A339911 -id:A339911 - cp's OEIS frontend

A339908 Odd squarefree numbers k > 1 for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

33, 65, 129, 141, 145, 161, 177, 201, 217, 249, 253, 321, 385, 393, 417, 501, 537, 649, 681, 705, 721, 737, 849, 865, 897, 913, 973, 993, 1041, 1057, 1081, 1101, 1121, 1135, 1149, 1169, 1177, 1281, 1329, 1345, 1401, 1441, 1457, 1473, 1497, 1509, 1537, 1561, 1569, 1585, 1633, 1689, 1729, 1761, 1793, 1821, 1837, 1841

Offset: 1

Antti Karttunen, Dec 21 2020

nonn

All terms can be found in A339911. Also, all nonmultiples of 3 certainly occur in A339912 also.

The first term of the form 4u+3 is 1135.

Antti Karttunen, Table of n, a(n) for n = 1..26693; all terms < 2^20

Cf. A000010, A001222.
Subsequence of A056911 and of A339907, A339911.
Cf. also A339912.
Cf. A339909 (a subsequence).

isA339908(n) = ((n>1)&&(n%2)&&issquarefree(n)&&(bigomega(eulerphi(n))

A339910 After 1, numbers k > 1 such that k has fewer prime divisors than k-1, when they are counted with multiplicity.

Original entry on oeis.org

1, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 41, 43, 46, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 82, 83, 85, 89, 91, 93, 97, 101, 103, 105, 106, 107, 109, 111, 113, 115, 118, 121, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 153, 155, 157, 161, 163, 166, 167, 169, 173, 177, 179

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

List of starts of nondecreasing runs of values of A001222 (= bigomega, number of prime factors, when counted with multiplicity).

Differs from A303578 (a similar sequence computed for tau, the number of divisors) for the first time at n=44, where a(44) = 105, a term missing from A303578.

Antti Karttunen, Table of n, a(n) for n = 1..28208; all terms <= 65537

Cf. A001222.
One more than A322839.
Cf. A339911, A339912 for subsequences.
Cf. also A303578.

1 + {0}~Join~Position[Sign@ Differences[Array[PrimeOmega, 180]], -1][[All, 1]] (* Michael De Vlieger, Dec 22 2020 *)

isA339910(n) = ((1==n)||(bigomega(n)

A339912 Numbers k > 1 for which bigomega(k) < bigomega(k-1)/2, where bigomega gives the number of prime factors, counted with multiplicity.

Original entry on oeis.org

13, 17, 19, 29, 31, 33, 37, 41, 43, 49, 53, 61, 65, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 121, 127, 129, 131, 137, 139, 145, 149, 151, 157, 161, 163, 169, 173, 177, 181, 191, 193, 197, 199, 201, 209, 211, 217, 223, 229, 233, 239, 241, 251, 253, 257, 265, 269, 271, 277, 281, 283, 289, 293, 301, 305, 307, 311, 313

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12313; all terms <= 65537

Cf. A001222.
Subsequence of A339911 and of A339910.
Cf. also A339908.

Select[Range[3, 313, 2], PrimeOmega[#] < PrimeOmega[# - 1]/2 &] (* Michael De Vlieger, Dec 22 2020 *)
Flatten[Position[Partition[PrimeOmega[Range[400]],2,1],?(#[[2]]<#[[1]]/2&),1,Heads->False]]+1 (* _Harvey P. Dale, Jan 11 2024 *)

isA339912(n) = ((n>1)&&((2*bigomega(n))

cp's OEIS Frontend

A339908 Odd squarefree numbers k > 1 for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

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A339910 After 1, numbers k > 1 such that k has fewer prime divisors than k-1, when they are counted with multiplicity.

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A339912 Numbers k > 1 for which bigomega(k) < bigomega(k-1)/2, where bigomega gives the number of prime factors, counted with multiplicity.

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Mathematica

PARI