A060680 -id:A060680 - 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

A060682 Number of distinct differences between consecutive divisors of n (ordered by size).

Original entry on oeis.org

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

Offset: 2

Labos Elemer, Apr 19 2001

nonn

Number of all differences for n is d(n)-1 = A000005(n)-1. Increments are not necessarily different, so a(n)<=d(n)-1.

			For n=70, divisors={1,2,5,7,10,14,35,70}; differences={1,3,2,3,4,21,35}; a(70) = number of distinct differences = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
A. Balog, P. Erdős and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.
Jason Earls, Smarandache iterations of the first kind on functions involving divisors and prime factors, in Smarandache Notions Journal (2004), Vol. 14.1, page 264.

Cf. A000005, A060680, A060681, A060683.

Cf. A193829.

import Data.List (nub, genericLength)
a060682 = genericLength . nub . a193829_row
-- Reinhard Zumkeller, Jun 25 2015

a[n_ ] := Length[Union[Drop[d=Divisors[n], 1]-Drop[d, -1]]]

a(n) = my(d=divisors(n)); #vecsort(vector(#d-1, k, d[k+1] - d[k]),,8); \\ Michel Marcus, Jul 04 2017

Edited by Dean Hickerson, Jan 22 2002

A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 5, 3, 4, 2, 5, 2, 4, 3, 7, 2, 5, 2, 6, 4, 4, 2, 7, 3, 4, 5, 6, 2, 5, 2, 11, 4, 4, 3, 7, 2, 4, 4, 10, 2, 6, 2, 6, 5, 4, 2, 11, 3, 6, 4, 6, 2, 7, 4, 10, 4, 4, 2, 7, 2, 4, 6, 13, 4, 6, 2, 6, 4, 6, 2, 11, 2, 4, 5, 6, 3, 6, 2, 14, 7, 4, 2, 10

Offset: 1

Gus Wiseman, Aug 18 2022

nonn

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.
A multiset is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {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 prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}. These have lengths (3,1,2), with Heinz number 30, so a(18564) = 30.

Positions of prime terms are A073491, complement A073492.
Positions of terms with bigomega 2-4 are A073493-A073495.
Applying bigomega gives A287170, firsts A066205, even bisection A356229.
These are the Heinz numbers of the rows of A356226.
Minimal/maximal prime indices are A356227/A356228.
A version for standard compositions is A356230, firsts A356232/A356603.
A001221 counts distinct prime factors, with sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A055932, A060680-A060683, A193829, A286470, A328166, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Length/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A001222(a(n)) = A287170(n).
A055396(a(n)) = A356227(n).
A061395(a(n)) = A356228(n).

A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 2, 5, 11, 12, 13, 2, 7, 15, 16, 17, 18, 19, 4, 5, 3, 7, 2, 11, 23, 24, 25, 2, 13, 27, 4, 7, 29, 30, 31, 32, 3, 11, 2, 17, 35, 36, 37, 2, 19, 3, 13, 8, 5, 41, 6, 7, 43, 4, 11, 45, 2, 23, 47, 48, 49, 2, 25, 3, 17, 4, 13, 53, 54, 5, 11, 8

Offset: 1

Gus Wiseman, Aug 28 2022

Row-products are the positive integers 1, 2, 3, ...

			The first 16 rows:
   1 =
   2 = 2
   3 = 3
   4 = 4
   5 = 5
   6 = 6
   7 = 7
   8 = 8
   9 = 9
  10 = 2 * 5
  11 = 11
  12 = 12
  13 = 13
  14 = 2 * 7
  15 = 15
  16 = 16
The factorization of 18564 is 18564 = 12*7*221, so row 18564 is {12,7,221}.

Row-lengths are A287170, firsts A066205, even bisection A356229.
Applying bigomega to all parts gives A356226, statistics A356227-A356232.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices.
A056239 adds up the prime indices, row sums of A112798.
A132747 counts non-isolated divisors, complement A132881.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Cf. A000005, A001222, A060680-A060683, A073491-A073495, A193829, A330103, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@#&/@Split[primeMS[n],#1>=#2-1&],{n,100}]

A328028 Nonprime numbers n whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

Original entry on oeis.org

1, 4, 6, 9, 10, 12, 14, 15, 21, 22, 24, 25, 26, 30, 33, 34, 35, 36, 38, 39, 45, 46, 48, 49, 51, 55, 57, 58, 60, 62, 63, 65, 69, 70, 72, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 105, 106, 108, 111, 115, 118, 119, 120, 121, 122, 123, 129, 132, 133, 134

Offset: 1

Gus Wiseman, Oct 06 2019

nonn

			The proper divisors of 18 are {2, 3, 6, 9}, and {3, 6} are a consecutive divisible pair, so 18 does not belong to the sequence.
The proper divisors of 60 are {2, 3, 4, 5, 6, 10, 12, 15, 20, 30}, and none of {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 10}, {10, 12}, {12, 15}, {15, 20}, or {20, 30} are divisible pairs, so 60 belongs to the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 0's or 2's in A328026.
1 and positions of 1's in A328194.
The version including primes is A328161.
Partitions with no consecutive divisibilities are A328171.
Numbers whose proper divisors have no consecutive successions are A088725.
Contains A001358.
Cf. A000005, A060680, A060775, A067513, A163870, A328162, A328189.

filter:= proc(n) local D,i;
  if isprime(n) then return false fi;
  D:= sort(convert(numtheory:-divisors(n) minus {1,n}, list));
  for i from 1 to nops(D)-1 do if (D[i+1]/D[i])::integer then return false fi od:
  true
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 11 2019

Select[Range[100],!PrimeQ[#]&&!MatchQ[DeleteCases[Divisors[#],1|#],{_,x_,y_,_}/;Divisible[y,x]]&]

A356228 Greatest size of a gapless submultiset of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 3, 1, 2, 1, 2, 3, 1, 1, 5, 2, 2, 1, 2, 1, 4, 1, 3, 1, 1, 1, 4, 1, 1, 2, 6, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 3, 2, 2, 2, 1, 4, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless intervals: {2,3}, {5,5,6}, {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 prime indices of 700 are {1,1,3,3,4}, with maximal gapless submultisets {1,1}, {3,3,4}, so a(700) = 3.
The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 3.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-maxima of A356226, firsts A356232.
The smallest instead of greatest size is A356227.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356069 counts gapless divisors.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A328162, A328457, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Max@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333766(A356230(n)).

a(n) = A061395(A356231(n)).

A356229 Number of maximal gapless submultisets of the prime indices of 2n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Gus Wiseman, Aug 16 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {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.
This is a bisection of A287170, but is important in its own right because the even numbers are exactly those whose prime indices begin with 1.

			The prime indices of 2*9282 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(9282) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

This is the even (bisected) case of A287170, firsts A066205.
Alternate row-lengths of A356226, minima A356227(2n), maxima A356228(2n).
A001221 counts distinct prime factors, sum A001414.
A001222 counts prime indices, listed by A112798, sum A056239.
A003963 multiplies together the prime indices of n.
A073093 counts the prime indices of 2n.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
Cf. A000005, A060680, A060681, A132747, A132881, A286470, A289509, A356230, A356231, A356232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Split[primeMS[2n],#1>=#2-1&]],{n,100}]

A287170(n) = { my(f=factor(n)); if(#f~==0, return (0), return(#f~ - sum(i=1, #f~-1, if (primepi(f[i, 1])+1 == primepi(f[i+1, 1]), 1, 0)))); };
A356229(n) = A287170(2*n); \\ Antti Karttunen, Jan 19 2025

a(n) = A287170(2n).

Data section extended to a(105) by Antti Karttunen, Jan 19 2025

A328026 Number of divisible pairs of consecutive divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 2, 2, 2, 3, 4, 1, 2, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 2, 2, 3, 2, 4, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 2, 6, 2, 4, 1, 4, 2, 2, 1, 2, 1, 2, 3, 4, 2, 4, 1, 4, 4, 2, 1, 2, 2, 2, 2, 6, 1, 2, 2, 4, 2, 2, 2, 2, 1, 3, 4, 6, 1, 4, 1, 6, 2

Offset: 1

Gus Wiseman, Oct 03 2019

nonn

The number m = 2^n, n >= 0, is the smallest for which a(m) = n. - Marius A. Burtea, Nov 20 2019

			The divisors of 500 are {1,2,4,5,10,20,25,50,100,125,250,500}, with consecutive divisible pairs {1,2}, {2,4}, {5,10}, {10,20}, {25,50}, {50,100}, {125,250}, {250,500}, so a(500) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of 1's are A000040.
Positions of 0's and 2's are A328028.
Positions of terms > 2 are A328189.
Successive pairs of consecutive divisors are counted by A129308.
Cf. A000005, A027750, A033676, A060680, A060681, A060682, A060683, A193829, A328023, A328024, A328194.

f:=func;  g:=func; [g(n):n in [1..100]]; // Marius A. Burtea, Nov 20 2019

Table[Length[Split[Divisors[n],!Divisible[#2,#1]&]]-1,{n,100}]

a(n) = {my(d=divisors(n), nb=0); for (i=2, #d, if ((d[i] % d[i-1]) == 0, nb++)); nb;} \\ Michel Marcus, Oct 05 2019

a(p^k) = k for any prime number p and k >= 0. - Rémy Sigrist, Oct 05 2019

Data section extended up to a(105) by Antti Karttunen, Feb 23 2023

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 1, 2, 4, 1, 3, 1, 1, 1, 1, 1, 4, 2, 1, 3, 1, 1, 3, 1, 5, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {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 prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 1.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-minima of A356226, firsts A356232.
The greatest instead of smallest size is A356228.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Min@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333768(A356230(n)).

a(n) = A055396(A356231(n)).

A328161 Numbers n that are prime or whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 72, 73, 74, 77, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91

Offset: 1

Gus Wiseman, Oct 06 2019

nonn

			The proper divisors of 18 are {2, 3, 6, 9}, and {3, 6} are a consecutive divisible pair, so 18 does not belong to the sequence.
The proper divisors of 60 are {2, 3, 4, 5, 6, 10, 12, 15, 20, 30}, and none of {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 10}, {10, 12}, {12, 15}, {15, 20}, or {20, 30} are divisible pairs, so 60 belongs to the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Equals the union of A328028 and A000040.
Complement of A328189.
One, primes, and positions of 1's in A328194.
Partitions with no consecutive divisibilities are A328171.
Cf. A000005, A060680, A060775, A067513, A088725, A163870, A328162.

filter:= proc(n) local D,i;
  if isprime(n) then return true fi;
  D:= sort(convert(numtheory:-divisors(n) minus {1,n}, list));
  for i from 1 to nops(D)-1 do if (D[i+1]/D[i])::integer then return false fi od:
  true
end proc:
select(filter, [$1..100]); # Robert Israel, Oct 11 2019

Select[Range[100],!MatchQ[DeleteCases[Divisors[#],1|#],{_,x_,y_,_}/;Divisible[y,x]]&]

cp's OEIS Frontend

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

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A060682 Number of distinct differences between consecutive divisors of n (ordered by size).

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A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

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A356234 Irregular triangle read by rows where row n is the ordered factorization of n into maximal gapless divisors.

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A328028 Nonprime numbers n whose proper divisors (greater than 1 and less than n) have no consecutive divisibilities.

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A356228 Greatest size of a gapless submultiset of the prime indices of n.

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A356229 Number of maximal gapless submultisets of the prime indices of 2n.

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