A176881 -id:A176881 - cp's OEIS frontend

A270650 Min(i, j), where p(i)*p(j) is the n-th term of A006881.

1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 3, 2, 1, 1, 3, 2, 1, 4, 1, 3, 1, 2, 4, 2, 1, 3, 1, 2, 3, 1, 4, 1, 2, 2, 4, 1, 2, 1, 5, 3, 1, 3, 1, 2, 4, 1, 2, 1, 2, 3, 5, 1, 2, 1, 4, 3, 1, 5, 2, 1, 3, 4, 1, 2, 6, 1, 3, 2, 6, 2, 5, 1, 4, 1, 3, 2, 1, 1, 4, 2, 3, 1

Offset: 1

Clark Kimberling, Apr 25 2016

			A006881 = (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, ... ), the increasing sequence of all products of distinct primes.  The first 4 factorizations are 2*3, 2*5, 2*7, 3*5, so that (a(1), a(2), a(3), a(4)) = (1,1,1,2).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A006881, A096916, A070647, A270652, A270003.

mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G. Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)
PrimePi[u1]  (* A270650 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)
PrimePi[v1]  (* A270652 *)
d = v1 - u1  (* A176881 *)
Map[PrimePi[FactorInteger[#][[1, 1]]] &, Select[Range@ 240, And[SquareFreeQ@ #, PrimeOmega@ # == 2] &]] (* Michael De Vlieger, Apr 25 2016 *)

A270652 Max(i,j), where p(i)*p(j) is the n-th term of A006881.

Original entry on oeis.org

2, 3, 4, 3, 4, 5, 6, 5, 7, 4, 8, 6, 9, 7, 5, 8, 10, 11, 6, 9, 12, 5, 13, 7, 14, 10, 6, 11, 15, 8, 16, 12, 9, 17, 7, 18, 13, 14, 8, 19, 15, 20, 6, 10, 21, 11, 22, 16, 9, 23, 17, 24, 18, 12, 7, 25, 19, 26, 10, 13, 27, 8, 20, 28, 14, 11, 29, 21, 7, 30, 15, 22

Offset: 1

Clark Kimberling, Apr 25 2016

			A006881 = (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, ... ), the increasing sequence of all products of distinct primes.  The first 4 factorizations are 2*3, 2*5, 2*7, 3*5, so that (a(1), a(2), a(3), a(4)) = (2,3,4,3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A006881, A096916, A270650, A070647, A270003.

mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G. Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)
PrimePi[u1]  (* A270650 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)
PrimePi[v1]  (* A270652 *)
d = v1 - u1  (* A176881 *)
Map[PrimePi[FactorInteger[#][[-1, 1]]] &, Select[Range@ 240, And[SquareFreeQ@ #, PrimeOmega@ # == 2] &]] (* Michael De Vlieger, Apr 25 2016 *)

from math import isqrt
from sympy import primepi, primerange, primefactors
def A270652(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: 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+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    return primepi(max(primefactors(bisection(f,n,n)))) # Chai Wah Wu, Oct 23 2024

A324331 a(n) = (n-1)^2 - phi(n)*sigma(n), where phi is A000010 and sigma is A000203.

Original entry on oeis.org

-1, -2, -4, -5, -8, 1, -12, -11, -14, 9, -20, 9, -24, 25, 4, -23, -32, 55, -36, 25, 16, 81, -44, 49, -44, 121, -44, 57, -56, 265, -60, -47, 64, 225, 4, 133, -72, 289, 100, 81, -80, 529, -84, 169, 64, 441, -92, 225, -90, 541, 196, 249, -104, 649, 36, 145, 256, 729, -116, 793

Offset: 1

Michel Marcus, Feb 23 2019

For squarefree semiprimes n = p*q a(n)=(p-q)^2 is a square. But the converse, a(n) is prime, can happen: see A324332.

Brian Alspach, Research problems, Problem 18, Discrete Math 40 (1982), page 126.

Cf. A000010, A000203, A006881, A069249, A176881.

Cf. A324332, A324333, A324334.

Table[(n-1)^2 - EulerPhi[n]*DivisorSigma[1, n], {n, 1, 60}] (* Vaclav Kotesovec, Feb 23 2019 *)

PARI
```
a(n) = (n-1)^2 - eulerphi(n)*sigma(n);
```

a(A006881(n)) = A176881(n)^2.

a(n) = A069249(n) - 2*n + 1. - Amiram Eldar, Dec 04 2023

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A270650 Min(i, j), where p(i)*p(j) is the n-th term of A006881.

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A270652 Max(i,j), where p(i)*p(j) is the n-th term of A006881.

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A324331 a(n) = (n-1)^2 - phi(n)*sigma(n), where phi is A000010 and sigma is A000203.

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