275252 Sequence of pairwise relatively prime numbers of class P_7 (see comment in A275246).

16, 33, 85, 133, 299, 1189, 1457, 1961, 2537, 4453, 5561, 6319, 7663, 11009, 13081, 14017, 15481, 21823, 24883, 26123, 29177, 35657, 40301, 43039, 45173, 55189, 59881, 62857, 67519, 76693, 85039, 87391, 91709, 109999, 116843, 120983, 128083, 145097, 151979

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Jul 21 2016

nonn

Cf. A000040, A001248, A089581, A275246, A275248, A275249, A275251.

A275253 Sequence of pairwise relatively prime numbers of class P_8 (see comment in A275246).

Original entry on oeis.org

18, 95, 119, 253, 377, 1643, 1739, 2419, 2623, 5893, 5963, 7081, 7979, 13493, 13589, 14933, 15707, 25217, 25777, 28103, 29503, 40723, 42593, 44719, 45571, 61279, 61423, 64829, 68021, 86147, 86267, 88579, 92881, 118829, 118961, 127349, 130177, 153583, 155011

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Jul 21 2016

nonn

Cf. A000040, A001248, A089581, A275246, A275248, A275249, A275251, A275252.

A275407 Let e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n. The sequence lists 1 followed by numbers n for which e_n(2i-1)=e_n(2i), for all i>=1.

Original entry on oeis.org

1, 6, 35, 36, 143, 210, 216, 323, 667, 858, 1147, 1225, 1260, 1296, 1763, 1938, 2491, 3599, 4002, 4757, 5005, 5148, 5767, 6882, 7350, 7387, 7560, 7776, 9797, 10578, 11021, 11305, 11628, 12317, 14946, 16637, 19043, 20449, 21594, 22499, 23345, 24012, 25591, 28542

Offset: 1

Vladimir Shevelev, Jul 26 2016

nonn

There exists a permutation alpha of the sequence such that {alpha(a(n))} is a completely multiplicative function.

This sequence corresponds to the fixed points of A061898. - Rémy Sigrist, Feb 15 2023

			15 is not in the sequence, since 15 = 3*5 and the prime index of 5 is odd.
5148 is in the sequence, since 5148 = 2^2*3^2*11*13 and
(1) 3 is the next prime after 2,
(2) the exponents of 2 and 3 are equal,
(3) the prime index of 3 is even,
(4) 13 is the next prime after 11,
(5) the exponents of 11 and 13 are equal,
(6) the prime index of 13 is even.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000040, A001248, A061898, A089581, A275246, A275248, A275249, A275251, A275252, A275253.

inA275407Q:=If[EvenQ[Length[#]],Apply[And,Join[Map[#[[1]]+1==#[[2]]&&EvenQ[#[[2]]]&,PrimePi[#[[1]]]],Map[#[[1]]==#[[2]]&,#[[2]]]]]&[Map[Partition[#,2]&,Transpose[#]]],False]&[FactorInteger[#]]&;
Join[{1},Select[Range[10000],inA275407Q]] (* Peter J. C. Moses, Jul 29 2016 *)

isok(n) = {f = factor(n); nbpok = 0; for (k=1, #f~, ip = primepi(f[k, 1]); if ((ip % 2) && (kk = vecsearch(f[,1]~, prime(ip+1))) && (f[kk, 2] == f[k,2]), nbpok++;)); nbpok == #f~/2;} \\ Michel Marcus, Jul 27 2016

def is_A275407(n):
    L = list(factor(n))
    if is_odd(len(L)): return False
    for i in range(0,len(L)//2+1,2):
        if L[i][1] != L[i+1][1]: return False
        if L[i][0] != previous_prime(L[i+1][0]): return False
        if is_even(len(prime_range(1, L[i+1][0]))): return False
    return True
[n for n in (2..5000) if is_A275407(n)] # Peter Luschny, Jul 27 2016

More terms from Peter J. C. Moses, Jul 26 2016

A279119 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies gcd(i, j)=1.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Dec 06 2016

nonn

Also, for n>1, a(n) equals the index of the class of n relatively to the algorithm described in A275246 (i.e., if a(n)=k, then n is of class P_k).
From Rémy Sigrist, Dec 21 2016: (Start)
For any prime p, the sequence b_p(n)=a(p*n) is a bijection from A000027 to A001477:
- b_p is injective: b_p(n)=b_p(m) implies p*n=p*m or gcd(p*n,p*m)=1; as p>1, gcd(p*n,p*m)>1, so p*n=p*m and n=m.
- b_p is surjective: by contradiction: let k be the least number such that b_p(n) never equals k; we have a set of k terms (i_1,...,i_k) such that b_p(i_j) = j-1 for any j between 1 and k; let l be the least value such that p^l > max({1, i_1,...,i_k}). Then, by definition of a, a(p^l)=k, and b_p(p^(l-1))=k, which is a contraction.
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A008578, A001248, A089581, A275246, A275248, A275249, A275251, A275252, A275253.

g = vector(76, i, 1); for (n=1, #g, a = 0; while (gcd(g[a+1],n)>1, a++); g[a+1] *= n; print1 (a ", "))

a(2*n) = n-1 for any n>0.
a(n)=0 iff n belongs to A008578.
a(n)=1 iff n belongs to A001248.
a(n)=2 iff n belongs to A089581.
a(n)=3 iff n belongs to A275246.
a(n)=4 iff n belongs to A275248.
a(n)=5 iff n belongs to A275249.
a(n)=6 iff n belongs to A275251.
a(n)=7 iff n belongs to A275252.
a(n)=8 iff n belongs to A275253.

A085087 a(1) = 1; for n>1, a(n) = a(n-1)*n if n is prime, a(n) = a(n-1)/n if n is composite dividing a(n-1) else a(n) = a(n-1).

Original entry on oeis.org

1, 2, 6, 6, 30, 5, 35, 35, 35, 35, 385, 385, 5005, 5005, 5005, 5005, 85085, 85085, 1616615, 1616615, 1616615, 1616615, 37182145, 37182145, 37182145, 37182145, 37182145, 37182145, 1078282205, 1078282205, 33426748355, 33426748355

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 02 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..500

Cf. A034386, A089581, A099788.

a = {1}; Do[AppendTo[a, If[PrimeQ[n], a[[-1]]*n, If[Divisible[a[[-1]], n], a[[-1]]/n, a[[-1]]]]], {n, 2, 32}]; a (* Ivan Neretin, May 21 2015 *)

print1(k=1); for(n=2,99, if(isprime(n), k*=n, if(k%n==0, k/=n)); print1(", "k)) \\ Charles R Greathouse IV, May 21 2015

Let k be the number of members of A089581 that are <= n; then a(n) = A034386(n)/A099788(k). - David Wasserman, Jan 25 2005

More terms from David Wasserman, Jan 25 2005

A135232 Sum of the products of the first n prime pairs.

Original entry on oeis.org

6, 41, 184, 507, 1174, 2321, 4084, 6575, 10174, 14931, 20698, 28085, 37882, 48903, 61220, 77857, 96900, 119399, 144990, 173881, 206280, 243143, 282346, 329399, 381382, 437069, 497560, 565151, 638050, 715887, 798806, 894283, 993504, 1105051

Offset: 1

Cino Hilliard, Feb 14 2008

nonn

			The sum of the products of the first 3 prime pairs is 2*3+5*7+11*13 = 184, the 3rd entry in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A089581.

Table[Sum[Prime[2*k - 1]*Prime[2*k], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Oct 04 2016 *)

g(n)=s=0;forstep(x=1,n*2,2,s+=prime(x)*prime(x+1);print1(s,","))

a(n) = Sum_{i=1..n} A089581(i). - R. J. Mathar, Feb 19 2008

A275474 Numbers n for which e_n(2i)=e_n(2i+1), for all i>=1, where e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 30, 32, 60, 64, 77, 120, 128, 154, 221, 225, 240, 256, 308, 437, 442, 450, 480, 512, 616, 874, 884, 899, 900, 960, 1024, 1155, 1232, 1517, 1748, 1768, 1798, 1800, 1920, 2021, 2048, 2310, 2464, 3034, 3127, 3315, 3375, 3496, 3536, 3596, 3600

Offset: 1

Vladimir Shevelev, Jul 29 2016

nonn

There exists a permutation alpha of the sequence such that {alpha(a(n))} is a completely multiplicative function.

Numbers which are the product of zero or more of {2, 3*5, 7*11, 13*17, 19*23, ...} with multiplicity. - Charles R Greathouse IV, Jul 30 2016

			1 is a member, since all e_1(k)=0;
Powers 2^m, m>=1, are members, since e_2^m(k)=0, for all k>=2;
15 is a member, since e_15(2)*e_15(3)=1;
n = 2983500 is a member, since e_n(1)=2, e_n(2)=e_n(3)=3 and e_n(6)=e_n(7)=1, all other e_n(k)=0.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A001248, A089581, A275246, A275248, A275249, A275251, A275252, A275253, A275407.

is(n)=my(f=factor(n>>valuation(n,2))); if (#f~%2, return(0)); for(i=1,#f~/2, if(f[2*i-1,2]!=f[2*i,2] || nextprime(f[2*i-1,1]+1)!=f[2*i,1], return(0))); for(i=1,#f~/2, if(primepi(f[2*i,1])%2==0, return(0))); 1 \\ Charles R Greathouse IV, Jul 30 2016

list(lim)=my(v=List([1,2]),p=3,pStart=2,pEnd,start=2,end,nStart,t); lim\=1; forprime(q=5,sqrtint(lim+1)+1, p=if(p, listput(v,p*q); 0, q)); end=pEnd=#v; for(n=2,logint(lim,2), nStart=end+1; for(i=start,end, for(j=pStart,pEnd, t=v[i]*v[j]; if(t>lim, break); listput(v, t))); start=nStart; end=#v); Set(v) \\ Charles R Greathouse IV, Jul 30 2016

More terms from Peter J. C. Moses, Jul 29 2016

A307360 A sequence in which every divisor other than 1 is used at most three times.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

Joshua R. Tint, Apr 04 2019

In other words, for every k > 1, there are at most 3 multiples of k in the sequence. - Rémy Sigrist,  Apr 08 2019
The sequence begins at 1. The smallest integer greater than the last term which is not divisible by a divisor already used three times (excluding one) is added to the sequence.
Contains all prime numbers (A000040), given that the prime numbers only have the divisors of themselves and one, by definition, therefore the only divisor which could exist in the sequence already to disqualify the number from inclusion in the sequence would be the prime number itself, but a number cannot have a divisor higher than itself (the prime numbers), so given that the sequence increases, the divisor could not exist in the sequence, and any prime number would be included.
Terms are {1} or primes or squares of primes (A000430) or numbers of the form prime(2k + 1) * prime(2k + 2) (A089581) where k >= 0. - David A. Corneth, Apr 09 2019

			For instance, 8 is not in the sequence because 2, 4, and 6 are all divisible by 2 and appear previously in the sequence. The sequence, then, skips to nine. After 9, no more numbers divisible by three appear in the sequence, given that after 3 and 6, it is the third number divisible by three to appear in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Aaron Greicius, The Prime Numbers, Lecture Fall 2012.

See A166684 for the variant in which every divisor other than one is used at most twice.

Union of {1}, A000430 and A089581.

N:= 1000: # for terms <= N
M:= Vector(N):
Candidates:= {$2..N}:
A[1]:= 1:
for n from 2 while Candidates <> {} do
  A[n]:= min(Candidates):
  Candidates:= Candidates minus {A[n]};
  for d in numtheory:-divisors(A[n]) minus {1} do
     M[d]:= M[d]+1;
     if M[d] = 3 then Candidates:= Candidates minus {seq(i,i=2*d..N, d)} fi;
  od;
od:
seq(A[i],i=1..n-1); # Robert Israel, Apr 09 2019

Select[Range@ 229, Or[# == 1, PrimeQ@ #, PrimeQ@ Sqrt@ #, And[SquareFreeQ@ #, If[PrimeNu@ # == 2, And[OddQ@ First@ #, Apply[SameQ, (# - {1, 2})/2]] &@ PrimePi[FactorInteger[#][[All, 1]]], False]]] &] (* Michael De Vlieger, Apr 11 2019 *)

is(n) = if(n==1, return(1)); my(f=factor(n)); if(f[, 2] == [1]~ || f[, 2] ==[2]~, return(1)); if(f[,2] == [1,1]~ && nextprime(f[1,1]+1) == f[2,1] && primepi(f[1,1]) % 2 == 1, return(1)); 0 \\ David A. Corneth, Apr 09 2019

More terms from Jinyuan Wang, Apr 07 2019

cp's OEIS Frontend

A275252 Sequence of pairwise relatively prime numbers of class P_7 (see comment in A275246).

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A275253 Sequence of pairwise relatively prime numbers of class P_8 (see comment in A275246).

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A275407 Let e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n. The sequence lists 1 followed by numbers n for which e_n(2*i-1)=e_n(2*i), for all i>=1.

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A279119 Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies gcd(i, j)=1.

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A085087 a(1) = 1; for n>1, a(n) = a(n-1)*n if n is prime, a(n) = a(n-1)/n if n is composite dividing a(n-1) else a(n) = a(n-1).

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A135232 Sum of the products of the first n prime pairs.

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A275474 Numbers n for which e_n(2*i)=e_n(2*i+1), for all i>=1, where e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n.

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A307360 A sequence in which every divisor other than 1 is used at most three times.

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A275407 Let e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n. The sequence lists 1 followed by numbers n for which e_n(2i-1)=e_n(2i), for all i>=1.

A275474 Numbers n for which e_n(2i)=e_n(2i+1), for all i>=1, where e_n(k)>=0 denote the exponent of prime(k) in the prime power representation of n.