A342258 -id:A342258 - cp's OEIS frontend

A342259 Numbers k such that omega(k-1) = omega(k)-1 and omega(k+1) = omega(k)+1, where omega(m) is the number of distinct primes dividing m, A001221(m).

65, 104, 129, 164, 194, 272, 284, 314, 344, 384, 398, 464, 524, 608, 614, 626, 662, 692, 734, 758, 824, 968, 1025, 1094, 1172, 1238, 1280, 1304, 1364, 1424, 1448, 1454, 1532, 1544, 1595, 1658, 1664, 1682, 1724, 1754, 1832, 1868, 1869, 1934, 1952, 2000, 2001, 2012

Offset: 1

Hugo Pfoertner, Mar 07 2021

nonn

			a(1)=65: 64=2^6 (1 distinct prime 2), 65=5*13 (2 distinct primes 5 and 13), 66=2*3*11 (3 distinct primes 2, 3, and 11).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001221, A342258.

seq[nmax_] := Module[{om}, om[n_] := om[n] = PrimeNu[n]; Select[Range[2, nmax], om[# - 1] == om[#] - 1 && om[# + 1] == om[#] + 1 &]]; seq[2500] (* Amiram Eldar, Sep 19 2024 *)

for(n=3,2100,my(om=omega(n));if(omega(n-1)==om-1&&omega(n+1)==om+1,print1(n,", ")))

A342246 Numbers k such that k-1, k and k+1 are all composite with four, five and six (not necessarily distinct) prime factors respectively.

Original entry on oeis.org

11151, 13455, 23375, 26271, 31311, 33776, 36125, 40375, 45495, 46375, 48411, 49049, 49167, 61335, 63125, 74151, 77895, 78111, 78351, 80271, 82575, 83511, 84591, 86031, 87375, 88749, 90207

Offset: 1

Sean Lestrange, Mar 07 2021

nonn

			For k=11151 we have 11150 = 2 * 5^2 * 223 which is composite with four prime factors, 11151 = 3^3 * 7 * 59 which is composite with five prime factors and 11152 = 2^4 * 17 * 41 which is composite with six prime factors.

Dumitru Damian, Table of n, a(n) for n = 1..10456

Subsequence of A342258.

Cf. A001222, A014613, A014614, A046306.

SequencePosition[PrimeOmega[Range[100000]],{4,5,6}][[;;,1]]+1 (* Harvey P. Dale, Jul 30 2024 *)

for(n=3,100000,if(bigomega(n-1)==4&&bigomega(n)==5&&bigomega(n+1)==6,print1(n,", "))) \\ Hugo Pfoertner, Mar 07 2021

# The following SageMath algorithm will generate all terms up to 100000
L=[]
for n in [1..100000]:
    sum1, sum2, sum3 = 0,0,0
    for f in list(factor(n)):
        sum1+=f[1]
    for f in list(factor(n+1)):
        sum2+=f[1]
    for f in list(factor(n+2)):
        sum3+=f[1]
    if sum1==4 and sum2==5:
        if sum3==6:
            L.append(n+1)
print(L)

cp's OEIS Frontend

A342259 Numbers k such that omega(k-1) = omega(k)-1 and omega(k+1) = omega(k)+1, where omega(m) is the number of distinct primes dividing m, A001221(m).

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