Ilya Gutkovskiy - cp's OEIS frontend

A385757 a(n) is the smallest number having n smaller numbers with the same number of prime factors (counted with multiplicity).

3, 5, 7, 11, 13, 17, 19, 23, 26, 31, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205

Offset: 1

Ilya Gutkovskiy, Jul 08 2025

nonn

a(n) is the least number k such that A335097(k) = n.

			The smallest number having 9 smaller numbers (4, 6, 9, 10, 14, 15, 21, 22 and 25) with the same number of prime factors (counted with multiplicity) is 26, so a(9) is 26.

Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001222, A081376, A338483, A335097.

a[n_]:=Module[{k = 1, m, cnt}, While[True,m = PrimeOmega[k];cnt = Sum[Boole[PrimeOmega[i] == m], {i, 2, k - 1}];If[cnt == n, Return[k]];k++ ]];Array[a,65] (* James C. McMahon, Jul 13 2025 *)

a(n) = my(k=2, m=bigomega(k)); while (sum(i=2, k-1, bigomega(i) == m) !=n, k++; m=bigomega(k)); k; \\ Michel Marcus, Jul 09 2025

A385752 a(n) = Sum_{k=0..n} Stirling1(n,k) * (n!/k!)^2.

Original entry on oeis.org

1, 1, -3, 46, -1967, 179351, -29861639, 8200834972, -3456505906559, 2118756407303197, -1811589861406160699, 2089746219541021377546, -3164800617505630505525903, 6151223064132377579849537011, -15052264342298428131766095419839, 45616620088948927404807879986431576, -168785206495071742797011703980958673919

Offset: 0

Ilya Gutkovskiy, Jul 08 2025

sign

Cf. A119391, A192554, A385750, A385751.

Table[Sum[StirlingS1[n, k] (n!/k!)^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Sum[Log[1 + x]^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

Sum_{n>=0} a(n) * x^n / n!^3 = Sum_{k>=0} log(1 + x)^k / k!^3.

A385750 a(n) = Sum_{k=0..n} Stirling2(n,k) * (n!/k!)^2.

Original entry on oeis.org

1, 1, 5, 64, 1681, 78651, 5891041, 653545390, 101785047169, 21431911982437, 5927319770834701, 2101574777340578156, 935265924020629176625, 512945332353359967175999, 341342159773993944429746793, 272012935493149854994361194426, 256689188247205271953044107166721, 284051735653584424779666013789038985

Offset: 0

Ilya Gutkovskiy, Jul 08 2025

nonn

Cf. A000110, A064618, A119392, A119400, A385751, A385752.

Table[Sum[StirlingS2[n, k] (n!/k!)^2, {k, 0, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Sum[(Exp[x] - 1)^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{k>=0} (x^k / k!^2) * Product_{j=1..k} 1 / (1 - j*x).

Sum_{n>=0} a(n) * x^n / n!^3 = Sum_{k>=0} (exp(x) - 1)^k / k!^3.

A385751 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (n!/k!)^2.

Original entry on oeis.org

1, 1, 5, 100, 5137, 539851, 101035441, 30669875230, 14117057058945, 9364637252286181, 8603755430968248301, 10603853731438585516856, 17077610933602804111318705, 35160631271792580418277658415, 90839446923946068488317221868825, 289828370988497912073923950177143826, 1126236403418687405801564385561640043521

Offset: 0

Ilya Gutkovskiy, Jul 08 2025

nonn

Cf. A119390, A320502, A385750, A385752.

Table[Sum[Abs[StirlingS1[n, k]] (n!/k!)^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Sum[(-Log[1 - x])^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

Sum_{n>=0} a(n) * x^n / n!^3 = Sum_{k>=0} (-log(1 - x))^k / k!^3.

a(n) ~ (sqrt(2*Pi/3) * exp(3*log(n)^(1/3) - 3*n) * n^(3*n + 1/2) / log(n)) * (1 - 2/(9*log(n)^(1/3)) + (gamma - 4/81)/log(n)^(2/3) - (40/2187 + 11*gamma/9)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 09 2025

A385571 Number of subsets of the first n twin primes (A001097) whose sum is a twin prime.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 22, 37, 64, 119, 227, 450, 884, 1673, 3019, 5113, 8877, 16119, 30250, 57282, 109178, 210547, 412261, 819165, 1641582, 3298602, 6600608, 13150469, 26176431, 52289653, 104981405, 210846036, 420699038, 828442946, 1610436120, 3102364760

Offset: 1

Ilya Gutkovskiy, Jul 03 2025

nonn

			a(5) = 8 subsets: {3}, {5}, {7}, {11}, {13}, {3, 5, 11}, {5, 11, 13} and {7, 11, 13}.

David Radcliffe, Table of n, a(n) for n = 1..501
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A001097, A071810.

a[n_] := Module[{t = {} , p = 2}, While[Length[t] < n, p = NextPrime[p]; If[PrimeQ[p - 2] || PrimeQ[p + 2], AppendTo[t, p]]]; Count[Total /@ Subsets[t], ?(PrimeQ[#] && (PrimeQ[# - 2] || PrimeQ[# + 2]) &)]]; Array[a, 20] (* _Amiram Eldar, Jul 03 2025 *)

a(23)-a(25) from Amiram Eldar, Jul 03 2025

a(26)-a(36) from David Radcliffe, Jul 04 2025

A385349 Product of odd proper divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 3, 1, 7, 15, 1, 1, 27, 1, 5, 21, 11, 1, 3, 5, 13, 27, 7, 1, 225, 1, 1, 33, 17, 35, 27, 1, 19, 39, 5, 1, 441, 1, 11, 2025, 23, 1, 3, 7, 125, 51, 13, 1, 729, 55, 7, 57, 29, 1, 225, 1, 31, 3969, 1, 65, 1089, 1, 17, 69, 1225, 1, 27, 1, 37, 5625

Offset: 1

Ilya Gutkovskiy, Jun 26 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.

Cf. A007955, A007956, A091570 (similar for sum), A136655, A385350 (fixed points).

a:= n-> mul(`if`(d::odd, d, 1), d=numtheory[divisors](n) minus {n}):
seq(a(n), n=1..75);  # Alois P. Heinz, Jun 27 2025

a[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Table[a[n], {n, 75}]

a(n) = my(m = n >> valuation(n,2), d = numdiv(m)); if(d % 2, sqrtint(m)^d, m^(d/2)) / if(m < n, 1, n); \\ Amiram Eldar, Jun 27 2025

from math import isqrt
from sympy import divisor_count
def A385349(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    k = isqrt(m)**d if d&1 else m**(d>>1)
    return k//n if n&1 else k # Chai Wah Wu, Jun 27 2025

a(n) = Product_{d|n, d < n, d odd} d.

A385350 Numbers j such that the product of odd proper divisors of j is j.

Original entry on oeis.org

1, 15, 21, 27, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299

Offset: 1

Ilya Gutkovskiy, Jun 26 2025

Fixed points of A385349.
Odd terms in A007422.
Also 1 with odd numbers with exactly 4 divisors. - David A. Corneth, Jun 26 2025

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A007422, A030513, A073582, A385349.

q:= n-> n=1 or n::odd and numtheory[tau](n)=4:
select(q, [$1..500])[];  # Alois P. Heinz, Jun 26 2025

A385349[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Select[Range[300], A385349[#] == # &]

isok(k) = vecprod(select((x->((x%2)==1) && (xMichel Marcus, Jun 26 2025

is(n) = (n == 1) || (bitand(n, 1) && numdiv(n) == 4) \\ David A. Corneth, Jun 26 2025

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A385350(n):
    def f(x): return int(n-1+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jun 27 2025

A384812 If n = Product prime(i)^e(i) then a(n) = Sum prime(e(i)).

Original entry on oeis.org

0, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 5, 2, 4, 4, 7, 2, 5, 2, 5, 4, 4, 2, 7, 3, 4, 5, 5, 2, 6, 2, 11, 4, 4, 4, 6, 2, 4, 4, 7, 2, 6, 2, 5, 5, 4, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 4, 2, 7, 2, 4, 5, 13, 4, 6, 2, 5, 4, 6, 2, 8, 2, 4, 5, 5, 4, 6, 2, 9, 7, 4, 2, 7, 4, 4, 4, 7, 2, 7

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

nonn

Cf. A001222, A001414, A008472, A066328, A181819, A366988.

f:= proc(n) local t;
  add(ithprime(t[2]),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jul 25 2025

Table[Plus @@ (Prime[#[[2]]] & /@ FactorInteger[n]), {n, 1, 90}]

a(n) = my(f=factor(n)); sum(k=1, #f~, prime(f[k,2])); \\ Michel Marcus, Jun 10 2025

A384815 Sum of the cubes of the exponents in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 8, 1, 2, 1, 27, 8, 2, 1, 9, 1, 2, 2, 64, 1, 9, 1, 9, 2, 2, 1, 28, 8, 2, 27, 9, 1, 3, 1, 125, 2, 2, 2, 16, 1, 2, 2, 28, 1, 3, 1, 9, 9, 2, 1, 65, 8, 9, 2, 9, 1, 28, 2, 28, 2, 2, 1, 10, 1, 2, 9, 216, 2, 3, 1, 9, 2, 3, 1, 35, 1, 2, 9, 9, 2, 3, 1, 65, 64, 2, 1, 10, 2, 2, 2, 28, 1, 10

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. L. Duncan, A class of additive arithmetical functions, The American Mathematical Monthly, Vol. 69, No. 1 (1962), pp. 34-36.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001222, A001620, A005064, A090885, A360970.

Join[{0}, Table[Plus @@ (#[[2]]^3 & /@ FactorInteger[n]), {n, 2, 90}]]

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]^3); \\ Michel Marcus, Jun 10 2025

If n = Product (p_j^k_j) then a(n) = Sum (k_j^3).
From Amiram Eldar, Jul 03 2025: (Start)
Additive with a(p^e) = e^3.
Sum_{k=1..n} a(k) ~ n * log(log(n)) + B_3 * n + O(n/log(n)), where B_3 = gamma + Sum_{p prime} ((1-1/p)*Sum_{m>=1} m^3/p^m + log(1-1/p)) = 16.17021843694072992072..., and gamma is Euler's constant (A001620) (Duncan, 1962). (End)

A384816 Sum of the cubes of the indices of distinct prime factors of n.

Original entry on oeis.org

0, 1, 8, 1, 27, 9, 64, 1, 8, 28, 125, 9, 216, 65, 35, 1, 343, 9, 512, 28, 72, 126, 729, 9, 27, 217, 8, 65, 1000, 36, 1331, 1, 133, 344, 91, 9, 1728, 513, 224, 28, 2197, 73, 2744, 126, 35, 730, 3375, 9, 64, 28, 351, 217, 4096, 9, 152, 65, 520, 1001, 4913, 36, 5832, 1332, 72, 1, 243, 134, 6859, 344, 737, 92

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

nonn

Index entries for sequences computed from indices in prime factorization.

Cf. A000720, A005064, A066328, A332385.

Table[Plus @@ (PrimePi[#[[1]]]^3 & /@ FactorInteger[n]), {n, 70}]
nmax = 70; CoefficientList[Series[Sum[k^3 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = my(f=factor(n)[,1]); sum(k=1, #f~, primepi(f[k])^3); \\ Michel Marcus, Jun 10 2025

If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^3), where pi = A000720.

G.f.: Sum_{k>=1} k^3 * x^prime(k) / (1 - x^prime(k)).

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User: Ilya Gutkovskiy

A385757 a(n) is the smallest number having n smaller numbers with the same number of prime factors (counted with multiplicity).

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A385752 a(n) = Sum_{k=0..n} Stirling1(n,k) * (n!/k!)^2.

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A385750 a(n) = Sum_{k=0..n} Stirling2(n,k) * (n!/k!)^2.

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A385751 a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (n!/k!)^2.

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A385571 Number of subsets of the first n twin primes (A001097) whose sum is a twin prime.

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A385349 Product of odd proper divisors of n.

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A385350 Numbers j such that the product of odd proper divisors of j is j.

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A384812 If n = Product prime(i)^e(i) then a(n) = Sum prime(e(i)).

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