A067513 -id:A067513 - cp's OEIS frontend

A334111 Irregular triangle where row n gives all terms k for which A064097(k) = n.

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 13, 14, 15, 17, 18, 20, 24, 32, 19, 21, 22, 25, 26, 27, 28, 30, 34, 36, 40, 48, 64, 23, 29, 31, 33, 35, 37, 38, 39, 41, 42, 44, 45, 50, 51, 52, 54, 56, 60, 68, 72, 80, 96, 128, 43, 46, 49, 53, 55, 57, 58, 61, 62, 63, 65, 66, 70, 73, 74, 75, 76, 78, 81, 82, 84

Offset: 0

Antti Karttunen, Michael De Vlieger and Robert G. Wilson v, Apr 14 2020

Applying map k -> (p-1)*(k/p) to any term k on any row n > 1, where p is any prime factor of k, gives one of the terms on preceding row n-1.
Any prime that appears on row n is 1 + {some term on row n-1}.
The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A064097 is completely additive.
A001221(k) gives the number of terms on the row above that are immediate descendants of k.
A067513(k) gives the number of terms on the row below that lead to k.

			Rows 0-6 of the irregular table:
0 |   1;
1 |   2;
2 |   3, 4;
3 |   5, 6, 8;
4 |   7, 9, 10, 12, 16;
5 |  11, 13, 14, 15, 17, 18, 20, 24, 32;
6 |  19, 21, 22, 25, 26, 27, 28, 30, 34, 36, 40, 48, 64;

Michael De Vlieger, Table of n, a(n) for n = 0..14422 (rows 0 <= n <= 17, flattened)
Index entries for sequences that are permutations of the natural numbers

Cf. A001221, A064097, A067513, A333123, A334144.
Cf. A105017 (left edge), A000079 (right edge), A175125 (row lengths).
Cf. also A058812, A334100.

f[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[1, 1]] &, n, # != 1 &]; SortBy[ Range@70, f]
(* Second program *)
With[{nn = 8}, Values@ Take[KeySort@ PositionIndex@ Array[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[1, 1]] &, #, # > 1 &] &, 2^nn], nn + 1]] // Flatten (* Michael De Vlieger, Apr 18 2020 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A064097(n) = if(1==n,0,1+A064097(A060681(n)));
for(n=0, 10, for(k=1,2^n,if(A064097(k)==n, print1(k,", "))));

A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

Original entry on oeis.org

2, 12, 2, 120, 2, 252, 2, 240, 2, 132, 2, 32760, 2, 12, 2, 8160, 2, 14364, 2, 6600, 2, 276, 2, 65520, 2, 12, 2, 3480, 2, 85932, 2, 16320, 2, 12, 2, 69090840, 2, 12, 2, 541200, 2, 75852, 2, 2760, 2, 564, 2, 2227680, 2, 132, 2, 6360

Offset: 1

Paul Curtz, Dec 18 2012

nonn

There is an integer sequence b(n) = A053657(n)/2^(n-1) = 1, 1, 6, 6, 360, 360, 45360, 45360, 5443200, 5443200,... which consists of the duplicated entries of A202367.

The ratios of this sequence are b(n+1)/b(n) = 1, 6, 1, 60, 1, 126 .... = a(n)/2, which is a variant of A036283.

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

Cf. A006953, A007395 (bisections).

Cf. A006863, A027760, A067513, A322312, A322315 (rgs-transform).

A185633 := proc(n)
    A053657(n+1)/A053657(n) ;
end proc: # R. J. Mathar, Dec 19 2012

max = 52; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, max}]]]; a[n_, k_] := Denominator[Coefficient[s, x^n*z^k]]; A053657 = Prepend[LCM @@@ Table[a[n, k], {n, max}, {k, n}], 1]; a[n_] := A053657[[n+1]]/A053657[[n]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Dec 20 2012 *)

A185633(n) = if(n%2,2,denominator(bernfrac(n)/(n))); \\ Antti Karttunen, Dec 03 2018

A185633(n) = { my(m=1); fordiv(n, d, if(isprime(1+d), m *= (1+d)^(1+valuation(n,1+d)))); (m); }; \\ Antti Karttunen, Dec 03 2018

a(n) = A053657(n+1)/A053657(n).
a(2*n) = 2*A036283(n).
From Antti Karttunen, Dec 03 2018: (Start)
a(n) = Product_{d|n} [(1+d)^(1+A286561(n,1+d))]^A010051(1+d) - after Peter J. Cameron's Mar 25 2002 comment in A006863.
A007947(a(n)) = A027760(n)
A001221(a(n)) = A067513(n).
A181819(a(n)) = A322312(n).
(End)

Name edited by Antti Karttunen, Dec 03 2018

A046886 Number of divisors d of 2n satisfying (d+1) = prime or number of prime factors of the denominator of the even Bernoulli numbers.

Original entry on oeis.org

2, 3, 3, 3, 3, 5, 2, 4, 4, 4, 3, 5, 2, 4, 5, 4, 2, 7, 2, 5, 4, 4, 3, 6, 3, 4, 4, 4, 3, 8, 2, 4, 5, 3, 4, 8, 2, 3, 4, 6, 3, 7, 2, 5, 6, 4, 2, 7, 2, 5, 4, 4, 3, 8, 4, 6, 3, 4, 2, 9, 2, 3, 6, 4, 4, 7, 2, 4, 5, 6, 2, 9, 2, 4, 6, 3, 3, 8, 2, 6, 5, 4, 3, 7, 3, 4, 4, 6, 3, 11, 2, 4, 3, 3, 4, 8, 2, 5, 7, 6, 2, 6, 2, 5

Offset: 1

Wouter Meeussen, Jan 23 2001

nonn

From von Staudt-Clausen theorem.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Th. 118.
Hans Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000146, A027642.

A046886 := proc(n)
    A067513(2*n) ;
end proc:
seq(A046886(n),n=1..100) ; # R. J. Mathar, Aug 07 2022

Length[ Select[ Divisors[ 2n ], PrimeQ[ #+1 ]& ] ] or Length[ FactorInteger[ Denominator@BernoulliB[ 2k ] ] ]
Table[Count[Divisors[2n],?(PrimeQ[#+1]&)],{n,110}] (* or *) PrimeOmega/@ Denominator[BernoulliB[2*Range[110]]] (* _Harvey P. Dale, Mar 19 2015 *)

a(n) = A067513(2n). - R. J. Mathar, Aug 07 2022

A290694 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, -1, 2, 0, 0, 1, -2, 3, 0, 0, -1, 14, -9, 24, 0, 0, 1, -10, 75, -48, 20, 0, 0, -1, 62, -135, 312, -300, 720, 0, 0, 1, -42, 903, -1680, 2800, -2160, 630, 0, 0, -1, 254, -1449, 40824, -21000, 27360, -17640, 4480

Offset: 0

Peter Luschny, Aug 24 2017

Consider a family of integrals I_m(n) = Integral_{x=0..1} P'(n, x)^m with P'(n,x) = Sum_{k=0..n}(-1)^(n-k)*Stirling2(n, k)*k!*x^k (see A278075 for the coefficients).
I_1(n) are the Bernoulli numbers A164555/A027642, I_2(n) are the Bernoulli median numbers A212196/A181131, I_3(n) are the numbers A291449/A291450.
The coefficients of the polynomials P_n(x)^m are for m = 1 A290694/A290695 and for m = 2 A291447/A291448.
Only omega(Clausen(n)) = A001221(A160014(n,1)) = A067513(n) coefficients are rational numbers if n is even. For odd n > 1 there are two rational coefficients.
Let C_k(n) = [x^k] P_n(x), k > 0 and n even. Conjecture: k is a prime factor of Clausen(n) <=> k = denominator(C_k(n)) <=> k does not divide Stirling2(n, k-1)*(k-1)!. (Note that by a comment in A019538 Stirling2(n, k-1)*(k-1)! is the number of chain topologies on an n-set having k open sets.)

			Triangle starts:
[0, 1]
[0, 0,  1]
[0, 0, -1,   2]
[0, 0,  1,  -2,    3]
[0, 0, -1,  14,   -9,  24]
[0, 0,  1, -10,   75, -48,   20]
[0, 0, -1,  62, -135, 312, -300, 720]
The first few polynomials are:
P_0(x) = x.
P_1(x) =  (1/2)*x^2.
P_2(x) = -(1/2)*x^2 + (2/3)*x^3.
P_3(x) =  (1/2)*x^2 - 2*x^3 + (3/2)*x^4.
P_4(x) = -(1/2)*x^2 + (14/3)*x^3 - 9*x^4 + (24/5)*x^5.
P_5(x) =  (1/2)*x^2 - 10*x^3 + (75/2)*x^4 - 48*x^5 + 20*x^6.
P_6(x) = -(1/2)*x^2 + (62/3)*x^3 - 135*x^4 + 312*x^5 - 300*x^6 + (720/7)*x^7.
Evaluated at x = 1 this gives an additive decomposition of the Bernoulli numbers:
B(0) =     1 =    1.
B(1) =   1/2 =  1/2.
B(2) =   1/6 = -1/2 +  2/3.
B(3) =     0 =  1/2 -    2 + 3/2.
B(4) = -1/30 = -1/2 + 14/3 -    9 + 24/5.
B(5) =     0 =  1/2 -   10 + 75/2 -   48 +  20.
B(6) =  1/42 = -1/2 + 62/3 -  135 +  312 - 300 + 720/7.

Peter Luschny, Illustrating A290694.

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Cf. A160014, A278075.

BG_row := proc(m, n, frac, val) local F, g, v;
F := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n):
g := x -> int(F(n,x)^m, x):
`if`(val = "val", subs(x=1, g(x)), [seq(coeff(g(x),x,j), j=0..m*n+1)]):
`if`(frac = "num", numer(%), denom(%)) end:
seq(BG_row(1, n, "num", "val"), n=0..16);         # A164555
seq(BG_row(1, n, "den", "val"), n=0..16);         # A027642
seq(print(BG_row(1, n, "num", "poly")), n=0..12); # A290694 (this seq.)
seq(print(BG_row(1, n, "den", "poly")), n=0..12); # A290695
# Alternatively:
T_row := n -> numer(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 6 do T_row(n) od;

T[n_, k_] := If[k > 0, Numerator[StirlingS2[n, k - 1]*(k - 1)! / k], 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n+1}] // Flatten

T(n, k) = Numerator(Stirling2(n, k - 1)*(k - 1)!/k) if k > 0 else 0; for n >= 0 and 0 <= k <= n+1.

A141197 a(n) = the number of divisors of n that are each one less than a power of a prime.

Original entry on oeis.org

1, 2, 2, 3, 1, 4, 2, 4, 2, 3, 1, 6, 1, 3, 3, 5, 1, 5, 1, 4, 3, 3, 1, 8, 1, 3, 2, 5, 1, 7, 2, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 6, 1, 4, 3, 3, 1, 10, 2, 3, 2, 5, 1, 5, 1, 6, 2, 3, 1, 10, 1, 3, 4, 5, 1, 6, 1, 3, 2, 5, 1, 11, 1, 2, 3, 3, 2, 6, 1, 8, 2, 3, 1, 9, 1, 2, 2, 6, 1, 8, 2, 4, 3, 2, 1, 11, 1, 3, 2, 5, 1, 5, 1

Offset: 1

Leroy Quet, Jun 12 2008

nonn

A067513(n) <= a(n) <= A000005(n). [From Reinhard Zumkeller, Oct 06 2008]

a(A185208(n)) = 1. - Reinhard Zumkeller, Nov 01 2012

			The divisors of 9 are 1,3,9. 1 is one less than 2, a power of a prime. 3 is one less than 4, a power of a prime. And 9 is one less than 10, not a power of a prime. There are therefore 2 such divisors that are each one less than a power of a prime. So a(9)=2.

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

Cf. A141198.

Cf. A049073.

a141197 = sum . map (a010055 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Nov 01 2012

a[n_] := Select[Divisors[n], PrimeNu[# + 1] == 1 &] // Length; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Aug 17 2013 *)
Table[DivisorSum[n, 1 &, PrimePowerQ[# + 1] &], {n, 103}] (* Michael De Vlieger, Aug 29 2017 *)

a(n) = sum (A010055(A027750(n,k)): k=1..A000005(n)). - Reinhard Zumkeller, Nov 01 2012

Added more terms. - Steven Bi (chenhsi(AT)stanford.edu), Dec 22 2008
Added more terms (Terms 27 - 50). Steven Bi (chenhsi(AT)stanford.edu), Jan 09 2009
Corrected and extended by Ray Chandler, Jun 25 2009

A322976 Number of divisors d of n such that d+2 is prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is added to each, as 1+2 = 3, 3+2 = 5, 5+2 = 7, etc, the only sums that are primes are: [3, 5, 7, 11, 13, 17, 23, 29, 37, 47, 79, 101, 107, 137, 167, 191, 233, 317, 947, 1487, 2081, 3467], thus (a10395) = 22.

Antti Karttunen, Table of n, a(n) for n = 1..10395
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A067513, A072627, A100821, A322358, A322975, A322977, A322978.

Array[DivisorSum[#, 1 &, PrimeQ[# + 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322976(n) = sumdiv(n, d, isprime(d+2));

a(n) = Sum_{d|n} A010051(d+2).

a(A000040(n)) = 1 + A100821(n).

A328338 Numbers whose third-largest divisor is prime.

Original entry on oeis.org

6, 8, 10, 14, 15, 20, 21, 22, 26, 27, 28, 33, 34, 35, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 68, 69, 74, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 99, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134, 141, 142, 143, 145, 146, 148

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

			The sequence of terms together with their divisors begins:
   6: {1,2,3,6}
   8: {1,2,4,8}
  10: {1,2,5,10}
  14: {1,2,7,14}
  15: {1,3,5,15}
  20: {1,2,4,5,10,20}
  21: {1,3,7,21}
  22: {1,2,11,22}
  26: {1,2,13,26}
  27: {1,3,9,27}
  28: {1,2,4,7,14,28}
  33: {1,3,11,33}
  34: {1,2,17,34}
  35: {1,5,7,35}
  38: {1,2,19,38}
  39: {1,3,13,39}
  44: {1,2,4,11,22,44}
  46: {1,2,23,46}
  51: {1,3,17,51}
  52: {1,2,4,13,26,52}

Antti Karttunen, Table of n, a(n) for n = 1..10000

A subset of A002808 and superset of A006881.
Numbers whose third-smallest divisor is prime are A119313.
Third-smallest divisor is A292269.
Cf. A000005, A020639, A027750, A033676, A060775, A067513, A088725.

q:= n-> (l-> nops(l)>2 and isprime(l[-3]))(
         sort([numtheory[divisors](n)[]])):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 19 2019

Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[-3]]]&]

isA328338(n) = { my(u=numdiv(n)); ((u>2)&&isprime(divisors(n)[u-2])); }; \\ Antti Karttunen, Oct 17 2019

A067847 Number of divisors d of n! such that d+1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 9, 15, 27, 40, 63, 100, 180, 248, 459, 696, 1019, 1299, 2495, 3219, 6011, 7986, 11247, 17029, 32567, 39501, 53017, 79775, 97251, 126981, 245675, 301054, 582853, 668274, 921616, 1382954, 1811949, 2097059, 4088346, 6142187, 8417004

Offset: 1

Vladeta Jovovic, Feb 14 2002

nonn

			a(5)=9 because the divisors of 5!=120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 60 and 120 and the primes are 2, 3, 5, 7, 11, 13, 31, 41 and 61

Ray Chandler, Table of n, a(n) for n=1..42

Cf. A067513, A156190.

Table[Count[Divisors[n!]+1,?PrimeQ],{n,40}] (* _Harvey P. Dale, Apr 19 2019 *)

a(n)=s=0;fordiv (n!,i,if(isprime(n!/i+1),s=s+1));return(s);

a(n) = A067513(A000142(n)).

More terms from Sascha Kurz, Mar 19 2002
More terms from Naohiro Nomoto, Mar 13 2002
More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004
a(28)=118584 from Vladeta Jovovic, Apr 07 2004
Corrected a(28)=126981 and added a(29)-a(31) by Ray Chandler, Feb 03 2009
a(32)-a(42) from Ray Chandler, Feb 05 2009

A085020 a(n) = Sum_{d|n, (d+1) prime} (d + 1).

Original entry on oeis.org

2, 5, 2, 10, 2, 12, 2, 10, 2, 16, 2, 30, 2, 5, 2, 27, 2, 31, 2, 21, 2, 28, 2, 30, 2, 5, 2, 39, 2, 54, 2, 27, 2, 5, 2, 86, 2, 5, 2, 62, 2, 55, 2, 33, 2, 52, 2, 47, 2, 16, 2, 63, 2, 31, 2, 39, 2, 64, 2, 133, 2, 5, 2, 27, 2, 102, 2, 10, 2, 87, 2, 159, 2, 5, 2, 10, 2, 91, 2, 79, 2, 88, 2, 102, 2, 5

Offset: 1

Jason Earls, Jun 18 2003

			a(18) = 31 because the divisors of 18 are [1, 2, 3, 6, 9, 18] and 2 + 3 + 7 + 19 = 31.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A067513.

Cf. A008472. [Peter Luschny, May 04 2009]

T := proc(n,k) local i; numtheory[divisors](n); select(isprime, map(i->i+k, %)); add(i,i=%) end: seq(T(n+1,1),n=0..20); # Peter Luschny, May 04 2009

a[n_] := Sum[If[PrimeQ[d+1], d+1, 0], {d, Divisors[n]}]; Array[a, 100] (* Jean-François Alcover, Jun 04 2019 *)

a(n) = sumdiv(n, d, if (isprime(q=d+1), q)); \\ Michel Marcus, Aug 14 2017

A332999 Maximum indegree in the graph formed by a subset of numbers in range 1 .. n with edge relation k -> k - k/p, where p is any of the prime factors of k.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 3, 3, 3, 3, 1, 3, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 4, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 4

Offset: 1

Antti Karttunen, Apr 05 2020

nonn

			For n=15 we have five alternative paths from 15 to 1: {15, 10, 5, 4, 2, 1}, {15, 10, 8, 4, 2, 1}, {15, 12, 8, 4, 2, 1},  {15, 12, 6, 4, 2, 1},  {15, 12, 6, 3, 2, 1}. These form a lattice illustrated below:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1
With edges going from 15 towards 1, the maximum indegree is 3, which occurs at node 4, therefore a(15) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..31031

Cf. A332992 (max. outdegree), A333123, A334144, A334184.

Cf. A067513 for the maximal indegree in the whole semilattice (see A334111).

With[{s = Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 105]}, Array[If[# == 1, 0, Max@ Tally[#][[All, -1]] &@ Union[Join @@ Map[Partition[#, 2, 1] &, s[[#]] ]][[All, -1]] ] &, Length@ s]] (* Michael De Vlieger, May 02 2020 *)

A332999(n) = { my(m = Map(), nodes = List([n]), x, xps, s=0, u, v); while(#nodes, x = nodes[#nodes]; listpop(nodes); xps = factor(x)[, 1]~; for(i=1,#xps, u=x-(x/xps[i]); if(!mapisdefined(m,u,&v), v=0; listput(nodes,u)); mapput(m,u,v+1); s = max(s,v+1))); (s); };

cp's OEIS Frontend

A334111 Irregular triangle where row n gives all terms k for which A064097(k) = n.

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A185633 For odd n, a(n) = 2; for even n, a(n) = denominator of Bernoulli(n)/n; The number 2 alternating with the elements of A006953.

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A046886 Number of divisors d of 2n satisfying (d+1) = prime or number of prime factors of the denominator of the even Bernoulli numbers.

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A290694 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

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Maple

Mathematica

Formula

A141197 a(n) = the number of divisors of n that are each one less than a power of a prime.

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Haskell

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Extensions

A322976 Number of divisors d of n such that d+2 is prime.

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Mathematica

PARI

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A328338 Numbers whose third-largest divisor is prime.

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