A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).

3, 5, 11, 17, 26, 31, 35, 38, 39, 41, 49, 57, 58, 59, 65, 67, 69, 77, 83, 86, 87, 94, 109, 119, 127, 129, 133, 146, 148, 157, 158, 179, 191, 202, 206, 211, 217, 235, 237, 241, 244, 253, 274, 277, 278, 283, 284, 287, 291, 298, 303, 319, 326, 331, 333, 334, 353

Offset: 1

Author

Gus Wiseman, Mar 14 2024

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
     3: {2}        67: {19}        158: {1,22}
     5: {3}        69: {2,9}       179: {41}
    11: {5}        77: {4,5}       191: {43}
    17: {7}        83: {23}        202: {1,26}
    26: {1,6}      86: {1,14}      206: {1,27}
    31: {11}       87: {2,10}      211: {47}
    35: {3,4}      94: {1,15}      217: {4,11}
    38: {1,8}     109: {29}        235: {3,15}
    39: {2,6}     119: {4,7}       237: {2,22}
    41: {13}      127: {31}        241: {53}
    49: {4,4}     129: {2,14}      244: {1,1,18}
    57: {2,8}     133: {4,8}       253: {5,9}
    58: {1,10}    146: {1,21}      274: {1,33}
    59: {17}      148: {1,1,12}    277: {59}
    65: {3,6}     157: {37}        278: {1,34}
		

Crossrefs

For prime factors instead of divisors on both sides we get A319899.

For prime factors on LHS we get A370802, for distinct prime factors A371177.

The RHS is A370820, for prime factors instead of divisors A303975.

For (greater than) instead of (equal) we get A371166.

For (less than) instead of (equal) we get A371167.

Partitions of this type are counted by A371172.

Other inequalities: A370348 (A371171), A371168 (A371173), A371169, A371170.

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, A112798 indices, length A001222.

A239312 counts divisor-choosable partitions, ranks A368110.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A370320 counts non-divisor-choosable partitions, ranks A355740.

A370814 counts divisor-choosable factorizations, complement A370813.

Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

Programs

Mathematica

Select[Range[100],Length[Divisors[#]] == Length[Union@@Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

Formula

A000005(a(n)) = A370820(a(n)).