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

1,1

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.

Links

Table of n, a(n) for n=1..57.

Formula

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

Example

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}

Mathematica

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

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.

Sequence in context: A309427 A211435 A049752 * A286264 A347027 A147350

Adjacent sequences: A371162 A371163 A371164 * A371166 A371167 A371168

Keyword

nonn

Author

Gus Wiseman, Mar 14 2024

Status

approved