A060681 -id:A060681 - cp's OEIS frontend

A328023 Heinz number of the multiset of differences between consecutive divisors of n.

1, 2, 3, 6, 7, 20, 13, 42, 39, 110, 29, 312, 37, 374, 261, 798, 53, 2300, 61, 3828, 903, 1426, 79, 18648, 497, 2542, 2379, 21930, 107, 86856, 113, 42294, 4503, 5546, 2247, 475800, 151, 7906, 8787, 370620, 173, 843880, 181, 249798, 92547, 12118, 199, 5965848

Offset: 1

Gus Wiseman, Oct 02 2019

nonn

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

			The sequence of terms together with their prime indices begins:
            1: ()
            2: (1)
            3: (2)
            6: (2,1)
            7: (4)
           20: (3,1,1)
           13: (6)
           42: (4,2,1)
           39: (6,2)
          110: (5,3,1)
           29: (10)
          312: (6,2,1,1,1)
           37: (12)
          374: (7,5,1)
          261: (10,2,2)
          798: (8,4,2,1)
           53: (16)
         2300: (9,3,3,1,1)
           61: (18)
         3828: (10,5,2,1,1)
For example, the divisors of 6 are {1,2,3,6}, with differences {1,1,3}, with Heinz number 20, so a(6) = 20.

The sorted version is A328024.
a(n) is the Heinz number of row n of A193829, A328025, or A328027.
Cf. A000005, A027750, A056239, A060682, A060683, A112798, A129308, A193829.

Table[Times@@Prime/@Differences[Divisors[n]],{n,100}]

A056239(a(n)) = n - 1. In words, the integer partition with Heinz number a(n) is an integer partition of n - 1.
A055396(a(n)) = A060680(n).
A061395(a(n)) = A060681(n).
A001221(a(n)) = A060682(n).
A001222(a(n)) = A000005(n).

A074845 Numbers k such that S(k) = largest difference between consecutive divisors of k (ordered by size), where S(k) is the Kempner function (A002034).

Original entry on oeis.org

6, 8, 9, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Jason Earls, Sep 10 2002

It appears that terms > 6 are simply given by: composite k such that k^2 doesn't divide A000254(k). - Benoit Cloitre, Mar 09 2004
It appears that A011776(a(k)) = 2. - Gionata Neri, Jul 31 2017
It appears that this sequence consists of the numbers k such that A045763(k) > 0 and k does not divide A070251(k). - Isaac Saffold, Jun 01 2018

Michael De Vlieger, Table of n, a(n) for n = 1..670 (a(n) < 10^4, from b-file at A002034).

Cf. A002034, A060681.

Select[Range@ 514, Function[n, Module[{m = 1}, While[! Divisible[m!, n], m++]; m] == Max@ Differences@ Divisors@ n]] (* Michael De Vlieger, Jul 31 2017 *)

K(n) = my(s=1); while(s!%n>0, s++); s;
dd(n) = my(vd=divisors(n)); vecmax(vector(#vd-1, k, vd[k+1] - vd[k]));
isok(n) = K(n) == dd(n); \\ Michel Marcus, Aug 03 2017

A323075 The fixed point reached when map x -> 1+(x-(largest divisor d < x)) is iterated, starting from x = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 7, 5, 7, 3, 11, 7, 13, 5, 11, 7, 17, 3, 19, 11, 11, 7, 23, 13, 11, 5, 19, 11, 29, 7, 31, 17, 23, 3, 29, 19, 37, 11, 19, 11, 41, 7, 43, 23, 31, 13, 47, 11, 43, 5, 29, 19, 53, 11, 31, 29, 19, 7, 59, 31, 61, 17, 43, 23, 53, 3, 67, 29, 47, 19, 71, 37, 73, 11, 29, 19, 67, 11, 79, 41, 31, 7, 83, 43, 47, 23, 59, 31, 89, 13

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

After a(1) = 1, the fixed point reached is always a prime. Question: Do all odd primes occur infinitely often?

Yes. All odd primes occur infinitely often. A060681(2*k) = k + 1 and so for each k > 1 there exists an integer m such that a(m) = p where p is an odd prime. - David A. Corneth, Jan 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..20669

Cf. A000040, A000079, A000918, A060681, A323076, A323079, A323164, A323165 (ordinal transform).

Cf. also A039650, A039654.

{1}~Join~Array[FixedPoint[1 + (# - Divisors[#][[-2]]) &, #] &, 89, 2] (* Michael De Vlieger, Jan 04 2019 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323075(n) = { my(nn = 1+A060681(n)); if(nn==n,n,A323075(nn)); };

If n == (1+A060681(n)), then a(n) = n, otherwise a(n) = a(1+A060681(n)).
a(2^k * p - 2^(k+1) + 2) = a(A000079(k) * p - A000918(k+1)) = p for k >= 0. - David A. Corneth, Jan 08 2019
a(1) = 1, and for n > 1, a(n) = A000040(A323164(n)). - Antti Karttunen, Jan 08 2019

A328024 Heinz numbers of multisets representing the differences between some positive integer's consecutive divisors.

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 20, 29, 37, 39, 42, 53, 61, 79, 107, 110, 113, 151, 173, 181, 199, 239, 261, 271, 281, 312, 317, 349, 359, 374, 397, 421, 457, 497, 503, 541, 557, 577, 593, 613, 701, 733, 769, 787, 798, 857, 863, 903, 911, 953, 983, 1021, 1061, 1069, 1151

Offset: 1

Gus Wiseman, Oct 02 2019

nonn

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

There is exactly one entry with any given sum of prime indices A056239.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     3: {2}
     6: {1,2}
     7: {4}
    13: {6}
    20: {1,1,3}
    29: {10}
    37: {12}
    39: {2,6}
    42: {1,2,4}
    53: {16}
    61: {18}
    79: {22}
   107: {28}
   110: {1,3,5}
   113: {30}
   151: {36}
   173: {40}
   181: {42}
   199: {46}
   239: {52}
   261: {2,2,10}
   271: {58}
   281: {60}
   312: {1,1,1,2,6}
For example, the divisors of 8 are {1,2,4,8}, with differences {1,2,4}, with Heinz number 42, so 42 belongs to the sequence.

A permutation of A328023.
Also the set of possible Heinz numbers of rows of A193829, A328025, or A328027.
Cf. A001222, A000005, A027750, A056239, A060680, A060681, A060682, A112798.

nn=1000;
Select[Union[Table[Times@@Prime/@Differences[Divisors[n]],{n,nn}]],#<=nn&]

A328450 Numbers that are a smallest number with k pairs of successive divisors, for some k.

Original entry on oeis.org

1, 2, 6, 12, 60, 72, 180, 360, 420, 840, 1260, 2520, 3780, 5040, 13860, 27720, 36960, 41580, 55440, 83160, 166320, 277200, 360360, 471240, 491400, 720720, 1081080, 1113840, 2162160, 2827440, 3341520, 4324320, 5405400, 6126120

Offset: 0

Gus Wiseman, Oct 15 2019

nonn

A sorted version of A287142.

			The divisors of 72 are {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, with pairs of successive divisors {{1, 2}, {2, 3}, {3, 4}, {8, 9}}, and no smaller number has 4 successive pairs, so 72 belongs to the sequence.

Sorted positions of first appearances in A129308.
The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a longest run of length n is A328449(n).
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A072627, A181063, A199970, A287142, A328165.

dat=Table[Count[Differences[Divisors[n]],1],{n,10000}];
Sort[Table[Position[dat,i][[1,1]],{i,Union[dat]}]]

A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 5, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1

Offset: 1

Gus Wiseman, Oct 17 2019

nonn

By convention, a(1) = 1, and a(p) = 0 for p prime.

			The non-singleton runs of the nontrivial divisors of 1260 are: {2,3,4,5,6,7} {9,10} {14,15} {20,21} {35,36}, so a(1260) = 6.

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

Positions of first appearances are A328459.
Positions of 0's and 1's are A088723.
The version that looks at all divisors is A055874.
The number of successive pairs of divisors > 1 of n is A088722(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A033676, A060681, A060775, A070824, A088725, A129308, A181063, A199970, A328165, A163870, A328194, A328448, A328449.

Table[Switch[n,1,1,?PrimeQ,0,,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,100}]

A328458(n) = if(1==n,n,my(rl=0,pd=0,m=0); fordiv(n, d, if(1(1+pd), m = max(m,rl); rl=0); pd=d; rl++)); max(m,rl)); \\ Antti Karttunen, Feb 23 2023

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

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

Original entry on oeis.org

2, 6, 15, 79, 68, 121, 162, 445, 416, 971, 836, 987, 2888, 1891, 1650, 5637, 5518, 4834, 9237, 8152, 10045, 21550, 20248, 20179, 29914, 36070, 24237, 53355, 52873, 34206, 103134, 90190, 63755, 147861, 98103, 117467, 209102, 206423, 124954, 237847, 369223

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

			We need the first 15 prime gaps (1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6) before we reach the 3rd appearance of 6, so a(6) = 15.

The first appearances are at A038664, seconds A356221.
Diagonal of A356222.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts divisors with gapless prime indices, complement A356225.
A356226 = gapless interval lengths of prime indices, run-lengths A287170.
Cf. A000005, A028334, A029709, A060681, A119313, A137921, A193829, A274121.

nn=1000;
gaps=Differences[Array[Prime,nn]];
Table[Position[gaps,2*n][[n,1]],{n,Select[Range[nn],Length[Position[gaps,2*#]]>=#&]}]

A358764 Largest difference between consecutive divisors of A276086(n), where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 4, 5, 10, 15, 30, 45, 20, 25, 50, 75, 150, 225, 100, 125, 250, 375, 750, 1125, 500, 625, 1250, 1875, 3750, 5625, 6, 7, 14, 21, 42, 63, 28, 35, 70, 105, 210, 315, 140, 175, 350, 525, 1050, 1575, 700, 875, 1750, 2625, 5250, 7875, 3500, 4375, 8750, 13125, 26250, 39375, 42, 49, 98, 147

Offset: 0

Antti Karttunen, Dec 02 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A002110, A053669, A060681, A276084, A276086, A324895.

Cf. also A353528, A353529.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A358764(n) = A060681(A276086(n));

A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
A358764(n) = if(!n, n, if(n%6, A276086(n-1), my(p=A053669(n)); ((p-1)/p)*A276086(n)));

a(n) = A060681(A276086(n)).
a(n) = A276086(n) - A324895(n).
For n >= 1, a(n) = A276086(n) - (A276086(n) / A053669(n)).
When n > 0 and A276084(n) is:
  < 2 (i.e., when n is not a multiple of 6),      then a(n) = A276086(n-1),
  2 (n is multiple of  6, but not of      30),    then a(n) = 2*A276086(n-5),
  3 (multiple of      30, but not of     210),    then a(n) = A276086(n-27),
  4 (multiple of     210, but not of    2310),    then a(n) = A276086(n-203),
  5 (multiple of    2310, but not of   30030),    then a(n) = 2*A276086(n-2307),
  6 (multiple of   30030, but not of  510510),    then a(n) = 8*A276086(n-30029),
  7 (multiple of  510510, but not of 9699690),    then a(n) = A276086(n-510505),
  8 (multiple of 9699690, but not of A002110(9)), then a(n) = A276086(n-9699479).

A359427 Dirichlet inverse of A358764.

Original entry on oeis.org

1, -2, -3, -2, -9, 8, -5, 6, -6, 6, -45, -4, -25, -30, -21, -130, -225, -70, -125, -130, -345, -570, -1125, -480, -544, -1150, -1812, -3550, -5625, 222, -7, 530, 249, 858, 27, 418, -35, 430, 45, 610, -315, 1520, -175, 2650, -48, 3450, -1575, 2060, -850, 804, -1275, -250, -7875, 4288, -3565, 6150, -12375

Offset: 1

Antti Karttunen, Jan 02 2023

Antti Karttunen, Table of n, a(n) for n = 1..11550
Index entries for sequences related to primorial base

Cf. A060681, A276086, A358764, A359428.
Cf. A056911 (positions of odd terms), A323239 (parity of terms), A337945.
Cf. also A342417.

up_to = 11550;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A358764(n) = A060681(A276086(n));
v359427 = DirInverseCorrect(vector(up_to,n,A358764(n)));
A359427(n) = v359427[n];

a(n) = A359428(n) - A358764(n).

A366387 Divide n by its smallest prime factor, then multiply with the index of that same prime; a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 6, 5, 5, 6, 6, 7, 10, 8, 7, 9, 8, 10, 14, 11, 9, 12, 15, 13, 18, 14, 10, 15, 11, 16, 22, 17, 21, 18, 12, 19, 26, 20, 13, 21, 14, 22, 30, 23, 15, 24, 28, 25, 34, 26, 16, 27, 33, 28, 38, 29, 17, 30, 18, 31, 42, 32, 39, 33, 19, 34, 46, 35, 20, 36, 21, 37, 50, 38, 44, 39, 22, 40, 54, 41, 23, 42, 51

Offset: 1

Antti Karttunen, Oct 23 2023

nonn

Cf. A000720, A020639, A032742, A055396.
Cf. A196050 (number of iterations needed to reach 1), A324923.
Cf. also A366385, and A060681 (divide by the smallest prime p, then multiply with p-1),

Array[PrimePi[#2]*#1/#2 & @@ {#, FactorInteger[#][[1, 1]]} &, 85] (* Michael De Vlieger, Oct 23 2023 *)

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A366387(n) = { my(spf=A020639(n)); primepi(spf)*(n/spf); };

a(n) = A032742(n) * A055396(n) = (n/A020639(n)) * A000720(A020639(n)).

a(2n) = n, a(3*(2n+1)) = 2*(2n+1) = 4n + 2.

cp's OEIS Frontend

A328023 Heinz number of the multiset of differences between consecutive divisors of n.

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A074845 Numbers k such that S(k) = largest difference between consecutive divisors of k (ordered by size), where S(k) is the Kempner function (A002034).

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A323075 The fixed point reached when map x -> 1+(x-(largest divisor d < x)) is iterated, starting from x = n.

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A328024 Heinz numbers of multisets representing the differences between some positive integer's consecutive divisors.

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A328450 Numbers that are a smallest number with k pairs of successive divisors, for some k.

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A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n.

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A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

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A358764 Largest difference between consecutive divisors of A276086(n), where A276086 is the primorial base exp-function.

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