A303661 -id:A303661 - cp's OEIS frontend

A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A268390 in lacking 210.
First differs from A238748 in lacking 210.
These are the Heinz numbers of the partitions counted by A367588.

			The terms together with their prime indices begin:
     6: {1,2}         57: {2,8}        106: {1,16}
    10: {1,3}         58: {1,10}       111: {2,12}
    14: {1,4}         62: {1,11}       115: {3,9}
    15: {2,3}         65: {3,6}        118: {1,17}
    21: {2,4}         69: {2,9}        119: {4,7}
    22: {1,5}         74: {1,12}       122: {1,18}
    26: {1,6}         77: {4,5}        123: {2,13}
    33: {2,5}         82: {1,13}       129: {2,14}
    34: {1,7}         85: {3,7}        133: {4,8}
    35: {3,4}         86: {1,14}       134: {1,19}
    36: {1,1,2,2}     87: {2,10}       141: {2,15}
    38: {1,8}         91: {4,6}        142: {1,20}
    39: {2,6}         93: {2,11}       143: {5,6}
    46: {1,9}         94: {1,15}       145: {3,10}
    51: {2,7}         95: {3,8}        146: {1,21}
    55: {3,5}        100: {1,1,3,3}    155: {3,11}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

The case of any multiplicities is A007774, counts A002133.
Partitions of this type are counted by A367588.
The case of distinct exponents is A367589, counts A182473.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A006881, A039956, A071625, A072233, A072774, A109297, A303661, A367580.

Select[Range[100], SameQ@@Last/@If[#==1, {}, FactorInteger[#]]&&PrimeNu[#]==2&]
Select[Range[200],PrimeNu[#]==2&&Length[Union[FactorInteger[#][[;;,2]]]]==1&] (* Harvey P. Dale, Aug 04 2025 *)

Union of A006881 and A303661. - Michael De Vlieger, Dec 01 2023

A355462 Powerful numbers divisible by exactly 2 distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500

Offset: 1

Amiram Eldar, Jul 03 2022

nonn

First differs from A286708 at n = 25.
Number of the form p^i * q^j, where p != q are primes and i,j > 1.
Numbers k such that A001221(k) = 2 and A051904(k) >= 2.
The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).
675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

			36 is a term since 36 = 2^2 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Intersection of A001694 and A007774.
Subsequence of A286708.
Subsequences: A085986, A143610, A162142, A179646, A179666, A179671, A179689, A179694, A179699, A179702, A179705, A189988, A189990, A189991, A190464, A190465, A303661.
Cf. A000005, A001221, A051904, A060355, A136141, A264828.

Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]

is(n) = {my(f=factor(n)); #f~ == 2 && vecmin(f[,2]) > 1};

Sum_{n>=1} 1/a(n) = ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.1583860791... .

cp's OEIS Frontend

A367590 Numbers with exactly two distinct prime factors, both appearing with the same exponent.

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