A318981 -id:A318981 - cp's OEIS frontend

A328335 Numbers whose consecutive prime indices are relatively prime.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88

Offset: 1

Gus Wiseman, Oct 14 2019

nonn

First differs from A302569 in having 105, which has prime indices {2, 3, 4}.
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), so these are Heinz numbers of partitions whose consecutive parts are relatively prime (A328172).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}

A superset of A302569.
Numbers whose prime indices are relatively prime are A289509.
Numbers with no consecutive prime indices relatively prime are A328336.
Cf. A000837, A056239, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188, A328220.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;GCD[x,y]>1]&]

A328336 Numbers with no consecutive prime indices relatively prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167

Offset: 1

Gus Wiseman, Oct 14 2019

nonn

First differs from A318978 in having 897, with prime indices {2, 6, 9}.
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), so these are Heinz numbers of partitions no consecutive parts relatively prime (A328187).
Besides the initial 1 this differs from A305078: 47541=897*prime(16) is in A305078 but not in this set. - Andrey Zabolotskiy, Nov 13 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {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}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}

Numbers with consecutive prime indices relatively prime are A328335.
Strict partitions with no consecutive parts relatively prime are A328220.
Numbers with relatively prime prime indices are A289509.
Cf. A000837, A056239, A078374, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;GCD[x,y]==1]&]

A328168 Numbers whose prime indices minus 1 are relatively prime.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 91, 93, 95, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 147

Offset: 1

Gus Wiseman, Oct 08 2019

nonn

A multiset is relatively prime if the GCD of its elements is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.
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), so these are Heinz numbers of partitions whose parts minus one are relatively prime. The enumeration of these partitions by sum is given by A328170.

			The sequence of terms together with their prime indices begins:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}
   51: {2,7}
   54: {1,2,2,2}
   57: {2,8}

Positions of 1's in A328167.
Numbers whose prime indices are relatively prime are A289509.
The version for prime indices plus 1 is A318981.
The GCD of prime indices is A289508.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A056239, A112798, A258409, A281116, A290103, A328163, A328169.

q:= n-> igcd(map(i-> numtheory[pi](i[1])-1, ifactors(n)[2])[])=1:
select(q, [$1..150])[];  # Alois P. Heinz, Oct 13 2019

Select[Range[100],GCD@@(PrimePi/@First/@If[#==1,{},FactorInteger[#]]-1)==1&]

A328167 GCD of the prime indices of n, all minus 1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 1, 2, 4, 1, 5, 3, 1, 0, 6, 1, 7, 2, 1, 4, 8, 1, 2, 5, 1, 3, 9, 1, 10, 0, 1, 6, 1, 1, 11, 7, 1, 2, 12, 1, 13, 4, 1, 8, 14, 1, 3, 2, 1, 5, 15, 1, 2, 3, 1, 9, 16, 1, 17, 10, 1, 0, 1, 1, 18, 6, 1, 1, 19, 1, 20, 11, 1, 7, 1, 1, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Oct 08 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

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.

			85 has prime indices {3,7}, so a(85) = GCD(2,6) = 2.

Positions of 0's are A000079.
Positions of 1's are A328168.
Positions of records (first appearances) are A006005.
The GCD of the prime indices of n is A289508(n).
The GCD of the prime indices of n, all plus 1, is A328169(n).
Looking at divisors instead of prime indices gives A258409.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A001222, A007359, A056239, A112798, A289509, A318978, A318981, A328163, A328164.

Table[GCD@@(PrimePi/@First/@If[n==1,{},FactorInteger[n]]-1),{n,100}]

A328169 GCD of the prime indices of n, all plus 1.

Original entry on oeis.org

0, 2, 3, 2, 4, 1, 5, 2, 3, 2, 6, 1, 7, 1, 1, 2, 8, 1, 9, 2, 1, 2, 10, 1, 4, 1, 3, 1, 11, 1, 12, 2, 3, 2, 1, 1, 13, 1, 1, 2, 14, 1, 15, 2, 1, 2, 16, 1, 5, 2, 1, 1, 17, 1, 2, 1, 3, 1, 18, 1, 19, 2, 1, 2, 1, 1, 20, 2, 1, 1, 21, 1, 22, 1, 1, 1, 1, 1, 23, 2, 3, 2

Offset: 1

Gus Wiseman, Oct 09 2019

nonn

Zeros are ignored when computing GCD, and the empty set has GCD 0.

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.

			85 has prime indices {3,7}, so a(85) = GCD(4,8) = 4.

Positions of 0's and 1's are A318981.
Positions of records (first appearances) appear to be A116974.
The GCD of the prime indices of n, all minus 1, is A328167(n).
The LCM of the prime indices of n, all plus 1, is A328219(n).
Partitions whose parts plus 1 are relatively prime are A318980.
Cf. A000837, A056239, A112798, A258409, A289508, A289509, A290103, A328168.

Table[GCD@@(PrimePi/@First/@If[n==1,{},FactorInteger[n]]+1),{n,100}]

a(n) = A289508(A003961(n)).

A328219 LCM of the prime indices of n, all plus 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 5, 2, 3, 4, 6, 6, 7, 10, 12, 2, 8, 6, 9, 4, 15, 6, 10, 6, 4, 14, 3, 10, 11, 12, 12, 2, 6, 8, 20, 6, 13, 18, 21, 4, 14, 30, 15, 6, 12, 10, 16, 6, 5, 4, 24, 14, 17, 6, 12, 10, 9, 22, 18, 12, 19, 12, 15, 2, 28, 6, 20, 8, 30, 20, 21, 6, 22, 26

Offset: 1

Gus Wiseman, Oct 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.

Sorted positions of first appearances are A328451.
LCM of prime indices is A290103.
LCM of prime indices minus 1 is A328456.
GCD of prime indices plus 1 is A328169.
Partitions whose parts plus 1 are relatively prime are A318980.
Numbers whose prime indices plus 1 are relatively prime are A318981,
Cf. A003961, A056239, A112798, A258409, A289508, A289509, A328167, A328168.

Table[If[n==1,1,LCM@@(PrimePi/@First/@FactorInteger[n]+1)],{n,100}]

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A328219(n) = A290103(A003961(n)); \\ Antti Karttunen, Oct 18 2019

a(n) = A290103(A003961(n)).

If n = A000040(i_1) * ... * A000040(i_k), then a(n) = lcm(1+i_1,...,1+i_k).

A328451 Sorted positions of first appearances in A328219, where if n = A000040(i_1) * ... * A000040(i_k), then A328219(n) = LCM(1+i_1,...,1+i_k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 13, 14, 15, 17, 19, 21, 26, 29, 35, 37, 38, 39, 42, 47, 51, 53, 58, 61, 65, 74, 78, 79, 87, 89, 91, 95, 101, 105, 106, 107, 111, 113, 119, 122, 127, 133, 141, 145, 151, 158, 159, 173, 174, 178, 181, 182, 183, 185, 195, 199, 202, 203, 214, 221

Offset: 1

Gus Wiseman, Oct 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.

Indices of 1's in the ordinal transform of A328219. - Antti Karttunen, Oct 18 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  26: {1,6}
  29: {10}
  35: {3,4}
  37: {12}
  38: {1,8}
  39: {2,6}
  42: {1,2,4}
  47: {15}

A subsequence of A005117.
Sorted positions of first appearances in A328219.
The GCD of the prime indices of n, all plus 1, is A328169(n).
The LCM of the prime indices of n, all minus 1, is A328456(n).
Partitions whose parts plus 1 are relatively prime are A318980.
Numbers whose prime indices plus 1 are relatively prime are A318981.
Cf. A001222, A056239, A112798, A289508, A289509, A290103, A328163, A328167, A328168, A328170.

dav=Table[If[n==1,1,LCM@@(PrimePi/@First/@FactorInteger[n]+1)],{n,100}];
Table[Position[dav,i][[1,1]],{i,dav//.{A___,x_,B___,x_,C___}:>{A,x,B,C}}]

up_to = 1024;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A328219(n) = A290103(A003961(n));
vord_trans = ordinal_transform(vector(up_to,n,A328219(n)));
for(n=1,up_to,if(1==vord_trans[n], print1(n,", "))); \\ Antti Karttunen, Oct 18 2019

A328456 LCM of the prime indices of 2n + 1, all minus 1; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 2, 1, 9, 10, 4, 6, 11, 5, 12, 13, 2, 14, 3, 6, 15, 4, 7, 16, 17, 3, 10, 18, 8, 19, 20, 2, 12, 21, 1, 22, 6, 9, 23, 15, 10, 14, 24, 4, 25, 26, 6, 27, 28, 11, 29, 8, 5, 6, 4, 12, 2, 30, 13, 31, 21, 2, 32, 33, 14, 20, 18, 3, 34

Offset: 0

Gus Wiseman, Oct 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 prime indices of 2 * 17 + 1 = 35, all minus 1, are {2,3}, with LCM 6, so a(17) = 6.

Positions of records (first appearances) are A006005.
The GCD of the prime indices of n, all minus 1, is A328167(n).
The LCM of the prime indices of n, all plus 1, is A328219(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Numbers whose prime indices minus 1 are relatively prime are A328168.
Cf. A056239, A112798, A258409, A289508, A289509, A290103, A318981, A328169, A328451.

Table[If[n==1,0,LCM@@(PrimePi/@First/@FactorInteger[n]-1)],{n,1,100,2}]

cp's OEIS Frontend

A328335 Numbers whose consecutive prime indices are relatively prime.

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A328336 Numbers with no consecutive prime indices relatively prime.

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A328168 Numbers whose prime indices minus 1 are relatively prime.

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A328167 GCD of the prime indices of n, all minus 1.

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A328169 GCD of the prime indices of n, all plus 1.

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A328219 LCM of the prime indices of n, all plus 1.

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A328451 Sorted positions of first appearances in A328219, where if n = A000040(i_1) * ... * A000040(i_k), then A328219(n) = LCM(1+i_1,...,1+i_k).

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A328456 LCM of the prime indices of 2n + 1, all minus 1; a(0) = 0.

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