A324696 -id:A324696 - cp's OEIS frontend

A324702 Lexicographically earliest sequence containing 2 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

2, 5, 13, 25, 43, 65, 101, 125, 169, 193, 215, 317, 325, 505, 557, 559, 625, 701, 845, 965, 1013, 1075, 1181, 1313, 1321, 1585, 1625, 1849, 2111, 2161, 2197, 2509, 2525, 2785, 2795, 3125, 3505, 3617, 4049, 4057, 4121, 4225, 4343, 4639, 4825, 5065, 5297, 5375

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.
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 2 and numbers whose prime indices belong to A324703.

			The sequence of terms together with their prime indices begins:
    2: {1}
    5: {3}
   13: {6}
   25: {3,3}
   43: {14}
   65: {3,6}
  101: {26}
  125: {3,3,3}
  169: {6,6}
  193: {44}
  215: {3,14}
  317: {66}
  325: {3,3,6}
  505: {3,26}
  557: {102}
  559: {6,14}
  625: {3,3,3,3}
  701: {126}
  845: {3,6,6}
  965: {3,44}

Prime indices > 1 are A324703.
Cf. A000002, A000720, A001222, A001462, A007097, A045965, A055396, A061395, A064989, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324700, A324701, A324704, A324705.

aQ[n_]:=Switch[n,0,False,1,False,2,True,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]-1]]];
Select[Range[100],aQ]

a(n) = A324703(n) - 1.

A324703 Lexicographically earliest sequence containing 3 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Original entry on oeis.org

3, 6, 14, 26, 44, 66, 102, 126, 170, 194, 216, 318, 326, 506, 558, 560, 626, 702, 846, 966, 1014, 1076, 1182, 1314, 1322, 1586, 1626, 1850, 2112, 2162, 2198, 2510, 2526, 2786, 2796, 3126, 3506, 3618, 4050, 4058, 4122, 4226, 4344, 4640, 4826, 5066, 5298, 5376

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.

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.

Prime indices of A324702.
Cf. A000002, A000720, A001222, A001462, A007097, A045965, A055396, A061395, A064989, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324700, A324701, A324704, A324705.

aQ[n_]:=Switch[n,0,False,3,True,,And@@Cases[FactorInteger[n-1],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[0,1000],aQ]

a(n) = A324702(n) + 1.

A324705 Lexicographically earliest sequence containing 1 and all composite numbers divisible by prime(m) for some m already in the sequence.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.

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 sequence of terms together with their prime indices begins:
   1: {}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  34: {1,7}
  35: {3,4}
  36: {1,1,2,2}

Complement of A324697.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324698, A324699, A324700, A324701, A324702, A324703, A324704.

aQ[n_]:=Switch[n,1,True,?PrimeQ,False,,!And@@Cases[FactorInteger[n],{p_,k_}:>!aQ[PrimePi[p]]]];
Select[Range[200],aQ]

A331784 Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 35, 37, 38, 39, 41, 43, 46, 47, 49, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 111, 113, 115, 119, 122, 127, 131, 133, 137, 139, 141, 142

Offset: 1

Gus Wiseman, Feb 01 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.

Conjecture: A331912(n)/a(n) -> 1 as n -> infinity.

			The sequence of terms together with their prime indices begins:
    1: {}        43: {14}       91: {4,6}      141: {2,15}
    2: {1}       46: {1,9}      94: {1,15}     142: {1,20}
    3: {2}       47: {15}       95: {3,8}      143: {5,6}
    5: {3}       49: {4,4}      97: {25}       145: {3,10}
    7: {4}       53: {16}       98: {1,4,4}    147: {2,4,4}
   11: {5}       57: {2,8}     101: {26}       149: {35}
   13: {6}       58: {1,10}    103: {27}       151: {36}
   14: {1,4}     59: {17}      106: {1,16}     157: {37}
   17: {7}       61: {18}      107: {28}       158: {1,22}
   19: {8}       65: {3,6}     109: {29}       159: {2,16}
   21: {2,4}     67: {19}      111: {2,12}     161: {4,9}
   23: {9}       69: {2,9}     113: {30}       163: {38}
   26: {1,6}     71: {20}      115: {3,9}      167: {39}
   29: {10}      73: {21}      119: {4,7}      169: {6,6}
   31: {11}      74: {1,12}    122: {1,18}     173: {40}
   35: {3,4}     77: {4,5}     127: {31}       178: {1,24}
   37: {12}      79: {22}      131: {32}       179: {41}
   38: {1,8}     83: {23}      133: {4,8}      181: {42}
   39: {2,6}     87: {2,10}    137: {33}       182: {1,4,6}
   41: {13}      89: {24}      139: {34}       183: {2,18}
For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.

Gus Wiseman, Plot of A331912(n)/A331784(n) for n = 1..3729.

Contains all prime numbers A000040.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with exactly one prime index in S are A331785.
Numbers S with at most one distinct prime index in S are A331912.
Numbers S with exactly one distinct prime index in S are A331913.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331914.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aQ[n_]:=Length[Cases[primeMS[n],_?aQ]]<=1;
Select[Range[100],aQ]

A324755 Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 3, 5, 6, 10, 7, 16, 14, 23, 23, 35, 34, 53, 54, 75, 80, 112, 115, 160, 169, 223, 244, 315, 339, 442, 478, 604, 664, 832, 910, 1131, 1245, 1524, 1689, 2054, 2263, 2743, 3039, 3634, 4042, 4809, 5343, 6326, 7035, 8276, 9217, 10795, 12011

Offset: 0

Gus Wiseman, Mar 16 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.

For example, (6,2) is such a partition because the prime indices of 6 are {1,2}, which do not all belong to the partition. On the other hand, (5,3) is not such a partition because the prime indices of 5 are {3}, and 3 belongs to the partition.

			The a(2) = 1 through a(10) = 10 integer partitions (A = 10):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)
            (22)       (33)   (43)  (44)    (54)   (55)
                       (42)   (52)  (62)    (63)   (64)
                       (222)        (422)   (72)   (73)
                                    (2222)  (333)  (82)
                                            (522)  (433)
                                                   (442)
                                                   (622)
                                                   (4222)
                                                   (22222)

The subset version is A324739, with maximal case A324762. The strict case is A324750. The Heinz number version is A324760. An infinite version is A324694.

Cf. A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324696, A324744.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@If[k==1,{},FactorInteger[k]]]]&]],{n,0,30}]

A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 8, 8, 11, 10, 15, 16, 19, 23, 27, 28, 35, 39, 47, 50, 63, 68, 77, 91, 102, 114, 130, 147, 169, 187, 213, 237, 268, 300, 336, 380, 422, 472, 525, 587, 647, 731, 810, 895, 996, 1102, 1227, 1355, 1498, 1661, 1818, 2020, 2221

Offset: 0

Gus Wiseman, Mar 15 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.

			The a(2) = 1 through a(17) = 15 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8   9   A   B    C    D    E    F    G    H
              42  43  62  54  64  65   75   76   86   87   97   98
                  52      63  73  83   84   85   95   96   A6   A7
                          72  82  542  93   94   A4   A5   C4   B6
                                       A2   A3   B3   B4   D3   C5
                                       642  B2   C2   C3   E2   D4
                                            643  752  D2   763  E3
                                            652  842  654  862  F2
                                                      762  943  854
                                                      843  A42  863
                                                      852       872
                                                                A43
                                                                A52
                                                                B42
                                                                6542

The subset version is A324739. The non-strict version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A304360, A305713, A306844.
Cf. A324696, A324737, A324742, A324744, A324764.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]

A324760 Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137, 139

Offset: 1

Gus Wiseman, Mar 17 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. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  41: {13}

The subset version is A324739, with maximal case A324762. The strict integer partition version is A324750. The integer partition version is A324755. An infinite version is A324694.

Cf. A000720, A001221, A007097, A056239, A112798, A289509, A290822, A306844, A324695, A324696, A324737, A324744.

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

A331785 Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 11, 14, 21, 26, 31, 34, 35, 38, 39, 43, 46, 51, 57, 58, 65, 69, 73, 74, 77, 82, 85, 87, 94, 95, 98, 101, 106, 111, 115, 118, 122, 123, 127, 134, 139, 141, 142, 143, 145, 147, 149, 158, 159, 163, 166, 167, 177, 178, 182, 183, 185, 187, 191, 194, 199

Offset: 1

Gus Wiseman, Feb 01 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.

			The sequence of terms together with their prime indices begins:
    1: {}         73: {21}       142: {1,20}     205: {3,13}
    2: {1}        74: {1,12}     143: {5,6}      206: {1,27}
    3: {2}        77: {4,5}      145: {3,10}     209: {5,8}
    5: {3}        82: {1,13}     147: {2,4,4}    213: {2,20}
   11: {5}        85: {3,7}      149: {35}       214: {1,28}
   14: {1,4}      87: {2,10}     158: {1,22}     217: {4,11}
   21: {2,4}      94: {1,15}     159: {2,16}     218: {1,29}
   26: {1,6}      95: {3,8}      163: {38}       226: {1,30}
   31: {11}       98: {1,4,4}    166: {1,23}     233: {51}
   34: {1,7}     101: {26}       167: {39}       235: {3,15}
   35: {3,4}     106: {1,16}     177: {2,17}     237: {2,22}
   38: {1,8}     111: {2,12}     178: {1,24}     238: {1,4,7}
   39: {2,6}     115: {3,9}      182: {1,4,6}    245: {3,4,4}
   43: {14}      118: {1,17}     183: {2,18}     249: {2,23}
   46: {1,9}     122: {1,18}     185: {3,12}     253: {5,9}
   51: {2,7}     123: {2,13}     187: {5,7}      262: {1,32}
   57: {2,8}     127: {31}       191: {43}       265: {3,16}
   58: {1,10}    134: {1,19}     194: {1,25}     266: {1,4,8}
   65: {3,6}     139: {34}       199: {46}       267: {2,24}
   69: {2,9}     141: {2,15}     201: {2,19}     269: {57}
For example, the prime indices of 77 are {4,5}, of which only 5 is in the sequence, so 77 is in the sequence.

Closed under A000040.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with at most one prime index in S are A331784.
Numbers S with at most one distinct prime index in S are A331912.
Numbers S with exactly one distinct prime index in S are A331913.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aQ[n_]:=n==1||Length[Select[primeMS[n],aQ]]==1;
Select[Range[100],aQ]

A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 312, 624, 936, 1872, 3744, 7488, 12480, 24960, 37440, 74880, 142848, 285696, 456192, 912384, 1548288, 3096576, 5308416, 10616832, 15925248, 31850496, 51978240, 103956480, 200835072, 401670144, 771489792, 1542979584, 2314469376

Offset: 1

Gus Wiseman, Mar 14 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.

			The a(1) = 1 through a(6) = 20 subsets:
  {}  {}   {}   {}     {}       {}
      {2}  {2}  {2}    {2}      {2}
           {3}  {3}    {3}      {3}
                {4}    {4}      {4}
                {2,4}  {5}      {5}
                {3,4}  {2,4}    {6}
                       {2,5}    {2,4}
                       {3,4}    {2,5}
                       {4,5}    {2,6}
                       {2,4,5}  {3,4}
                                {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {2,4,6}
                                {2,5,6}
                                {3,4,6}
                                {4,5,6}
                                {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 1..100

The maximal case is A324762. The case of subsets of {1...n} is A324738. The strict integer partition version is A324750. The integer partition version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324696, A324736, A324737, A324741, A324742, A324744, A324764.

Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A331913 Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 26, 27, 31, 32, 39, 49, 52, 53, 58, 59, 64, 65, 67, 74, 81, 82, 83, 86, 87, 91, 94, 97, 101, 103, 104, 111, 116, 117, 121, 122, 123, 125, 127, 128, 129, 131, 141, 142, 143, 145, 146, 148, 158, 164, 167, 172, 178

Offset: 1

Gus Wiseman, Feb 01 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.

			The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}          116: {1,1,10}
    2: {1}             53: {16}             117: {2,2,6}
    3: {2}             58: {1,10}           121: {5,5}
    4: {1,1}           59: {17}             122: {1,18}
    5: {3}             64: {1,1,1,1,1,1}    123: {2,13}
    7: {4}             65: {3,6}            125: {3,3,3}
    8: {1,1,1}         67: {19}             127: {31}
    9: {2,2}           74: {1,12}           128: {1,1,1,1,1,1,1}
   11: {5}             81: {2,2,2,2}        129: {2,14}
   16: {1,1,1,1}       82: {1,13}           131: {32}
   17: {7}             83: {23}             141: {2,15}
   19: {8}             86: {1,14}           142: {1,20}
   23: {9}             87: {2,10}           143: {5,6}
   25: {3,3}           91: {4,6}            145: {3,10}
   26: {1,6}           94: {1,15}           146: {1,21}
   27: {2,2,2}         97: {25}             148: {1,1,12}
   31: {11}           101: {26}             158: {1,22}
   32: {1,1,1,1,1}    103: {27}             164: {1,1,13}
   39: {2,6}          104: {1,1,1,6}        167: {39}
   49: {4,4}          111: {2,12}           172: {1,1,14}

Contains all prime powers A000961.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with at most one prime index in S are A331784.
Numbers S with exactly one prime index in S are A331785.
Numbers S with at most one distinct prime index in S are A331912.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.

aQ[n_]:=n==1||Length[Select[PrimePi/@First/@FactorInteger[n],aQ]]==1;
Select[Range[200],aQ]

cp's OEIS Frontend

A324702 Lexicographically earliest sequence containing 2 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

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A324703 Lexicographically earliest sequence containing 3 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

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A324705 Lexicographically earliest sequence containing 1 and all composite numbers divisible by prime(m) for some m already in the sequence.

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A331784 Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.

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A324755 Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

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A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

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A324760 Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.

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A331785 Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.

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A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

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