A289508 -id:A289508 - cp's OEIS frontend

A318990 Numbers of the form prime(x) * prime(y) where x divides y.

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 46, 49, 57, 58, 62, 65, 74, 82, 86, 87, 94, 106, 111, 115, 118, 121, 122, 129, 133, 134, 142, 146, 158, 159, 166, 169, 178, 183, 185, 194, 202, 206, 213, 214, 218, 226, 235, 237, 254, 259, 262, 267, 274, 278, 289

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of all dividing pairs (columns) begins:
  1  1  2  1  1  2  1  3  1  1  1  2  1  4  2  1  1  3  1  1  1  2  1  1
  1  2  2  3  4  4  5  3  6  7  8  6  9  4  8 10 11  6 12 13 14 10 15 16

Andrew Howroyd, Table of n, a(n) for n = 1..10000

A subset of A001358 (semiprimes), squarefree A006881.
The squarefree version is A339005.
The quotient is A358103 = A358104 / A358105.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A300912, A318991, A318992, A318993.
Cf. A000720, A027751, A032741, A215366, A296150.
Cf. A289508, A289509, A358106.

Select[Range[100],And[PrimeOmega[#]==2,Or[PrimePowerQ[#],Divisible@@Reverse[PrimePi/@FactorInteger[#][[All,1]]]]]&]

ok(n)={my(f=factor(n)); bigomega(f)==2 && (#f~==1 || primepi(f[2,1]) % primepi(f[1,1]) == 0)} \\ Andrew Howroyd, Oct 26 2018

A300912 Numbers of the form prime(x)*prime(y) where x and y are relatively prime.

Original entry on oeis.org

4, 6, 10, 14, 15, 22, 26, 33, 34, 35, 38, 46, 51, 55, 58, 62, 69, 74, 77, 82, 85, 86, 93, 94, 95, 106, 118, 119, 122, 123, 134, 141, 142, 143, 145, 146, 155, 158, 161, 166, 177, 178, 187, 194, 201, 202, 205, 206, 209, 214, 215, 217, 218, 219, 221, 226, 249

Offset: 1

Gus Wiseman, Sep 06 2018

nonn

			The sequence of all relatively prime pairs (columns) begins:
  1  1  1  1  2  1  1  2  1  3  1  1  2  3  1  1  2  1  4  1  3  1  2  1  3
  1  2  3  4  3  5  6  5  7  4  8  9  7  5 10 11  9 12  5 13  7 14 11 15  8

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A001358, A000837, A001221, A001222, A006881, A289508, A289509, A318990.

Select[Range[100],And[PrimeOmega[#]==2,GCD@@PrimePi/@If[#==1,{},FactorInteger[#]][[All,1]]==1]&]
With[{nn=40},Join[{4},Take[Prime[#[[1]]]Prime[#[[2]]]&/@Select[Subsets[ Range[ nn],{2}],CoprimeQ@@#&]//Union,Floor[Prime[nn]/2]]]] (* Harvey P. Dale, Jan 04 2021 *)

ok(n)={my(f=factor(n)); bigomega(f)==2 && gcd(apply(primepi, f[,1]))==1} \\ Andrew Howroyd, Oct 26 2018

A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 2, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 2, 1, 78, 1

Offset: 2

Ivan Neretin, May 29 2015

nonn

a(n) = 1 for even n; a(p) = p-1 for prime p.
a(n) is even for odd n (since all divisors of n are odd).
It appears that a(n) = A052409(A005179(n)), i.e., it is the largest integer power of the smallest number with exactly n divisors. - Michel Marcus, Nov 10 2015
Conjecture: GCD of all (p-1) for prime p|n. - Thomas Ordowski, Sep 14 2016
Conjecture is true, because the set of numbers == 1 (mod g) is closed under multiplication. - Robert Israel, Sep 14 2016
Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n. See A328163 and A328164. - Gus Wiseman, Oct 16 2019

			65 has divisors 1, 5, 13, and 65, hence a(65) = gcd(1-1,5-1,13-1,65-1) = gcd(0,4,12,64) = 4.

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A049559, A057237, A060680, A063994, A187730, A027750.
Cf. A084190 (similar but with LCM).
Looking at prime indices instead of divisors gives A328167.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000005, A060681, A060683, A193829, A289508.

a258409 n = foldl1 gcd $ map (subtract 1) $ tail $ a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015

f:= n -> igcd(op(map(`-`,numtheory:-factorset(n),-1))):
map(f, [$2..100]); # Robert Israel, Sep 14 2016

Table[GCD @@ (Divisors[n] - 1), {n, 2, 100}]

a(n) = my(g=0); fordiv(n, d, g = gcd(g, d-1)); g; \\ Michel Marcus, May 29 2015

a(n) = gcd(apply(x->x-1, divisors(n))); \\ Michel Marcus, Nov 10 2015

a(n)=if(n%2==0, return(1)); if(n%3==0, return(2)); if(n%5==0 && n%4 != 1, return(2)); gcd(apply(p->p-1, factor(n)[,1])) \\ Charles R Greathouse IV, Sep 19 2016

A316431 Least common multiple divided by greatest common divisor of the integer partition with Heinz number n > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 6, 1, 1, 2, 1, 3, 2, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 12, 2, 1, 8, 3, 3, 1, 4, 1, 5, 6, 9, 1, 2, 1, 3, 14, 6, 1, 2, 15, 4, 4, 10, 1, 6, 1, 11, 2, 1, 2, 10, 1, 7, 18, 12, 1, 2, 1, 12, 6, 8, 20, 6, 1, 3, 1, 13, 1, 4, 21, 14, 5, 5, 1, 6, 6, 9, 22, 15, 24, 2, 1, 4, 10, 3, 1, 14, 1, 6, 12

Offset: 2

Gus Wiseman, Jul 02 2018

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

			63 is the Heinz number of (4,2,2), which has LCM 4 and GCD 2, so a(63) = 4/2 = 2.
91 is the Heinz number of (6,4), which has LCM 12 and GCD 2, so a(91) = 12/2 = 6.

Cf. A005117, A056239, A074761, A289508, A289509, A290103, A296150, A316429, A316430, A316437.

Table[With[{pms=Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]},LCM@@pms/GCD@@pms],{n,2,100}]

A316431(n) = if(1==n,1,my(pis = apply(p -> primepi(p), factor(n)[, 1]~)); lcm(pis)/gcd(pis)); \\ Antti Karttunen, Sep 06 2018

a(n) = A290103(n)/A289508(n).

a(n) = a(A005117(n)). - David A. Corneth, Sep 06 2018

More terms from Antti Karttunen, Sep 06 2018

A356237 Heinz numbers of integer partitions with a neighborless singleton.

Original entry on oeis.org

2, 3, 5, 7, 10, 11, 13, 14, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93

Offset: 1

Gus Wiseman, Aug 24 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts, and a singleton if it appears only once.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also numbers that, for some prime index x, are not divisible by prime(x)^2, prime(x - 1), or prime(x + 1). Here, a prime index of n is a number m such that prime(m) divides n.

			The terms together with their prime indices begin:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  26: {1,6}
  28: {1,1,4}

The complement is counted by A355393.
These partitions are counted by A356235.
Not requiring a singleton gives A356734.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356236 counts partitions with a neighborless part, complement A355394.
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,Or@@Table[Count[ptn,x]==1&&!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 4, 2, 8, 7, 14, 14, 21, 18, 33, 32, 50, 54, 72, 67, 103, 110, 145, 155, 201, 196, 271, 293, 372, 400, 493, 512, 647, 704, 858, 924, 1115, 1167, 1436, 1560, 1854, 2022, 2368, 2510, 3005, 3255, 3804, 4144, 4792, 5116, 5989, 6514, 7486

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(14) = 7 partitions are (932), (8321), (7421), (653), (6521), (6431), (5432).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303139.

Table[Select[IntegerPartitions[n],UnsameQ@@#&&!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,20}]

A328335 Numbers whose consecutive prime indices are relatively prime.

Original entry on oeis.org

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]&]

A332004 Number of compositions (ordered partitions) of n into distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 0, 2, 2, 4, 8, 12, 16, 24, 52, 64, 88, 132, 180, 344, 416, 616, 816, 1176, 1496, 2736, 3232, 4756, 6176, 8756, 11172, 15576, 24120, 30460, 41456, 55740, 74440, 97976, 130192, 168408, 256464, 315972, 429888, 558192, 749920, 958264, 1274928, 1621272, 2120288, 3020256

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

Moebius transform of A032020.

Ranking these compositions using standard compositions (A066099) gives the intersection of A233564 (strict) with A291166 (relatively prime). - Gus Wiseman, Oct 18 2020

			a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].
From _Gus Wiseman_, Oct 18 2020: (Start)
The a(1) = 1 through a(8) = 16 compositions (empty column indicated by dot):
  (1)  .  (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
          (2,1)  (3,1)  (2,3)  (5,1)    (2,5)    (3,5)
                        (3,2)  (1,2,3)  (3,4)    (5,3)
                        (4,1)  (1,3,2)  (4,3)    (7,1)
                               (2,1,3)  (5,2)    (1,2,5)
                               (2,3,1)  (6,1)    (1,3,4)
                               (3,1,2)  (1,2,4)  (1,4,3)
                               (3,2,1)  (1,4,2)  (1,5,2)
                                        (2,1,4)  (2,1,5)
                                        (2,4,1)  (2,5,1)
                                        (4,1,2)  (3,1,4)
                                        (4,2,1)  (3,4,1)
                                                 (4,1,3)
                                                 (4,3,1)
                                                 (5,1,2)
                                                 (5,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A007360, A032020, A108700, A302698.
A000740 is the non-strict version.
A078374 is the unordered version (non-strict: A000837).
A101271*6 counts these compositions of length 3 (non-strict: A000741).
A337561/A337562 is the pairwise coprime instead of relatively prime version (non-strict: A337462/A101268).
A289509 gives the Heinz numbers of relatively prime partitions.
A333227/A335235 ranks pairwise coprime compositions.
Cf. A001523, A178472, A216652, A289508, A291166, A333228.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&GCD@@#<=1&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)

A333226 Least common multiple of the n-th composition in standard order.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 2, 2, 1, 5, 4, 6, 3, 6, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 6, 5, 4, 4, 3, 6, 6, 3, 4, 6, 2, 2, 6, 2, 2, 2, 5, 4, 6, 3, 6, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 1, 7, 6, 10, 5, 12, 4, 4, 4, 12, 3, 6, 6, 3, 6, 6, 3, 10, 4, 6, 6, 6, 2, 2

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

The version for binary indices is A271410.
The version for prime indices is A290103.
Positions of first appearances are A333225.
Let q(k) be the k-th composition in standard order:
- The terms of q(k) are row k of A066099.
- The sum of q(k) is A070939(k).
- The product of q(k) is A124758(k).
- The GCD of q(k) is A326674(k).
- The LCM of q(k) is A333226(k).
Cf. A000120, A029931, A048793, A074971, A076078, A233564, A285572, A289508, A289509, A324837, A333227, A333492.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[LCM@@stc[n],{n,100}]

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288

Offset: 1

Gus Wiseman, Apr 20 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.

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

			The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

cp's OEIS Frontend

A318990 Numbers of the form prime(x) * prime(y) where x divides y.

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A300912 Numbers of the form prime(x)*prime(y) where x and y are relatively prime.

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A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

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A316431 Least common multiple divided by greatest common divisor of the integer partition with Heinz number n > 1.

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A356237 Heinz numbers of integer partitions with a neighborless singleton.

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A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

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A328335 Numbers whose consecutive prime indices are relatively prime.

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A332004 Number of compositions (ordered partitions) of n into distinct and relatively prime parts.

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