A302242 -id:A302242 - cp's OEIS frontend

A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 15, 16, 17, 20, 22, 24, 26, 29, 30, 31, 32, 33, 34, 40, 41, 43, 44, 47, 48, 51, 52, 55, 58, 59, 60, 62, 64, 66, 67, 68, 73, 79, 80, 82, 83, 85, 86, 88, 93, 94, 96, 101, 102, 104, 109, 110, 113, 116, 118, 120, 123, 124, 127

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
08: {{},{},{}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
13: {{1,2}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
20: {{},{},{2}}
22: {{},{3}}
24: {{},{},{},{1}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
32: {{},{},{},{},{}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Join@@primeMS/@primeMS[#]&]

A325033 Sum of sums of the multisets of prime indices of each prime index of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 3, 1, 3, 2, 3, 0, 4, 2, 3, 2, 3, 3, 4, 1, 4, 3, 3, 2, 4, 3, 5, 0, 4, 4, 4, 2, 4, 3, 4, 2, 6, 3, 5, 3, 4, 4, 5, 1, 4, 4, 5, 3, 4, 3, 5, 2, 4, 4, 7, 3, 5, 5, 4, 0, 5, 4, 8, 4, 5, 4, 5, 2, 6, 4, 5, 3, 5, 4, 6, 2, 4, 6, 9, 3, 6, 5, 5

Offset: 1

Gus Wiseman, Mar 25 2019

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.

			91 has prime indices {4,6} with prime indices {{1,1},{1,2}} with sum a(91) = 5.

Cf. A000720, A056239, A109082, A112798, A302242.

Cf. A324926, A325032, A325034, A325035.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Plus@@Join@@primeMS/@primeMS[n],{n,100}]

Totally additive with a(prime(n)) = A056239(n).

A302590 Squarefree numbers whose prime indices are prime numbers.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 41, 51, 55, 59, 67, 83, 85, 93, 109, 123, 127, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 241, 249, 255, 277, 283, 295, 327, 331, 335, 341, 353, 367, 381, 401, 415, 431, 451, 461, 465, 471, 509, 527, 537, 545, 547, 561, 563

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.
From David A. Corneth, Feb 05 2021: (Start)
Product_{p in A006450} (p + 1)/p where primepi(p) <= 10^k for k = 3..9 respectively is
3221793975627545730894469494385382768...
3962097386916566795581118542505513350...
4423525010102788492232765893521739629...
4739349879225654126399615785205666552...
4969363158706022367680967716958174889...
5144436325229538304870684054018856517...
5282263225826916578696019016723107071... (End)

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
001: {}
003: {{1}}
005: {{2}}
011: {{3}}
015: {{1},{2}}
017: {{4}}
031: {{5}}
033: {{1},{3}}
041: {{6}}
051: {{1},{4}}
055: {{2},{3}}
059: {{7}}
067: {{8}}
083: {{9}}
085: {{2},{4}}
093: {{1},{5}}
109: {{10}}
123: {{1},{6}}
127: {{11}}
155: {{2},{5}}
157: {{12}}
165: {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302568.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[600],SquareFreeQ[#]&&And@@PrimeQ/@primeMS[#]&]

ok(n)={issquarefree(n) && !#select(p->!isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Intersection of A005117 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A320325 Numbers whose product of prime indices is a perfect power.

Original entry on oeis.org

7, 9, 14, 18, 19, 21, 23, 25, 27, 28, 36, 38, 42, 46, 49, 50, 53, 54, 56, 57, 63, 72, 76, 81, 84, 92, 97, 98, 100, 103, 106, 108, 112, 114, 115, 121, 125, 126, 131, 133, 144, 147, 151, 152, 159, 161, 162, 168, 169, 171, 175, 183, 184, 185, 189, 194, 195, 196

Offset: 1

Gus Wiseman, Oct 10 2018

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.

			The terms together with their corresponding multiset multisystems (A302242):
   7: {{1,1}}
   9: {{1},{1}}
  14: {{},{1,1}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  36: {{},{},{1},{1}}
  38: {{},{1,1,1}}
  42: {{},{1},{1,1}}
  46: {{},{2,2}}
  49: {{1,1},{1,1}}
  50: {{},{2},{2}}
  53: {{1,1,1,1}}
  54: {{},{1},{1},{1}}
  56: {{},{},{},{1,1}}
  57: {{1},{1,1,1}}
  63: {{1},{1},{1,1}}
  72: {{},{},{},{1},{1}}
  76: {{},{},{1,1,1}}
  81: {{1},{1},{1},{1}}

Cf. A000720, A001222, A001597, A003963, A056239, A064573, A112798, A302242, A305551, A306017, A319056, A319066, A320322, A320323, A320324.

Select[Range[100],GCD@@FactorInteger[Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]][[All,2]]>1&]

A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 91, 161, 299, 329, 377, 611, 667, 1261, 1363, 1937, 2021, 2093, 2117, 2639, 4277, 4669, 7567, 8671, 8827, 9541, 13559, 14053, 14147, 14819, 15617, 16211, 17719, 23989, 24017, 26273, 27521, 28681, 29003, 31349, 31913, 36569, 44551, 44603, 46483, 48691

Offset: 1

Gus Wiseman, Oct 13 2018

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. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
     1: {}
     7: {{1,1}}
    13: {{1,2}}
    91: {{1,1},{1,2}}
   161: {{1,1},{2,2}}
   299: {{2,2},{1,2}}
   329: {{1,1},{2,3}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
   667: {{2,2},{1,3}}
  1261: {{3,3},{1,2}}
  1363: {{1,3},{2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2093: {{1,1},{2,2},{1,2}}
  2117: {{1,3},{2,4}}
  2639: {{1,1},{1,2},{1,3}}
  4277: {{1,1},{1,2},{2,3}}
  4669: {{1,1},{2,2},{1,3}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320456, A320458, A320459, A320462, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

A302797 Squarefree numbers whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions with pairwise coprime parts.

Original entry on oeis.org

1, 2, 6, 10, 14, 15, 22, 26, 30, 33, 34, 35, 38, 46, 51, 55, 58, 62, 66, 69, 70, 74, 77, 82, 85, 86, 93, 94, 95, 102, 106, 110, 118, 119, 122, 123, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165, 166, 170, 177, 178, 186, 187, 190, 194, 201, 202

Offset: 1

Gus Wiseman, Apr 13 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1. A single number is not considered coprime unless it is equal to 1.

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

			Sequence of terms together with their sets of prime indices begins:
01 : {}
02 : {1}
06 : {1,2}
10 : {1,3}
14 : {1,4}
15 : {2,3}
22 : {1,5}
26 : {1,6}
30 : {1,2,3}
33 : {2,5}
34 : {1,7}
35 : {3,4}
38 : {1,8}
46 : {1,9}
51 : {2,7}
55 : {3,5}
58 : {1,10}
62 : {1,11}
66 : {1,2,5}
69 : {2,9}
70 : {1,3,4}

Cf. A001222, A003963, A005117, A007359, A051424, A056239, A275024, A289509, A302242, A302505, A302696, A302697, A302698, A302796, A302798.

Select[Range[100],Or[#===1,SquareFreeQ[#]&&CoprimeQ@@PrimePi/@FactorInteger[#][[All,1]]]&]

A304118 Number of z-blobs with least common multiple n > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-blob is a finite connected set S of pairwise indivisible positive integers greater than 1 such that no cap of S with at least two edges has clutter density -1.

If n is squarefree with k prime factors, then a(n) = A275307(k).

			The a(60) = 7 z-blobs together with the corresponding multiset systems (see A112798, A302242) are the following.
        (60): {{1,1,2,3}}
     (12,30): {{1,1,2},{1,2,3}}
     (20,30): {{1,1,3},{1,2,3}}
   (6,15,20): {{1,2},{2,3},{1,1,3}}
  (10,12,15): {{1,3},{1,1,2},{2,3}}
  (12,15,20): {{1,1,2},{2,3},{1,1,3}}
  (12,20,30): {{1,1,2},{1,1,3},{1,2,3}}
The a(120) = 14 z-blobs together with the corresponding multiset systems are the following.
       (120): {{1,1,1,2,3}}
     (24,30): {{1,1,1,2},{1,2,3}}
     (24,60): {{1,1,1,2},{1,1,2,3}}
     (30,40): {{1,2,3},{1,1,1,3}}
     (40,60): {{1,1,1,3},{1,1,2,3}}
   (6,15,40): {{1,2},{2,3},{1,1,1,3}}
  (10,15,24): {{1,3},{2,3},{1,1,1,2}}
  (12,15,40): {{1,1,2},{2,3},{1,1,1,3}}
  (12,30,40): {{1,1,2},{1,2,3},{1,1,1,3}}
  (15,20,24): {{2,3},{1,1,3},{1,1,1,2}}
  (15,24,40): {{2,3},{1,1,1,2},{1,1,1,3}}
  (20,24,30): {{1,1,3},{1,1,1,2},{1,2,3}}
  (24,30,40): {{1,1,1,2},{1,2,3},{1,1,1,3}}
  (24,40,60): {{1,1,1,2},{1,1,1,3},{1,1,2,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A303838.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
zlobQ[s_]:=Apply[And,Composition[Not,zreeQ]/@Apply[LCM,zptns[s],{2}]];
zswell[s_]:=Union[LCM@@@Select[Subsets[s],Length[zsm[#]]==1&]];
zkernels[s_]:=Table[Select[s,Divisible[w,#]&],{w,zswell[s]}];
zptns[s_]:=Select[stableSets[zkernels[s],Length[Intersection[#1,#2]]>0&],Union@@#==s&];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[If[n==1,0,Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={},zlobQ[#]]&]]],{n,100}]

A322338 Edge-connectivity of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 3, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Dec 04 2018

nonn

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

The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			2093 is the Heinz number of (9,6,4), corresponding to the multiset partition {{1,1},{1,2},{2,2}}, which can be made disconnected by removing only the part {1,2}, so a(2093) = 1.

Wikipedia, k-edge-connected graph

Cf. A001222, A003963, A013922, A054921, A056239, A095983, A112798, A302242, A304714, A304716, A305078, A305079, A322335, A322336, A322337.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[PrimeOmega[n]-Max@@PrimeOmega/@Select[Divisors[n],Length[csm[primeMS/@primeMS[#]]]!=1&],{n,100}]

A302540 Numbers whose prime indices other than 1 are prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 34, 36, 40, 41, 44, 45, 48, 50, 51, 54, 55, 59, 60, 62, 64, 66, 67, 68, 72, 75, 80, 81, 82, 83, 85, 88, 90, 93, 96, 99, 100, 102, 108, 109, 110, 118, 120, 121, 123, 124

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A291686, A302242, A302243, A302534, A302539.

Select[Range[400],#===1||And@@(#===1||PrimeQ[#]&)/@PrimePi/@FactorInteger[#][[All,1]]&]

ok(n)={!#select(p->p>2 && !isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Sum_{n>=1} 1/a(n) = 2 * Sum_{n>=1} 1/A076610(n) = 2 * Product_{p in A006450} p/(p-1) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A325032 Product of products of the multisets of prime indices of each prime index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 4, 1, 1, 2, 1, 3, 4, 1, 4, 2, 1, 1, 3, 2, 5, 1, 3, 4, 2, 1, 2, 1, 2, 2, 6, 1, 4, 3, 2, 4, 6, 1, 1, 4, 4, 2, 1, 1, 6, 1, 1, 3, 7, 2, 4, 5, 1, 1, 4, 3, 8, 4, 4, 2, 3, 1, 8, 2, 4, 1, 3, 2, 5, 2, 1, 6, 9, 1, 8, 4, 3

Offset: 1

Gus Wiseman, Mar 25 2019

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.

			94 has prime indices {1,15} with prime indices {{},{2,3}} with product a(94) = 6.

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A302242, A320325.

Cf. A324850, A324926, A324930, A325031, A325033, A325034, A325035.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Join@@primeMS/@primeMS[n],{n,100}]

Fully multiplicative with a(prime(n)) = A003963(n).

cp's OEIS Frontend

A302505 Numbers whose prime indices are squarefree and have disjoint prime indices.

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A325033 Sum of sums of the multisets of prime indices of each prime index of n.

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Formula

A302590 Squarefree numbers whose prime indices are prime numbers.

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PARI

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A320325 Numbers whose product of prime indices is a perfect power.

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A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

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A302797 Squarefree numbers whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions with pairwise coprime parts.

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A304118 Number of z-blobs with least common multiple n > 1.

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A322338 Edge-connectivity of the integer partition with Heinz number n.

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