A276078 -id:A276078 - cp's OEIS frontend

A377515 The largest divisor of n that is a term in A276078.

1, 2, 3, 2, 5, 6, 7, 2, 9, 10, 11, 6, 13, 14, 15, 2, 17, 18, 19, 10, 21, 22, 23, 6, 25, 26, 9, 14, 29, 30, 31, 2, 33, 34, 35, 18, 37, 38, 39, 10, 41, 42, 43, 22, 45, 46, 47, 6, 49, 50, 51, 26, 53, 18, 55, 14, 57, 58, 59, 30, 61, 62, 63, 2, 65, 66, 67, 34, 69, 70

Offset: 1

Amiram Eldar, Oct 30 2024

First differs from A327937 at n = 625 = 5^4: a(625) = 125, while A327937(625) = 625.

The number of these divisors is A377516(n), and their sum is A377517(n).

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

Cf. A000720, A276078, A327937, A377516, A377517, A377518.

f[p_, e_] := p^Min[PrimePi[p], e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(primepi(f[i,1]), f[i,2]));}

Multiplicative with a(p^e) = p^min(pi(p), e), where pi(n) = A000720(n).
a(n) = n if and only if n is in A276078.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((pi(p)+1)*s) - p^(pi(p)+1) - p^(pi(p)*s) + p^pi(p))/p^((pi(p)+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^pi(p) * (p+1))) = 0.80906238421914194523... .

A377516 The number of divisors of n that are terms in A276078.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 3, 4, 2, 4, 2, 4, 4, 2, 2, 6, 2, 4, 4, 4, 2, 4, 3, 4, 3, 4, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 4, 2, 8, 2, 4, 6, 4, 2, 4, 3, 6, 4, 4, 2, 6, 4, 4, 4, 4, 2, 8, 2, 4, 6, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 6, 4, 4, 8, 2, 4, 3, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 30 2024

The sum of these divisors is A377517(n), and the largest of them is A377515(n).

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

Cf. A000005, A000720, A013661, A276078, A377515, A377517, A377519.

f[p_, e_] := Min[PrimePi[p], e] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, min(primepi(f[i,1]), f[i,2]) + 1);}

a(n) = A000005(A377515(n)).
Multiplicative with a(p^e) = min(pi(p), e) + 1, where pi(n) = A000720(n).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^((pi(p)+1)*s)).

A377517 The sum of the divisors of n that are terms in A276078.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 13, 18, 12, 12, 14, 24, 24, 3, 18, 39, 20, 18, 32, 36, 24, 12, 31, 42, 13, 24, 30, 72, 32, 3, 48, 54, 48, 39, 38, 60, 56, 18, 42, 96, 44, 36, 78, 72, 48, 12, 57, 93, 72, 42, 54, 39, 72, 24, 80, 90, 60, 72, 62, 96, 104, 3, 84, 144, 68, 54

Offset: 1

Amiram Eldar, Oct 30 2024

First differs from A046897 at n = 27 = 3^3: a(27) = 13, while A046897(27) = 40.

The number of these divisors is A377516(n), and the largest of them is A377515(n).

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

Cf. A000203, A000720, A013661, A046897, A276078, A377515, A377516, A377520.

f[p_, e_] := (p^(Min[PrimePi[p], e] + 1) - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(primepi(f[i,1]), f[i,2]) + 1) - 1)/(f[i,1] - 1));}

a(n) = A000203(A377515(n)).
Multiplicative with a(p^e) = (p^(min(pi(p), e)+1) - 1)/(p - 1), where pi(n) = A000720(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((pi(p)+1)*s) - p^(pi(p)+1))/p^((pi(p)+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^(pi(p)+1)) = 1.18603586369737251334... .

A380085 The largest unitary divisor of n that is a term in A276078.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 3, 25, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 9, 37, 38, 39, 5, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A077610, A276078, A325127, A377515, A380086, A380087, A380088.

f[p_, e_] := p^If[e <= PrimePi[p], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i,2] <= primepi(f[i,1]), f[i,2], 0));}

Multiplicative with a(p^e) = p^e if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) = n if and only if n is in A276078.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(2*(k+1)) + p(k)^(2*k+1) - p(k)^(k+1) - p(k)^k + 1)/(p(k)^(2*k+1) * (p(k)+1)) = 0.76189494803691349595..., where p(k) = prime(k).

A380086 The number of unitary divisors of n that are terms in A276078.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 1, 2, 4, 2, 2, 2, 4, 4, 1, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 1, 2, 2, 8, 2, 1, 4, 4, 4, 2, 2, 4, 4, 2, 2, 8, 2, 2, 4, 4, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 4, 4, 2, 4, 2, 4, 4, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 4, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A005117, A034444, A077610, A276078, A325127, A377516, A380085, A380087, A380089.

f[p_, e_] := If[e <= PrimePi[p], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~,if(f[i,2] <= primepi(f[i,1]), 2, 1));}

a(n) = A034444(A380085(n)).
Multiplicative with a(p^e) = 2 if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) < A034444(n) if and only if n is in A276079.
a(n) = A034444(n) if and only if n is in A276078.
a(n) = A377516(n) if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((pi(p)+1)*s)).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(pi(p)+1))) = 0.85808348184674088116... .

A380087 The sum of the unitary divisors of n that are terms in A276078.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 1, 10, 18, 12, 4, 14, 24, 24, 1, 18, 30, 20, 6, 32, 36, 24, 4, 26, 42, 1, 8, 30, 72, 32, 1, 48, 54, 48, 10, 38, 60, 56, 6, 42, 96, 44, 12, 60, 72, 48, 4, 50, 78, 72, 14, 54, 3, 72, 8, 80, 90, 60, 24, 62, 96, 80, 1, 84, 144, 68, 18, 96, 144

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A005117, A034448, A077610, A276078, A377517, A380085, A380086, A380090.

f[p_, e_] := If[e <= PrimePi[p], p^e, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= primepi(f[i,1]), f[i,1]^f[i,2], 0) + 1);}

a(n) = A034448(A380085(n)).
Multiplicative with a(p^e) = p^e + 1 if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) < A034448(n) if and only if n is in A276079.
a(n) = A034448(n) if and only if n is in A276078.
a(n) = A377517(n) if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(k+2) + p(k)^(k+1) + p(k)^k - p(k) - 1)/(p(k)^(k+1) * (p(k)+1)) = 1.03676..., where p(k) = prime(k).

A048103 Numbers not divisible by p^p for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

N. J. A. Sloane

If a(n) = Product p_i^e_i then p_i > e_i for all i.
Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012
Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

			6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

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

Complement: A100716.
Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).
Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).
Cf. A276086 (permutation of this sequence, see also A376411, A376413).
Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A328387, A328832, A359547, A370114, A371083, A373848, A377871, A377992.
Disjoint union of {1}, A327934 and A358215.
Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)

isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,1] <= f[i,2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020

A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024

from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]A048103_list = list(islice(A048103_gen(),30)) # Chai Wah Wu, Jan 05 2023

;; With Antti Karttunen's IntSeq-library.
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012

For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

More terms from James Sellers, Apr 22 2000

A109298 Primal codes of finite idempotent functions on positive integers.

Original entry on oeis.org

1, 2, 9, 18, 125, 250, 1125, 2250, 2401, 4802, 21609, 43218, 161051, 300125, 322102, 600250, 1449459, 2701125, 2898918, 4826809, 5402250, 9653618, 20131375, 40262750, 43441281, 86882562, 181182375, 362364750, 386683451, 410338673, 603351125, 773366902, 820677346

Offset: 1

Jon Awbrey, Jul 06 2005

nonn

Finite idempotent functions are identity maps on finite subsets, counting the empty function as the idempotent on the empty set.
From Gus Wiseman, Mar 09 2019: (Start)
Also numbers whose ordered prime signature is equal to the distinct prime indices in increasing order. A prime index of n is a number m such that prime(m) divides n. The ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base. The case where the prime indices are taken in decreasing order is A324571.
Also numbers divisible by prime(k) exactly k times for each prime index k. These are a kind of self-describing numbers (cf. A001462, A304679).
Also Heinz numbers of integer partitions where the multiplicity of m is m for all m in the support (counted by A033461). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of distinct elements of A062457. For example, 43218 = prime(1)^1 * prime(2)^2 * prime(4)^4.
(End)

			Writing (prime(i))^j as i:j, we have the following table of examples:
Primal Codes of Finite Idempotent Functions on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = ` ` 2:2
` ` `18 = 1:1 2:2
` ` 125 = ` ` ` ` 3:3
` ` 250 = 1:1 ` ` 3:3
` `1125 = ` ` 2:2 3:3
` `2250 = 1:1 2:2 3:3
` `2401 = ` ` ` ` ` ` 4:4
` `4802 = 1:1 ` ` ` ` 4:4
` 21609 = ` ` 2:2 ` ` 4:4
` 43218 = 1:1 2:2 ` ` 4:4
`161051 = ` ` ` ` ` ` ` ` 5:5
`300125 = ` ` ` ` 3:3 4:4
`322102 = 1:1 ` ` ` ` ` ` 5:5
`600250 = 1:1 ` ` 3:3 4:4
From _Gus Wiseman_, Mar 09 2019: (Start)
The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2) has prime signature {1,2} and the distinct prime indices are also {1,2}.
       1: {}
       2: {1}
       9: {2,2}
      18: {1,2,2}
     125: {3,3,3}
     250: {1,3,3,3}
    1125: {2,2,3,3,3}
    2250: {1,2,2,3,3,3}
    2401: {4,4,4,4}
    4802: {1,4,4,4,4}
   21609: {2,2,4,4,4,4}
   43218: {1,2,2,4,4,4,4}
  161051: {5,5,5,5,5}
  300125: {3,3,3,4,4,4,4}
  322102: {1,5,5,5,5,5}
  600250: {1,3,3,3,4,4,4,4}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Awbrey, Riffs and Rotes

Cf. A076954, A106177, A108352, A108371, A109297, A109301.
Cf. A001156, A033461, A056239, A062457, A112798, A118914, A124010 (ordered prime signature), A181819, A276078, A304679.
Cf. A324524, A324525, A324570, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Select[Range[10000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]==k]&]

is(n) = my(f = factor(n)); for(i = 1, #f~, if(prime(f[i, 2]) != f[i, 1], return(0))); 1 \\ David A. Corneth, Mar 09 2019

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/prime(n)^n) = 1.6807104966... - Amiram Eldar, Jan 03 2021

Offset set to 1, missing terms inserted and more terms added by Alois P. Heinz, Mar 08 2019

A276076 Factorial base exp-function: digits in factorial base representation of n become the exponents of successive prime factors whose product a(n) is.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450, 3675, 7350, 11025, 22050, 6125, 12250, 18375, 36750, 55125, 110250, 343

Offset: 0

Antti Karttunen, Aug 18 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the digit in one-based position k of the factorial base representation of n. See the examples.

			   n  A007623   polynomial     encoded as             a(n)
   -------------------------------------------------------
   0       0    0-polynomial   (empty product)        = 1
   1       1    1*x^0          prime(1)^1             = 2
   2      10    1*x^1          prime(2)^1             = 3
   3      11    1*x^1 + 1*x^0  prime(2) * prime(1)    = 6
   4      20    2*x^1          prime(2)^2             = 9
   5      21    2*x^1 + 1*x^0  prime(2)^2 * prime(1)  = 18
   6     100    1*x^2          prime(3)^1             = 5
   7     101    1*x^2 + 1*x^0  prime(3) * prime(1)    = 10
and:
  23     321  3*x^2 + 2*x + 1  prime(3)^3 * prime(2)^2 * prime(1)
                                      = 5^3 * 3^2 * 2 = 2250.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Indranil Ghosh, Python program for computing this sequence.
Index entries for sequences related to factorial base representation.

Cf. A000040, A007623, A084558, A099563, A257687, A276009.
Cf. A276075 (a left inverse).
Cf. A276078 (same terms in ascending order).
Cf. also A000142, A001221, A001222, A002110, A007489, A008836, A019565, A033312, A034968, A048675, A051903, A059590, A060130, A076954, A246359, A248663, A262725, A276073, A276074, A351576, A351577, A351950, A351951, A351952, A351954.
Cf. also A275733, A275734, A275735, A275725 for other such encodings of factorial base related polynomials, and A276086 for a primorial base analog.

a[n_] := Module[{k = n, m = 2, r, p = 2, q = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, q *= p^r; p = NextPrime[p]; m++]; q]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

a(0) = 1, for n >= 1, a(n) = A275733(n) * a(A276009(n)).
Or: for n >= 1, a(n) = a(A257687(n)) * A000040(A084558(n))^A099563(n).
Other identities.
For all n >= 0:
A276075(a(n)) = n.
A001221(a(n)) = A060130(n).
A001222(a(n)) = A034968(n).
A051903(a(n)) = A246359(n).
A048675(a(n)) = A276073(n).
A248663(a(n)) = A276074(n).
a(A007489(n)) = A002110(n).
a(A059590(n)) = A019565(n).
For all n >= 1:
a(A000142(n)) = A000040(n).
a(A033312(n)) = A076954(n-1).
From Antti Karttunen, Apr 18 2022: (Start)
a(n) = A276086(A351576(n)).
A276085(a(n)) = A351576(n)
A003557(a(n)) = A351577(n).
A003415(a(n)) = A351950(n).
A069359(a(n)) = A351951(n).
A083345(a(n)) = A342001(a(n)) = A351952(n).
A351945(a(n)) = A351954(n).
A181819(a(n)) = A275735(n).
(End)
lambda(a(n)) = A262725(n+1), where lambda is Liouville's function, A008836. - Antti Karttunen and Peter Munn, Aug 09 2024

Name changed by Antti Karttunen, Apr 18 2022

A117144 Partitions of n in which each part k occurs at least k times.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322

Offset: 0

Emeric Deutsch, Mar 06 2006

nonn

The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019

			a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(9) = 5 integer partitions:
  1  11  111  22    221    222     2221     2222      333
              1111  11111  2211    22111    22211     22221
                           111111  1111111  221111    222111
                                            11111111  2211111
                                                      111111111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.

Cf. A324518, A324520, A324525, A324572.

g:=product((1-x^k+x^(k^2))/(1-x^k),k=1..100): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..66);
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Mar 09 2019 *)
nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)

G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).

cp's OEIS Frontend

A377515 The largest divisor of n that is a term in A276078.

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A377516 The number of divisors of n that are terms in A276078.

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A377517 The sum of the divisors of n that are terms in A276078.

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A380085 The largest unitary divisor of n that is a term in A276078.

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A380086 The number of unitary divisors of n that are terms in A276078.

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A380087 The sum of the unitary divisors of n that are terms in A276078.

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A048103 Numbers not divisible by p^p for any prime p.

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