A328167 -id:A328167 - cp's OEIS frontend

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

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

A328170 Number of integer partitions of n whose parts minus 1 are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 8, 12, 18, 27, 38, 53, 74, 102, 137, 184, 241, 317, 413, 536, 687, 880, 1112, 1405, 1765, 2215, 2755, 3424, 4229, 5216, 6402, 7847, 9572, 11662, 14148, 17139, 20688, 24940, 29971, 35969, 43044, 51438, 61311, 72985, 86678, 102807, 121675

Offset: 0

Gus Wiseman, Oct 09 2019

nonn

A partition is relatively prime if the GCD of its parts is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.

			The a(2) = 1 through a(9) = 18 partitions:
  (2)  (21)  (22)   (32)    (42)     (43)      (62)       (54)
             (211)  (221)   (222)    (52)      (332)      (63)
                    (2111)  (321)    (322)     (422)      (72)
                            (2211)   (421)     (431)      (432)
                            (21111)  (2221)    (521)      (522)
                                     (3211)    (2222)     (621)
                                     (22111)   (3221)     (3222)
                                     (211111)  (4211)     (3321)
                                               (22211)    (4221)
                                               (32111)    (4311)
                                               (221111)   (5211)
                                               (2111111)  (22221)
                                                          (32211)
                                                          (42111)
                                                          (222111)
                                                          (321111)
                                                          (2211111)
                                                          (21111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The Heinz numbers of these partitions are given by A328168.
Partitions whose parts are relatively prime are A000837.
Partitions whose parts plus 1 are relatively prime are A318980.
The GCD of the prime indices of n, all minus 1, is A328167(n).
Cf. A007359, A018783, A258409, A289509, A328163, A328164.

Table[Length[Select[IntegerPartitions[n],GCD@@(#-1)==1&]],{n,0,30}]

seq(n)=Vec(sum(d=1, n-1, moebius(d)*(-1/(1-x) + 1/prod(k=0, n\d, 1 - x*x^(k*d) + O(x*x^n)))), -(n+1)) \\ Andrew Howroyd, Oct 17 2019

G.f.: Sum_{d>=1} mu(d)*(-1/(1-x) + 1/(Prod_{k>=0} 1 - x^(k*d + 1))). - Andrew Howroyd, Oct 17 2019

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&]

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).

A328163 Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 2, 5, 5, 9, 5, 15, 9, 19, 16, 28, 16, 44, 21, 55, 38, 73, 34, 109, 46, 130, 73, 170, 66, 251, 78, 287, 137, 364, 119, 522, 135, 590, 236, 759, 190, 1042, 219, 1175, 425, 1460, 306, 2006, 347, 2277, 671, 2780, 471, 3734, 584, 4197, 1087

Offset: 0

Gus Wiseman, Oct 07 2019

nonn

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

			The a(2) = 1 through a(12) = 15 partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)      (B)     (C)
            (22)       (33)   (52)  (44)    (63)   (55)     (83)    (66)
                       (42)         (62)    (72)   (64)     (92)    (84)
                       (222)        (422)   (333)  (73)     (722)   (93)
                                    (2222)  (522)  (82)     (5222)  (A2)
                                                   (442)            (444)
                                                   (622)            (552)
                                                   (4222)           (633)
                                                   (22222)          (642)
                                                                    (822)
                                                                    (3333)
                                                                    (4422)
                                                                    (6222)
                                                                    (42222)
                                                                    (222222)

The complement to these partitions is counted by A328164.
The GCD of the divisors of n all minus 1 is A258409(n).
The GCD of the prime indices of n all minus 1 is A328167(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A018783, A175342, A279945, A289508, A328168, A328169.

Table[Length[Select[IntegerPartitions[n],GCD@@Differences[#]!=GCD@@(#-1)&]],{n,0,30}]

A328164 Number of integer partitions of n whose unsigned differences have the same GCD as the GCD of their parts all minus 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 7, 13, 17, 25, 33, 51, 62, 92, 116, 160, 203, 281, 341, 469, 572, 754, 929, 1221, 1466, 1912, 2306, 2937, 3548, 4499, 5353, 6764, 8062, 10006, 11946, 14764, 17455, 21502, 25425, 30949, 36579, 44393, 52132, 63042, 74000, 88709, 104098, 124448

Offset: 0

Gus Wiseman, Oct 07 2019

nonn

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

			The a(1) = 1 through a(8) = 17 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (61)       (71)
                    (1111)  (221)    (411)     (322)      (332)
                            (311)    (2211)    (331)      (431)
                            (2111)   (3111)    (421)      (521)
                            (11111)  (21111)   (511)      (611)
                                     (111111)  (2221)     (3221)
                                               (3211)     (3311)
                                               (4111)     (4211)
                                               (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The complement to these partitions is counted by A328163.
The GCD of the divisors of n all minus 1 is A258409(n).
The GCD of the prime indices of n all minus 1 is A328167(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A018783, A175342, A279945, A289508, A328168, A328169.

Table[Length[Select[IntegerPartitions[n],GCD@@Differences[#]==GCD@@(#-1)&]],{n,0,30}]

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

A328170 Number of integer partitions of n whose parts minus 1 are relatively prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328168 Numbers whose prime indices minus 1 are relatively prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A328163 Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A328164 Number of integer partitions of n whose unsigned differences have the same GCD as the GCD of their parts all minus 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords