A067513 -id:A067513 - cp's OEIS frontend

A067846 Least m such that A067513(m) = n.

1, 2, 4, 16, 12, 48, 36, 60, 120, 252, 180, 480, 360, 540, 720, 1080, 1620, 1260, 1680, 2160, 3600, 2520, 3780, 6720, 6480, 9240, 5040, 7560, 15840, 16380, 10080, 22680, 18480, 32400, 20160, 15120, 37800, 32760, 25200, 40320, 30240, 73920, 45360

Offset: 1

Vladeta Jovovic, Feb 14 2002

nonn

There are 57 terms less than 200000. - David Wasserman, Dec 20 2002

Robert G. Wilson v, Table of n, a(n) for n = 1..1276 (first 161 terms from Robert Israel)

Cf. A067513, A183063.

N:= 80: # to get a(1)..a(N)
A067513:= n -> nops(select(t -> isprime(t+1), numtheory:-divisors(n))):
A:= Vector(N):
count:= 1: A[1]:= 1:
for n from 2 by 2 while count < N do
  v:= A067513(n);
  if v <= N and A[v]=0 then A[v]:= n; count:= count+1; fi
od:
convert(A, list); # Robert Israel, Jun 08 2018

a067513(n) = sumdiv(n, d, isprime(d+1));
a(n) = {my(k=1); while(a067513(k) != n, k++); k;} \\ Michel Marcus, Jun 09 2018

More terms from David Wasserman, Dec 20 2002

A202727 Position of records in A067513.

Original entry on oeis.org

1, 2, 4, 12, 36, 60, 120, 180, 360, 540, 720, 1080, 1260, 1680, 2160, 2520, 3780, 5040, 7560, 10080, 15120, 25200, 30240, 45360, 55440, 60480, 75600, 85680, 100800, 110880, 131040, 166320, 196560, 257040, 332640, 393120, 514080, 655200, 665280, 786240, 831600

Offset: 1

Charles R Greathouse IV, Dec 23 2011

nonn

Numbers n such that n has more divisors d such that d+1 is prime than any smaller number.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A067513, A202728.

r=0;for(n=1,1e8,t=sumdiv(n,d,isprime(d+1));if(t>r,r=t;print1(n", ")))

A202728 Records in A067513.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 16, 18, 19, 20, 22, 23, 27, 28, 31, 36, 39, 41, 43, 45, 47, 48, 50, 52, 54, 57, 60, 62, 67, 73, 74, 76, 79, 82, 83, 85, 86, 88, 89, 93, 98, 101, 102, 109, 111, 117, 120, 134, 136, 143, 151, 158, 160, 166, 167, 192, 202, 221

Offset: 1

Charles R Greathouse IV, Dec 23 2011

nonn

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A202727, A067513.

r=0;for(n=1,1e8,t=sumdiv(n,d,isprime(d+1));if(t>r,r=t;print1(t", ")))

a(n) = A067513(A202727(n)).

A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.

Original entry on oeis.org

2, 14, 26, 34, 38, 62, 74, 86, 94, 98, 118, 122, 134, 142, 146, 158, 182, 194, 202, 206, 214, 218, 254, 266, 274, 278, 298, 302, 314, 326, 334, 338, 362, 386, 394, 398, 422, 434, 446, 454, 458, 482, 494, 514, 518, 526, 538, 542, 554, 566, 578

Offset: 1

N. J. A. Sloane

Alternative definition: let D(m) = set of divisors of m; sequence gives n such that the set 1 + D(n) contains only two primes, 2 and 3. E.g., n=98: D(98)={1,2,7,15,49,98}, 1+D = {2,3,8,16,50,99} of which only 2 terms are prime numbers: {2,3}. Observation by Labos Elemer, Jun 24 2002. This is a consequence of the von Staudt-Clausen theorem. - N. J. A. Sloane, Jan 04 2004
The fraction of Bernoulli numbers with denominator 6 is roughly 1/6, see Erdős-Wagstaff. But calculations by H. Cohen and G. Tenenbaum suggest that the fraction is closer to 1/7 (posting to Number Theory List around Dec 20 2005).
Simon Plouffe reports (Feb 13 2007) that at B_{9083002} the proportion is 0.151848915149418661363281... and still decreasing very slowly.
In his PhD thesis at the University of Illinois (see reference), Richard Sunseri proved that a higher proportion of Bernoulli denominators equal 6 than any other value.
Rado showed that for a given Bernoulli number B_n there exist infinitely many Bernoulli numbers B_m having the same denominator. As a special case, if n = 2p where p is an odd prime p == 1 (mod 3), then the denominator of the Bernoulli number B_n equals 6. - Bernd C. Kellner, Mar 21 2018
Conjecture: When the expression (p+q^b)/2 is required to be prime, p is prime, and q is a prime >=5, then all p values are prime congruent to 1 (mod 12) (A068228), if and only if the exponent b is a member of this set. - Richard R. Forberg, Apr 07 2025
There are additional exponential expressions conjectured for generating each of several known prime subsequences (e.g., Pythagorean primes, A002144) where the sequence is invariant to the exponent, if and only if the exponent is a member of this set.  See Forberg link. - Richard R. Forberg, Apr 25 2025

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 75.
C. J. Moreno and S. S. Wagstaff, Sums of Squares of Integers, CRC Press, 2005, Sect. 3.9.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 10.

T. D. Noe, Table of n, a(n) for n = 1..1000
Paul Erdős and Samuel S. Wagstaff, Jr., The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), pp. 104-112, MR 81c:10064.
Richard R. Forberg, A051222 and Exponential Expressions the Reproduce Certain Prime Subsets, 2025.
K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.
R. Rado, A note on the Bernoullian numbers, J. London Math. Soc. 9 (1934) 88-90.
Richard Sunseri, Zeros of p-adic L-functions and densities relating to Bernoulli numbers, PhD thesis, University of Illinois, 1979.
Index entries for sequences related to Bernoulli numbers.

Except for 2, all terms are even nontotient numbers. Proper subset of A005277: e.g., 50 and 90 are not here. - Labos Elemer
A112772 is a subsequence. - Bernd C. Kellner, Mar 21 2018
Cf. A045979, A000005, A067513, A002202, A005277.

di[x_] := Divisors[x]
dp[x_] := Part[di[x], Flatten[Position[PrimeQ[1+di[x]], True]]]+1
Do[s=Length[dp[n]]; If[Equal[s, 2], Print[n]], {n, 1, 10000}] (* Labos Elemer *)
Do[s=Denominator[BernoulliB[n]]; If[Equal[s, 6], Print[n]], {n, 1, 1000}] (* Labos Elemer *)
Do[s=1+Divisors[n];s1=Flatten[Position[PrimeQ[s], True]]; (*analogous [suitably modified] pairs of programs yield A051225-A051230*) s2=Part[s, s1];If[Equal[s2, {2, 3}], Print[n]], {n, 1, 100}] (* Labos Elemer *)
Select[Range[600],Denominator[BernoulliB[#]]==6&] (* Harvey P. Dale, Dec 08 2011 *)

for(n=1,10^3,if(denominator(bernfrac(n))==6,print1(n,", "))); \\ Joerg Arndt, Oct 28 2014

is(n)=if(n%2,return(0)); fordiv(n/2,d,if(isprime(2*d+1)&&d>1, return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2014

Additional comments and references from Sam Wagstaff, Dec 20 2005

A072627 Number of divisors d of n such that d-1 is prime.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 2, 1, 0, 0, 4, 0, 1, 1, 2, 0, 3, 0, 2, 1, 0, 0, 6, 0, 0, 1, 2, 0, 3, 0, 3, 1, 0, 0, 5, 0, 1, 1, 3, 0, 4, 0, 2, 1, 0, 0, 7, 0, 0, 1, 1, 0, 4, 0, 3, 1, 0, 0, 7, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 8, 0, 1, 1, 2, 0, 2, 0, 4, 1, 0, 0, 7, 0, 0, 1, 3, 0, 5, 0, 1, 1, 0, 0, 8, 0, 2, 1, 2, 0, 3, 0, 3, 1

Offset: 1

Labos Elemer, Jun 28 2002

nonn

Inverse Möbius transform of A010051 (when it is shifted one step right). - R. J. Mathar, Jan 25 2009, comment in parenthesis added by Antti Karttunen, Dec 27 2018

If n == 3 (mod 6) then a(n)=1; a(n) = 0 for all other odd n. - Robert Israel, Dec 27 2018

			n=240: a(240)=12 because primes of -1+d form are: {2,3,5,7,11,19,23,29,47,59,79,239}. These and only these divisors are present in any InvSigma of n, like:InvSig[240]= {114,135,158,177,203,209,239} with {2,3,19,3,5,2,79,3,59,7,29,11,19,239} p-divisors.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A000005.

Cf. also A067513.

a072627 = length . filter ((== 1) . a010051 . (subtract 1)) . a027749_row
-- Reinhard Zumkeller, Oct 01 2012

f:= n -> nops(select(t -> isprime(t-1), numtheory:-divisors(n))):
map(f, [$1..100]); # Robert Israel, Dec 27 2018

di[x_] := Divisors[x] dp[x_] := Part[di[x], Flatten[Position[PrimeQ[ -1+di[x]], True]]]-1 Table[Length[dp[w]], {w, 1, 128}]
Table[Count[Divisors[n],?(PrimeQ[#-1]&)],{n,110}] (* _Harvey P. Dale, Jul 04 2021 *)

a(n) = sumdiv(n, d, isprime(d-1)); \\ Michel Marcus, Dec 27 2018

If p is prime <> 3, then divisors(p)-1 = {1,p}-1 = {0,p-1}, so a(p) = 0.

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]]&]

A322312 a(n) = Product_{d|n, d+1 is prime} prime(1+A286561(n,d+1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

Original entry on oeis.org

2, 6, 2, 20, 2, 18, 2, 28, 2, 12, 2, 120, 2, 6, 2, 88, 2, 60, 2, 60, 2, 12, 2, 168, 2, 6, 2, 40, 2, 72, 2, 104, 2, 6, 2, 800, 2, 6, 2, 168, 2, 54, 2, 40, 2, 12, 2, 528, 2, 12, 2, 40, 2, 84, 2, 56, 2, 12, 2, 1440, 2, 6, 2, 136, 2, 72, 2, 20, 2, 24, 2, 2240, 2, 6, 2, 20, 2, 36, 2, 528, 2, 12, 2, 720, 2, 6, 2, 112, 2

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A067513, A185633, A286561, A322313 (rgs-transform), A322314.

Cf. also A293514, A322310.

A322312(n) = { my(m=1,p); fordiv(n,d,p=1+d; if(isprime(p), for(i=0,oo,if(n%(p^i),m *= prime(i);break)))); (m); };

a(n) = Product_{d|n} A000040(1+A286561(n,1+d))^A010051(1+d).
a(n) = A181819(A185633(n)).
For all n, A001222(a(n)) = A067513(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]]&]

A328189 Numbers n with at least one pair of consecutive divisible nontrivial divisors (greater than 1 and less than n).

Original entry on oeis.org

8, 16, 18, 20, 27, 28, 32, 40, 42, 44, 50, 52, 54, 56, 64, 66, 68, 75, 76, 78, 80, 81, 88, 92, 98, 99, 100, 102, 104, 110, 112, 114, 116, 117, 124, 125, 126, 128, 130, 136, 138, 140, 147, 148, 152, 153, 156, 160, 162, 164, 170, 171, 172, 174, 176, 184, 186

Offset: 1

Gus Wiseman, Oct 13 2019

nonn

			The nontrivial divisors of 42 are {2, 3, 6, 7, 14, 21}, with pairs of consecutive divisible divisors {3, 6} and {7, 14}, so 42 belongs to the sequence.

Complement of A328161.
Positions of terms greater than 1 in A328194.
Partitions with a pair of consecutive divisible parts are A328221.
Cf. A000005, A060680, A060775, A067513, A088725, A163870, A328028, A328162, A328171.

Select[Range[200],MatchQ[DeleteCases[Divisors[#],1|#],{_,x_,y_,_}/;Divisible[y,x]]&]
Select[Range[2,200],AnyTrue[Partition[Most[Rest[Divisors[#]]],2,1],Mod[#[[2]],#[[1]]] == 0&]&] (* Harvey P. Dale, Mar 14 2023 *)

A105017 Positions of records in A064097.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 19, 23, 43, 47, 94, 139, 235, 283, 517, 659, 1081, 1319, 2209, 2879, 5758, 8637, 13301, 20147, 30337, 49727, 61993, 103823, 135313, 247439, 366683, 606743, 811879, 1266767, 1739761, 2913671, 3797401, 5827343, 8288641, 16577282, 22784407, 37346483, 58003213, 81768767

Offset: 1

Hugo Pfoertner, Feb 17 2006

nonn

With a(1) = 1, a(n) is the smallest number m such that the number of iterations of k -> k - k/p, p being any prime factor of k, needed to reach 1 starting at k = m is equal to n-1. (See Example section.) - Jaroslav Krizek, Feb 15 2010

a(n) =~ sqrt(e^(5n/6)). - Robert G. Wilson v, Aug 11 2022

			a(6)=11 because m=11 requires 6-1 = 5 iterations of r -> r - (largest divisor d < r) to reach 1 (the 5 iterations are 11-1=10, 10-5=5, 5-1=4, 4-2=2, and 2-1=1) and 11 is the smallest such number m. - _Jaroslav Krizek_, Feb 15 2010

Robert G. Wilson v, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Program

Cf. A000079, A064097, A067513, A175125.

A105017 := proc()
    local maxa,a ;
    maxa := -999 ;
    for n from 1 do
        a := A064097(n) ;
        if a > maxa then
            printf("%d\n",n) ;
            maxa :=a ;
        end if;
    end do:
end proc:
A105017() ; # R. J. Mathar, Aug 07 2022

g[n_] := Block[{p = Select[1 + Divisors@n, PrimeQ]}, n*p/(p - 1)]; f[n_] := f[n] = Block[{lst = Union@Flatten[g@# & /@ f[n - 1]]}, If[ Length@ lst > 325, lst = Take[lst, 325 (* This limit must be increased for greater n's from the start. *) ]]; lst]; f[1] = {1}; f[0] = {0}; lst = {}; Do[ AppendTo[lst, Min[ f[n]]]; f[n - 1] =., {n, 44}]; lst (* Robert G. Wilson v, Aug 11 2022 *)

a=vectorsmall(10^7); a[1]=0;
for(n=2,#a,if(isprime(n),a[n]=1+a[n-1],f=factor(n);a[n]=a[f[1,1]]+a[n/f[1,1]])); \\ computes A064097
r=-oo; for(k=1,#a,if(a[k]>r,print1(k,", ");r=a[k])); \\ Hugo Pfoertner, Mar 16 2020

a(1)=1 inserted by Robert G. Wilson v, Mar 16 2020

cp's OEIS Frontend

A067846 Least m such that A067513(m) = n.

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A202727 Position of records in A067513.

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A202728 Records in A067513.

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A051222 Numbers k such that Bernoulli number B_{k} has denominator 6.

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Mathematica

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A072627 Number of divisors d of n such that d-1 is prime.

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Haskell

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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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Maple

Mathematica

A322312 a(n) = Product_{d|n, d+1 is prime} prime(1+A286561(n,d+1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

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

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