A275253 -id:A275253 - cp's OEIS frontend

A275246 Sequence of pairwise relatively prime numbers of class P_3 (see comment).

8, 15, 77, 221, 437, 899, 1517, 2021, 3127, 4087, 5183, 6557, 8633, 10403, 11663, 14351, 17947, 20711, 23707, 27221, 30967, 34571, 38021, 41989, 50621, 53357, 57599, 64507, 70747, 75067, 79523, 89951, 97343, 104927, 116939, 123197, 131753, 141367, 148987

Offset: 1

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

nonn

Consider the sequence P_0 of primes (A000040).
The simplest algorithm giving this sequence is the sieve of Eratosthenes. If we already know primes 2,3,...,p_n, then, by the algorithm of this sieve, the remaining numbers are not divisible by 2,3,...,p_(n-1)and to obtain p_(n+1) we should remove all remaining numbers divisible by p_n.
Note that we can also say that we remove all remaining numbers k for which GCD(k,p_n)>1. Although for generating the primes the algorithm is unchanged, in this form the algorithm we will apply in more general cases. Denote this algorithm by E*.
Remove 1 and the primes from the positive numbers. We get sequence
4,6,8,9,10,12,14,15,16,18,20,21,22,24,... (1)
By algorithm E*, keeping 4, we remove all even numbers; further keeping 4,9, we remove numbers divisible by 3, etc. We obtain sequence 4,9,25,49,...consisting of squares of primes (A001248). Denote this sequence by P_1. Removing P_1 from (1), we obtain sequence
6,8,10,12,14,15,16,18,20,21,22,24,26,... (2)
By algorithm E*, keeping 6, we remove all numbers divisible by 2 and 3; the least ramaning number is 35; keeping 6 and 35, we remove further all numbers divisible by 5 and 7, etc. We obtain sequence 6,35,143,...
(A089581). Denote this sequence by P_2.
The sequence P_3,...,P_8 are presented in A275246, A275248, A275249, A275251, A275252, A275253 respectively. All sequences {P_k} consist of pairwise relatively prime numbers, beginning with 2*(k+1)(which is a unique even number in sequence P_k).

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

k = 3; {2 (k + 1)}~Join~Map[Times @@ # &, Partition[Prime@ Range[k - 1, 78], 2, 2]] (* Michael De Vlieger, Jul 21 2016 *)

a(n)=if(n==1, 8,  prime(2*n-2)*prime(2*n-1)) \\ Bill McEachen, Nov 26 2024

a(n) = prime(2*n-2)*prime(2*n-1), n>1, a(1)=8. - Bill McEachen, Nov 26 2024

A275248 Sequence of pairwise relatively prime numbers of class P_4 (see comment in A275246).

Original entry on oeis.org

10, 21, 187, 247, 713, 1073, 1927, 2279, 3953, 4331, 6059, 7031, 9991, 10807, 13843, 14803, 20413, 20989, 26219, 28199, 34189, 34933, 41567, 44377, 52891, 54731, 61937, 66013, 74513, 76151, 86881, 91123, 103603, 106829, 122491, 125291, 139093, 142859, 155989

Offset: 1

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

nonn

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

k = 4; t = Union[{2 (k + 1)}, Select[# NextPrime[#, 2] &@ Prime@ Range@ 120, # > 2 (k + 1) &]]; Last@ Select[#, ListQ@ # &] &@ Table[If[Length@ t < n, 0, t = DeleteCases[t, k_ /; MemberQ[Select[Drop[t, n], ! CoprimeQ[#, t[[n]]] &], k]]], {n, Length@ t}] (* Michael De Vlieger, Jul 21 2016 *)

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.

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

cp's OEIS Frontend

A275246 Sequence of pairwise relatively prime numbers of class P_3 (see comment).

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A275248 Sequence of pairwise relatively prime numbers of class P_4 (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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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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PARI

Extensions

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.