A302796 -id:A302796 - cp's OEIS frontend

A338468 Odd squarefree numbers whose prime indices have no common divisor > 1.

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 205, 209, 215, 217, 219, 221, 231, 249, 253, 255, 265, 285, 287, 291, 295, 309, 323, 327, 329, 335, 341, 345, 355, 357, 381, 385, 391, 395, 403, 407, 411

Offset: 1

Gus Wiseman, Oct 29 2020

nonn

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.

Also Heinz numbers of relatively prime strict integer partitions with no 1's (A337452). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     15: {2,3}      145: {3,10}     249: {2,23}     355: {3,20}
     33: {2,5}      155: {3,11}     253: {5,9}      357: {2,4,7}
     35: {3,4}      161: {4,9}      255: {2,3,7}    381: {2,31}
     51: {2,7}      165: {2,3,5}    265: {3,16}     385: {3,4,5}
     55: {3,5}      177: {2,17}     285: {2,3,8}    391: {7,9}
     69: {2,9}      187: {5,7}      287: {4,13}     395: {3,22}
     77: {4,5}      195: {2,3,6}    291: {2,25}     403: {6,11}
     85: {3,7}      201: {2,19}     295: {3,17}     407: {5,12}
     93: {2,11}     205: {3,13}     309: {2,27}     411: {2,33}
     95: {3,8}      209: {5,8}      323: {7,8}      413: {4,17}
    105: {2,3,4}    215: {3,14}     327: {2,29}     415: {3,23}
    119: {4,7}      217: {4,11}     329: {4,15}     429: {2,5,6}
    123: {2,13}     219: {2,21}     335: {3,19}     435: {2,3,10}
    141: {2,15}     221: {6,7}      341: {5,11}     437: {8,9}
    143: {5,6}      231: {2,4,5}    345: {2,3,9}    447: {2,35}

A302568 is the prime or pairwise coprime version, counted by A007359.
A302697 is not required to be squarefree, counted by A302698 (ordered version: A337450).
A302796 allows evens, counted by A078374 (ordered version: A332004).
A337452 counts partitions with these Heinz numbers (ordered version: A337451).
A337984 is the pairwise coprime version, counted by A337485 (ordered version: A337697).
A005117 lists squarefree numbers.
A005408 lists odd numbers.
A056911 lists odd squarefree numbers.
A289509 lists Heinz numbers of relatively prime partitions, counted by A000837 (ordered version: A000740).
Cf. A000010, A007359, A051424, A055684, A056239, A101268, A289508, A302505, A302569, A302696, A302798, A337694.

Select[Range[1,100,2],SquareFreeQ[#]&&GCD@@PrimePi/@First/@FactorInteger[#]==1&]

A338552 Non-powers of primes whose prime indices have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507

Offset: 1

Gus Wiseman, Nov 03 2020

nonn

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.

Also Heinz numbers of non-constant, non-relatively prime partitions. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     21: {2,4}      183: {2,18}       305: {3,18}
     39: {2,6}      185: {3,12}       319: {5,10}
     57: {2,8}      189: {2,2,2,4}    321: {2,28}
     63: {2,2,4}    203: {4,10}       325: {3,3,6}
     65: {3,6}      213: {2,20}       333: {2,2,12}
     87: {2,10}     235: {3,15}       339: {2,30}
     91: {4,6}      237: {2,22}       351: {2,2,2,6}
    111: {2,12}     247: {6,8}        365: {3,21}
    115: {3,9}      259: {4,12}       371: {4,16}
    117: {2,2,6}    261: {2,2,10}     377: {6,10}
    129: {2,14}     267: {2,24}       387: {2,2,14}
    133: {4,8}      273: {2,4,6}      393: {2,32}
    147: {2,4,4}    299: {6,9}        399: {2,4,8}
    159: {2,16}     301: {4,14}       417: {2,34}
    171: {2,2,8}    303: {2,26}       427: {4,18}

A318978 allows prime powers, counted by A018783, with complement A289509.
A327685 allows nonprime prime powers.
A338330 is the coprime instead of relatively prime version.
A338554 counts the partitions with these Heinz numbers.
A338555 is the complement.
A000740 counts relatively prime compositions.
A000961 lists powers of primes, with complement A024619.
A051424 counts pairwise coprime or singleton partitions.
A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
A291166 ranks relatively prime compositions, with complement A291165.
A302696 gives the Heinz numbers of pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.

Select[Range[100],!(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]

Equals A024619 /\ A318978.

Complement of A000961 \/ A289509.

A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Gus Wiseman, Nov 03 2020

nonn

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.

Also Heinz numbers of partitions either constant or relatively prime (A338553). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

A327534 uses primes instead of prime powers.
A338331 is the pairwise coprime version, with complement A338330.
A338552 is the complement.
A338553 counts the partitions with these Heinz numbers.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A000961 lists powers of primes.
A018783 counts partitions whose prime indices are not relatively prime, with Heinz numbers A318978.
A051424 counts pairwise coprime or singleton partitions.
A291166 ranks relatively prime compositions, with complement A291165.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000740, A056239, A108572, A112798, A302569, A302796, A327685, A328677.

Select[Range[100],#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]

Equals A000961 \/ A289509.

Complement of A024619 /\ A318978.

A318718 Heinz numbers of strict integer partitions with a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Cf. A000009, A000837, A051424, A078374, A289509, A302696, A302796, A302797, A303280, A305713.

Select[Range[200],And[SquareFreeQ[#],GCD@@PrimePi/@FactorInteger[#][[All,1]]>1]&]

A302979 Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 15, 16, 22, 26, 30, 32, 33, 34, 35, 36, 38, 42, 46, 51, 55, 58, 62, 64, 66, 69, 70, 74, 77, 78, 82, 85, 86, 93, 94, 95, 100, 102, 105, 106, 110, 114, 118, 119, 122, 123, 128, 130, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165

Offset: 1

Gus Wiseman, Apr 16 2018

nonn

A prime index of n is a number m such that prime(m) divides n. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The number of uniform partitions of n with relatively prime parts is A078374(n).

			Sequence of all uniform relatively prime integer partitions begins (1), (11), (21), (111), (31), (41), (32), (1111), (51), (61), (321), (11111), (52), (71), (43), (2211).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796.

Select[Range[200],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]===1,SameQ@@FactorInteger[#][[All,2]]]&]

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A338468 Odd squarefree numbers whose prime indices have no common divisor > 1.

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A338552 Non-powers of primes whose prime indices have a common divisor > 1.

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A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

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A318718 Heinz numbers of strict integer partitions with a common divisor > 1.

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A302979 Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.

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