A062503 -id:A062503 - cp's OEIS frontend

A328401 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(i) = A328400(j) for all i, j.

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 5, 2, 2, 2, 6, 2, 5, 2, 5, 2, 2, 2, 7, 3, 2, 4, 5, 2, 2, 2, 8, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 5, 5, 2, 2, 9, 3, 5, 2, 5, 2, 7, 2, 7, 2, 2, 2, 5, 2, 2, 5, 10, 2, 2, 2, 5, 2, 2, 2, 11, 2, 2, 5, 5, 2, 2, 2, 9, 6, 2, 2, 5, 2, 2, 2, 7, 2, 5, 2, 5, 2, 2, 2, 12, 2, 5, 5, 3, 2, 2, 2, 7, 2

Offset: 1

Antti Karttunen, Oct 17 2019

nonn

Restricted growth sequence transform of A328400(n), or equally, of A007947(A181819(n)).
For all i, j:
  A101296(i) = A101296(j) => a(i) = a(j),
  a(i) = a(j) => A051903(i) = A051903(j) => A008966(i) = A008966(j),
  a(i) = a(j) => A051904(i) = A051904(j),
  a(i) = a(j) => A052409(i) = A052409(j),
  a(i) = a(j) => A072411(i) = A072411(j),
  a(i) = a(j) => A071625(i) = A071625(j),
  a(i) = a(j) => A267115(i) = A267115(j),
  a(i) = a(j) => A267116(i) = A267116(j).

			Numbers 2 (= 2^1), 3 (= 3^1), 6 = (2^1 * 3^1) and 30 (2^1 * 3^1 * 5^1) all have just one distinct exponent, 1, in the multisets of exponents that occur in their prime factorization, thus they all have the same value a(2) = a(3) = a(6) = a(30) = 2 in this sequence.
Number 4 (2^2), 9 (3^2) and 36 (2^2 * 3^2) all have just one distinct exponent, 2, in the multisets of exponents that occur in their prime factorization, thus they all have the same value a(4) = a(9) = a(36) = 3 in this sequence.
Numbers 12 = 2^2 * 3^1, 18 = 2^1 * 3^2, 60 = 2^2 * 3^1 * 5^1 and 300 = 2^2 * 3^1 * 5^2 all have both 1 and 2 and none other values occurring in the multisets of exponents in their prime factorization, thus they all have the value of a(12) = 5 that was allotted to 12 by the restricted growth sequence transform, as 12 is the smallest number with prime signature (1, 2).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A007947, A181819, A328400.
Cf. A005117 (gives indices of terms <= 2), A062503 (after its initial 1, gives indices of 3's).
Cf. A008966, A051903, A051904, A052409, A072411, A071625, A267115, A267116.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
v328401 = rgs_transform(vector(up_to, n, A007947(A181819(n)))); \\ Faster than with A328400(n).
A328401(n) = v328401[n];

A351009 Numbers k such that the k-th composition in standard order is a concatenation of distinct twins (x,x).

Original entry on oeis.org

0, 3, 10, 36, 43, 58, 136, 147, 228, 528, 547, 586, 676, 904, 2080, 2115, 2186, 2347, 2362, 2696, 2707, 2788, 3600, 3658, 3748, 8256, 8323, 8458, 8740, 8747, 8762, 9352, 10768, 10787, 11144, 14368, 14474, 14984, 32896, 33027, 33290, 33828, 33835, 33850, 34963

Offset: 1

Gus Wiseman, Feb 03 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and standard compositions begin:
    0:           0  ()
    3:          11  (1,1)
   10:        1010  (2,2)
   36:      100100  (3,3)
   43:      101011  (2,2,1,1)
   58:      111010  (1,1,2,2)
  136:    10001000  (4,4)
  147:    10010011  (3,3,1,1)
  228:    11100100  (1,1,3,3)
  528:  1000010000  (5,5)
  547:  1000100011  (4,4,1,1)
  586:  1001001010  (3,3,2,2)
  676:  1010100100  (2,2,3,3)
  904:  1110001000  (1,1,4,4)

The case of twins (binary weight 2) is A000120.
All terms are evil numbers A001969.
The version for Heinz numbers of partitions is A062503, counted by A035457.
These compositions are counted by A032020 interspersed with 0's.
Taking singles instead of twins gives A349051.
This is the strict (distinct twins) version of A351010 and A351011.
A011782 counts compositions.
A085207 represents concatenation using standard compositions.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351014 counts distinct runs in standard compositions, see A351015.
Cf. A003242, A027383, A035363, A088218, A106356, A122134, A238279, A344604, A349054, A351005, A351007.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]], 1],0]]//Reverse;
Select[Range[0,1000], UnsameQ@@Split[stc[#]]&&And@@(#==2&)/@Length/@Split[stc[#]]&]

A353025 Terms of A352991 which are perfect powers.

Original entry on oeis.org

1, 13527684, 34857216, 65318724, 73256481, 81432576, 139854276, 152843769, 157326849, 215384976, 245893761, 254817369, 326597184, 361874529, 375468129, 382945761, 385297641, 412739856, 523814769, 529874361, 537219684, 549386721, 587432169, 589324176, 597362481, 615387249, 627953481, 653927184

Offset: 1

Marco Ripà, Apr 17 2022

It appears that all terms are terms of A062503.
We note that a(n)=A352329(n) up to a(36)=A352329(36)=923187456, while the mentioned match does not hold starting from a(37)=14102987536 (since A352329(37)=1234608769).
There are no perfect powers among terms t which are permutations of 123_...(m - 1)_m for m == {2, 3, 5, 6} (mod 9). This is since 10 == 1 (mod 9) and also (1 + 0) == 1 (mod 9), so digit position has no effect. Hence, t == A134804(m) (mod 9). Now, if m is such that A134804(m) = {3, 6}, there is a lone factor of 3, which is not a perfect power (indeed).
Therefore, all terms are necessarily congruent modulo 9 to 0 or 1 (see Marco Ripà link).
All terms up to 10^34 are squares (in particular, there are 67 squares with no more than 17 digits). - Aldo Roberto Pessolano, May 12 2022

			75910168324 is a term since 75910168324 = 275518^2.

Aldo Roberto Pessolano, Table of n, a(n) for n = 1..71
Daniel J. Bernstein, Detecting Perfect Powers in Essentially Linear Time, Mathematics of Computation. Vol. 67, 233, 1253-1283 (1998).
Marco Ripà, On some conjectures concerning perfect powers, ResearchGate (2022).

Cf. A001292, A058183, A062503, A134804, A181129, A352329, A352991.

z = 1; Do[r = Range[k];
n = ToExpression[StringJoin[ToString[#] & /@ r]];
If[And[Mod[n, 9] != 3, Mod[n, 9] != 6], d = DigitCount[n];
  s = IntegerPart[Sqrt[10^(IntegerLength[n] - 1)]];
  f = IntegerPart[Sqrt[10^(IntegerLength[n])]];
  Do[y = x^2;
   If[DigitCount[y] == d, c = True;
    Do[If[Not[StringContainsQ[ToString[y], ToString[i]]],
      c = False], {i, 10, k}]; If[c, Print[z, " ", y]; z++]], {x, s,
    f}]], {k, 1, 10}] (* Aldo Roberto Pessolano, May 12 2022 *)

Digit sum of a(n) is always congruent to 0 or 1 modulo 9.

a(n) = m^2, where the integer m := m(n) is not a perfect power itself (conjectured).

A368886 The largest unitary divisor of n without an exponent 2 in its prime factorization (A337050).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 16, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 48, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 64, 65, 66, 67, 17, 69, 70

Offset: 1

Amiram Eldar, Jan 09 2024

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

Cf. A062503, A337050, A368884.

f[p_, e_] := If[e == 2, 1, p^e]; 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] == 2, 1, f[i,1]^f[i,2]));}

from math import prod
from sympy import factorint
def A368886(n): return prod(p**e for p, e in factorint(n).items() if e!=2) # Chai Wah Wu, Jan 09 2024

Multiplicative with a(p^2) = 1, and a(p^e) = p^e if e != 2.
a(n) = n / A368884(n).
a(n) >= 1, with equality if and only if n is in A062503.
a(n) <= n, with equality if and only if n is in A337050.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s) + 1/p^(3*s-3) - 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 + 1/p^4 - 1/p^5) = 0.78357388280736936739... .

A369307 The number of exponentially odd divisors d of n such that n/d is also exponentially odd.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 19 2024

First differs from A366308 at n = 32.

Dirichlet convolution of A295316 with itself.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A005117, A062503, A268335, A295316, A322483, A365173, A366308, A369308.

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

a(n) = vecprod(apply(x -> if(x%2, 2, x/2), factor(n)[,2]));

for(n=1, 100, print1(direuler(p=2, n, (1 - X^2 + X)^2/(1 - X^2)^2)[n], ", ")) \\ Vaclav Kotesovec, Jan 19 2024

from math import prod
from sympy import factorint
def A369307(n): return prod(2 if e&1 else e>>1 for e in factorint(n).values()) # Chai Wah Wu, Jan 19 2024

Multiplicative with a(p^e) = 2 is e is odd, and e/2 if e is even.
a(n) >= 1, with equality if and only if n is the square of a squarefree number (A062503).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s)^2 * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s)))^2.
From Vaclav Kotesovec, Jan 19 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (p^(2*s) + p^s - 1)^2 / ((p^s + 1)^2 * p^(2*s)).
Let f(s) = Product_{p prime} (p^(2*s) + p^s - 1)^2 / ((p^s + 1)^2 * p^(2*s)).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (2*p^2 + 2*p - 1) / (p^2*(p+1)^2)) = 0.49623881454854881762168565097162197963340069996226074849602334089041678...,
f'(1) = f(1) * Sum_{p prime} 2*(2*p + 1) * log(p) / ((p+1)*(p^2 + p - 1)) = f(1) * 1.49674466685934940187617305887881799198585080518913793200171026177150513...
and gamma is the Euler-Mascheroni constant A001620. (End)

A370239 The sum of divisors of n that are squares of squarefree numbers.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 5, 10, 1, 1

Offset: 1

Amiram Eldar, Feb 13 2024

The number of these divisors is A323308(n).

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

Cf. A005117, A036785, A048250, A062503, A071327, A078434, A323308, A370240.

f[p_, e_] := If[e == 1, 1, 1 + p^2]; 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] == 1, 1, 1 + f[i,1]^2));}

Multiplicative with a(p) = 1 and a(p^e) = 1 + p^2 for e >= 2.
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) = A071327(n) + 1 if and only if n is not in A036785.
Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(4*s-4).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 2*zeta(3/2)/Pi^2 = 0.5293779248... .

A374458 Squares of exponentially odd numbers (A268335).

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 64, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 576, 676, 729, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489

Offset: 1

Amiram Eldar, Jul 09 2024

Numbers whose exponents in their prime factorization are all congruent to 2 (mod 4).

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

Cf. A000290, A016825, A013662, A268335.

Similar sequences: A030140, A062503, A085986, A153158, A374291.

Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], OddQ] &]^2

is(k) = issquare(k) && if(k == 1, 1, my(e = factor(k)[, 2]); for(i = 1, #e, if(e[i] % 4 != 2, return(0))); 1);

a(n) = A000290(A268335(n)) = A268335(n)^2.
Sum_{n>=1} 1/a(n) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.54211628314015874165... .
Sum_{n>=1} 1/a(n)^s = zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)), for s > 1/2.

A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244, 245

Offset: 1

Amiram Eldar, Jul 28 2024

Subsequence of A304365 and differs from it by not having the terms 1, 144, 216, 324, 400, ... .
Subsequence of A038109 and differs from it by not having the terms 144, 324, 400, 576, 720, ... .
Numbers whose largest unitary divisor that is a square (A350388) is a square of squarefree number (A062503) that is larger than 1.
Each term is a product of two coprime numbers: an exponentially odd number (A268335) and a square of a squarefree number (A062503) that is larger than 1.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^3*(p+1))) - Product_{p prime}(1 - 1/(p*(p+1))) = A065466 - A065463 = 0.2432910611445097832029... .

			4 = 2^2 is a term because it has the exponent 2 in its prime factorization, and no higher even exponent.
144 = 2^4 * 3^2 is not a term because it has the exponent 4 in its prime factorization which is even and larger than 2.

Subsequence of A013929, A038109 and A304365.
A062503 \ {1} is a subsequence.
Cf. A065463, A065466, A268335, A335275, A350388, A375033.

q[n_] := Max[Select[FactorInteger[n][[;; , 2]], EvenQ]] == 2; Select[Range[250], q]

is(k) = {my(e = select(x -> !(x % 2), factor(k)[,2])); #e > 0 && vecmax(e) == 2;}

A375033(a(n)) = 2.

A376172 Numbers whose prime factorization has an even minimum exponent.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 72, 81, 100, 108, 121, 144, 169, 196, 200, 225, 256, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 625, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1369, 1372, 1444

Offset: 1

Amiram Eldar, Sep 13 2024

Numbers k such that A051904(k) is even.

The minimum exponent in the prime factorization of 1 is considered to be A051904(1) = 0, and therefore 1 is a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Subsequence of A001694.
Complement of A376173 within A001694.
Subsequences: A001248, A062503, A325240.
Cf. A051904.

seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # == 1 || EvenQ[Min[FactorInteger[#][[;; , 2]]]] &]; seq[2000]

is(k) = {my(f = factor(k), e = f[,2]); !(#e) || (ispowerful(f) && !(vecmin(e) % 2));}

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (-1)^(k+1) * s(k) = 1.70559662202357112914..., where s(k) = Product_{p prime} (1 + 1/(p^k*(p-1))).

A359580 Numbers that are either (an odd squarefree number squared) or twice such a number.

Original entry on oeis.org

1, 2, 9, 18, 25, 49, 50, 98, 121, 169, 225, 242, 289, 338, 361, 441, 450, 529, 578, 722, 841, 882, 961, 1058, 1089, 1225, 1369, 1521, 1681, 1682, 1849, 1922, 2178, 2209, 2450, 2601, 2738, 2809, 3025, 3042, 3249, 3362, 3481, 3698, 3721, 4225, 4418, 4489, 4761, 5041, 5202, 5329, 5618, 5929, 6050, 6241

Offset: 1

Antti Karttunen, Jan 07 2023

nonn

Numbers in whose prime factorization the exponent of 2 can be only 0 or 1, and the exponent of any odd prime can be only 0 or 2.

Cf. A359549 (characteristic function).
Positions of odd terms in A046692, A327276, A327278, A359548.
Cf. also A056911, A062503.

Select[Range[6000], (e = IntegerExponent[#, 2]) < 2 && SquareFreeQ[Sqrt[#/2^e]] &] (* Amiram Eldar, Jan 07 2023 *)
{#,2#}&/@(Select[Range[1,101,2],SquareFreeQ]^2)//Flatten//Sort (* Harvey P. Dale, May 27 2025 *)

PARI
```
isA359580(n) = A359549(n);
```

from itertools import count, islice
from sympy import factorint
def A359580_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n & n-1).bit_length())<=1 and all(e==2 for e in factorint(n>>m).values()),count(max(startvalue,1)))
A359580_list = list(islice(A359580_gen(),20)) # Chai Wah Wu, Jan 11 2023

Sum_{n>=1} 1/a(n) = 18/Pi^2. - Amiram Eldar, Jan 07 2023

cp's OEIS Frontend

A328401 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(i) = A328400(j) for all i, j.

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A351009 Numbers k such that the k-th composition in standard order is a concatenation of distinct twins (x,x).

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A353025 Terms of A352991 which are perfect powers.

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A368886 The largest unitary divisor of n without an exponent 2 in its prime factorization (A337050).

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A369307 The number of exponentially odd divisors d of n such that n/d is also exponentially odd.

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A370239 The sum of divisors of n that are squares of squarefree numbers.

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A374458 Squares of exponentially odd numbers (A268335).

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A375031 Numbers whose prime factorization has at least one exponent that equals 2 and no higher even exponent.

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