A275314 -id:A275314 - cp's OEIS frontend

A329071 a(n) = phi(A275314(n)) - mu(A275314(n)), where A275314(n) is Euler's gradus function.

0, 2, 3, 3, 5, 2, 7, 2, 5, 1, 11, 5, 13, 4, 7, 5, 17, 1, 19, 7, 6, 4, 23, 1, 6, 5, 7, 6, 29, 4, 31, 1, 13, 6, 11, 7, 37, 8, 7, 4, 41, 3, 43, 13, 6, 8, 47, 7, 13, 3, 19, 7, 53, 4, 7, 3, 11, 9, 59, 6, 61, 16, 11, 7, 17, 5, 67, 19, 20, 4, 71, 4, 73, 17, 11, 11

Offset: 1

Daniel Hoyt, Nov 03 2019

nonn

Cf. A000010, A008683, A275314.

gradus[n_] := 1 + Plus @@ ((First[#] - 1) * Last[#] & /@ FactorInteger[n]); a[n_] := EulerPhi[(g = gradus[n])] - MoebiusMu[g]; Array[a, 76] (* Amiram Eldar, Nov 03 2019 *)

g(n) = my(f = factor(n)); sum(k=1, #f~, (f[k, 1]-1)*f[k, 2])+ 1; \\ A275314
a(n) = my(gn = g(n)); eulerphi(gn) - moebius(gn); \\ Michel Marcus, Nov 04 2019

a(n) = phi(A275314(n)) - mu(A275314(n)) where phi is Euler's totient function (A000010) and mu is the Mobius function (A008683).

A352064 Irregular triangle T(n,k) where row n lists the positions of n in A275314.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 7, 15, 20, 27, 36, 48, 64, 14, 30, 40, 54, 72, 96, 128, 21, 25, 28, 45, 60, 80, 81, 108, 144, 192, 256, 42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512, 11, 35, 63, 75, 84, 100, 112, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024

Offset: 1

Michael De Vlieger, Mar 02 2022

A table by Leonhard Euler.
Let L(n-1) be a partition of (n-1) whose parts m are restricted to predecessors of primes. Row n lists the products (m+1) for all such partitions L(n-1).
Greatest term in row n is 2^(n-1).
Least term in row p prime is p.

			Triangle begins:
   1;
   2;
   3,  4;
   6,  8;
   5,  9, 12, 16;
  10, 18, 24, 32;
   7, 15, 20, 27,  36,  48,  64;
  14, 30, 40, 54,  72,  96, 128;
  21, 25, 28, 45,  60,  80,  81, 108, 144, 192, 256;
  42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512;
  ...
Illustration of relationship of terms of row n and partitions of (n-1) such that all parts m are restricted to prime predecessors. We show the partitions in parentheses, adding 1 to each part m below in brackets to take the product. The products are terms in row n in this sequence.
      1 = (1);
          [2]
row 2:     2;
.
      2 = (2),    (1+1);
          [3]     [2*2]
row 3:     3,       4;
.
      3 = (2+1),  (1+1+1);
          [3*2]   [2*2*2]
row 4:     6,       8;
.
      4 = (4),    (2+2),    (2+1+1),     (1+1+1+1);
          [5]     [3*3]     [3*2*2]      [2*2*2*2]
row 5:     5,       9,        12,           16;
.
      5 = (4+1),  (2+2+1),  (2+1+1+1),   (1+1+1+1+1);
          [5*2]   [3*3*2]   [3*2*2*2]    [2*2*2*2*2]
row 6:    10,      18,        24,           32;
etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10673 (rows n = 1..40, flattened)
Leonhard Euler, Tentamen novae theoriae mvsicae ex certissimis harmoniae principiis dilvcide expositae, Petropoli, ex typographia Academiae scientiarvm (1739), 41.

Cf. A001414, A006093, A275314, A280954.

With[{n = 12}, Take[#, n] &@ Values@ KeySort@ PositionIndex@ Array[Total[Flatten[ConstantArray[#1 - 1, #2] & @@@ FactorInteger[#]]] &, 2^n]] // TableForm (* syntactically simple, or, more efficiently *)
f[m_] := Block[{s = {Prime@ PrimePi[m + 1] - 1}}, KeySort@ Merge[#, Identity] &@ Join[{1 -> {}}, Reap[Do[If[# <= m, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 1, s = DeleteCases[s, 1]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]]] &@ Total[s], {i, Infinity}]][[-1, -1]] ]]; Map[Union[Times @@ # & /@ #] &, Values@ f[40] + 1] // Flatten

A280954(n) = length of row n.

A329277 a(n) is the fixed point reached by iterating Euler's gradus function A275314 starting at n.

Original entry on oeis.org

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

Offset: 1

Daniel Hoyt, Nov 11 2019

nonn

Cf. A275314, A329071.

gradus[n_] := 1 + Plus @@ ((First[#] - 1) * Last[#] & /@ FactorInteger[n]); a[n_] := FixedPoint[gradus, n]; Array[a, 100] (* Amiram Eldar, Nov 11 2019 *)

g(n) = my (f=factor(n)); 1+sum(k=1, #f~, f[k,2]*(f[k,1]-1))
a(n) = while (n!=n=g(n),); n \\ Rémy Sigrist, Dec 03 2019

from gmpy2 import is_prime
from sympy import factorint
def gradus(n):
    sum  = 0
    factors = factorint(n)
    for p,a in factors.items():
        sum += (p - 1)*a
    return sum + 1
if _name_ == "_main_":
    glist = []
    for x in range(1, 80):
        glist.append(gradus(x))
    while True:
        for p in glist:
            a = 0
            if not is_prime(p):
                glist = [gradus(x) for x in glist]
                a = 1
        if a == 0:
            break
    print(', '.join([str(x) for x in glist]))

A091137 The Hirzebruch numbers. a(n) = Product_{2 <= p <= n+1, p prime} p^floor(n / (p - 1)).

Original entry on oeis.org

1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 479001600, 958003200, 2615348736000, 5230697472000, 31384184832000, 62768369664000, 32011868528640000, 64023737057280000, 51090942171709440000, 102181884343418880000, 33720021833328230400000, 67440043666656460800000

Offset: 0

Henry Bottomley, Dec 19 2003

nonn

Largest number m such that number of times m divides k! is almost k/n for large k, i.e., largest m with A090624(m) = n.
This is always a relatively small multiple of n!, since the multiplicity with which a prime p divides n! is always <= n/(p-1); it is equal to floor(n/(p-1)) at least when n is a power of p. - Franklin T. Adams-Watters, May 31 2010
At least for most small n, a(n) = A002790(n) * n!; the first difference is n=15. It appears that A002790(n) * n! always divides a(n).
Conjecture: The denominators of the series reversion of the sequence with e.g.f. Polylog(2,x). - Benedict W. J. Irwin, Jan 05 2017
Not only is a(n) divisible by n!; a(n) is divisible by (n + 1)! as has been observed by Bedhouche and Bakir (see links and A363596). - Hal M. Switkay, Aug 15 2025

			Let n = 4. The partitions of 4 are [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. Thus a(4) = lcm([5, 4*2, 3*3, 3*2*2, 2*2*2*2]) = 720.

P. Curtz, Integration numérique ..., Note 12, C.C.S.A., Arcueil, 1969; see pp. 36, 56.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 3rd. ed., 1966; Lemma 1.7.3, p. 14. [From N. J. A. Sloane, Sep 06 2010]

Michael De Vlieger, Table of n, a(n) for n = 0..443
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See sigma_n p. 3.
Victor M. Buchstaber and Alexander P. Veselov, Todd polynomials and Hirzebruch numbers, arXiv:2310.07383 [math.AT], Oct. 2023.
Donghyun Kim and Jaeseong Oh, Extending the science fiction and the Loehr--Warrington formula, arXiv:2409.01041 [math.CO], 2024. See p. 32.

Starts similarly to A002207 especially for even n and all values of A002207 seen so far seem to divide a(n).

Cf. A002790, A000142, A090622, A090624, A091136, A171080, A238963, A275314, A368093, A368116, A363596.

A091137 := proc(n) local a,i,p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end:
seq(A091137(n), n = 0..47); # R. J. Mathar, Feb 23 2009

A027760[n_] := Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]; a[n_] := a[n] = A027760[n]*a[n-1]; a[0] = 1; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 04 2011 *)

a(n) = local(r); r=1; forprime(p=2, n+1, r*=p^(n\(p-1))); r
\\ Franklin T. Adams-Watters, May 31 2010

from math import prod
from sympy import primerange
def A091137(n): return prod(p**(n//(p-1)) for p in primerange(n+2))
# Chai Wah Wu, Apr 28 2023

def a(n): return lcm(product(r + 1 for r in p) for p in Partitions(n))
# Or, more efficient:
from functools import cache
@cache
def a_rec(n):
    if n == 0: return 1
    p = mul(s for s in map(lambda i: i + 1, divisors(n)) if is_prime(s))
    return p * a_rec(n - 1)
print([a_rec(n) for n in range(22)]) # Peter Luschny, Dec 12 2023

a(n) = Product_p {p prime} p^floor(n/(p-1)).
a(2n+1) = 2*a(2n).
a(n+1) = A027760(n+1)*a(n). - Paul Curtz, Aug 01 2008
From Peter Luschny, Dec 11 2023: (Start)
a(n) = lcm_{p in P(n)} Product_{r in p}(r + 1), where P(n) are the partitions of n.
a(n) = lcm(A238963row(n)).
a(n) = A368116(1, n), seen as the lcm of the product of the 1-shifted partitions.
a(n) = A368093(1, n), seen as the cumulative product of the Clausen numbers A160014(1, n).  (End)
a(n) = lcm({k: A275314(k) = n+1}). - Hal M. Switkay, Aug 13 2025
a(n) = (n + 1)! * A363596(n). - Hal M. Switkay, Aug 15 2025

New name using a formula of the author by Peter Luschny, Dec 11 2023

cp's OEIS Frontend

A329071 a(n) = phi(A275314(n)) - mu(A275314(n)), where A275314(n) is Euler's gradus function.

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A352064 Irregular triangle T(n,k) where row n lists the positions of n in A275314.

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A329277 a(n) is the fixed point reached by iterating Euler's gradus function A275314 starting at n.

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A091137 The Hirzebruch numbers. a(n) = Product_{2 <= p <= n+1, p prime} p^floor(n / (p - 1)).

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