A024619 -id:A024619 - cp's OEIS frontend

A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

12, 20, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 92, 99, 104, 116, 117, 124, 132, 136, 140, 148, 152, 153, 156, 164, 171, 172, 175, 176, 184, 188, 204, 207, 208, 212, 220, 228, 232, 236, 244, 248, 260, 261, 268, 272, 275, 276, 279, 280, 284, 292, 296, 297, 304, 308, 315, 316, 325, 328, 332, 333

Offset: 1

Michael De Vlieger, Feb 28 2023

Proper subsequence of A126706.

Numbers k such that there does not exist j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 12, since 12/6 < 3.
a(2) = 20, since 20/10 < 5.
a(3) = 28, since 28/14 < 7.
a(4) = 40, since 40/10 < 5, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A013929, A024619, A119288, A126706, A355432, A360768.

Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 < #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
f(n) = if (isprimepower(n) || (n==1), 1, my(f=factor(n)[, 1]); f[2]); \\ A119288
isok(k) = !isprimepower(k) && !issquarefree(k) && (k/rad(k) < f(k)); \\ Michel Marcus, Mar 01 2023

This sequence is { k in A126706 : k/A007947(k) < A119288(k) } = A126706 \ A360768.

A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 0, 3, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 1, 0, 0, 0, 0, 5, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 6, 0, 2, 0, 2, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

Note 1 is a power of a prime (A000961) but not a prime-power (A246655).

Positions of 1 are A006549.
Positions of 0 are A080765 = A024619 - 1, complement A181062 = A000961 - 1.
Positions of 2 are A120432 (except initial terms).
Sorted positions of first appearances appear to include A167236 - 1.
Positions of terms > 1 are A373677.
The restriction to primes minus 1 is A377289.
Below, A (B) indicates that A is the first-differences of B:
- This sequence is A377782 (A031218), which has restriction to primes A065514 (A377781).
- The opposite is A377780 (A000015), restriction A377703 (A345531).
- For nonsquarefree we have A378036 (A378033), opposite A378039 (A120327).
- For squarefree we have A378085 (A112925), restriction A378038 (A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
A378034 gives differences of A378032 (restriction of A378033).
Prime-powers between primes: A053607, A080101, A366833, A377057, A377286, A377287.
Cf. A053289, A343249, A375706, A377281, A377282.

Differences[Table[NestWhile[#-1&,n,#>1&&!PrimePowerQ[#]&],{n,100}]]

A378367 Greatest non prime power <= n, allowing 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 6, 6, 6, 10, 10, 12, 12, 14, 15, 15, 15, 18, 18, 20, 21, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 30, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 48, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 63, 65, 66, 66

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

			The greatest non prime power <= 7 is 6, so a(7) = 6.

Sequences obtained by subtracting each term from n are placed in parentheses below.
For prime we have A007917 (A064722).
For nonprime we have A179278 (A010051 almost).
For perfect power we have A081676 (A069584).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For non perfect power we have A378363.
The opposite is A378372, subtracting n A378371.
For prime power we have A031218 (A276781 - 1).
Subtracting from n gives (A378366).
A000015 gives the least prime power >= n (A378370).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n (A013632), weak version A007918 (A007920).
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A065514, A113646, A345531 (A377281), A377051, A377054, A377282, A377468 (A074984), A378358, A378457.
Cf. A356068.

Table[NestWhile[#-1&,n,PrimePowerQ[#]&],{n,100}]

a(n) = n - A378366(n).

a(n) = A361102(A356068(n)). - Ridouane Oudra, Aug 22 2025

A222085 Sum of the least divisors of n whose LCM is equal to n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 8, 12, 10, 14, 10, 9, 31, 18, 21, 20, 12, 11, 14, 24, 24, 31, 16, 40, 14, 30, 11, 32, 63, 15, 20, 13, 25, 38, 22, 17, 20, 42, 19, 44, 18, 18, 26, 48, 52, 57, 43, 21, 20, 54, 66, 17, 22, 23, 32, 60, 15, 62, 34, 20, 127, 19, 23, 68

Offset: 1

Paolo P. Lava, Feb 11 2013

nonn

			The divisors of 20 are 1, 2, 4, 5, 10, 20 while the least divisors of 20 whose LCM is equal to 20 are 1, 2, 4, 5. Then a(20) = 1+2+4+5 = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084.

with(numtheory);
A222085:=proc(q)
local a,b,c,j,n,v; print(1);
for n from 2 to q do a:=ifactors(n)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n)); v:=0;
  for j from 1 to b do v:=v+a[j]; if a[j]=c then break; fi; od; print(v);
od; end:
A222085(100000000);

s[n_] := Module[{sum=0, L=1}, Do[sum+=d; L = LCM[L, d]; If[L == n, Break[]], {d, Divisors[n]}]; sum]; Array[s, 67] (* Amiram Eldar, Nov 05 2019 *)

a(n)=my(s,L=1);fordiv(n,d,s+=d;L=lcm(L,d);if(L==n,return(s))) \\ Charles R Greathouse IV, Feb 14 2013

A268376 Numbers n for which A001222(n) > A267116(n).

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 24, 26, 30, 33, 34, 35, 36, 38, 39, 40, 42, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 66, 69, 70, 72, 74, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 100, 102, 104, 105, 106, 108, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141

Offset: 1

Antti Karttunen, Feb 03 2016

Numbers n such that in their prime factorization n = p_1^e_1 * ... * p_k^e_k, there is at least one pair of exponents e_i and e_j (i and j distinct), such that their base-2 representations have at least one shared digit-position in which both exponents have 1-bit.

			n = 6 = 2^1 * 3^1 is included as both exponents, 1 and 1 ("1" in binary) have both 1-bit in position 0 of their binary representations.
n = 24 = 2^3 * 3^1 is included as both exponents, 1 and 3 ("01" and "11" in binary) have both 1-bit in position 0 of their binary representations.
n = 36 = 2^2 * 3^2 is included as both exponents, 2 and 2 ("10" in binary) have both 1-bit in position 1 of their binary representations.
n = 60 = 2^2 * 3^1 * 5^1 is included as the exponents of 3 and 5, both of which are 1, have both 1-bit in position 1 of their binary representations.

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

Cf. A001222, A267116.
Indices of nonzeros in A268374.
Subsequence of A002808 and A024619.
Cf. A268375 (complement).
Cf. A260730 (subsequence).
Cf. also A267117.
Differs from A067582(n+1) for the first time at n=25, where a(n) = 60, a value which is missing from A067582.

Select[Range@ 144, PrimeOmega@ # > BitOr @@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, Feb 04 2016 *)

A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

Original entry on oeis.org

0, 1, 2, 6, 12, 0, 60, 420, 840, 0, 2520, 0, 27720, 0, 0, 360360, 720720, 0, 12252240, 0, 0, 0, 232792560, 0, 5354228880, 0, 26771144400, 0, 80313433200, 0, 2329089562800, 72201776446800, 0, 0, 0, 0, 144403552893600, 0, 0, 0, 5342931457063200, 0

Offset: 0

Gus Wiseman, Oct 16 2019

nonn

Wikipedia, Run (cards)

Positions of 0's are 0 followed by A024619 - 1.
The version that looks only at all divisors > 1 is A328448.
The longest run of divisors of n has length A055874.
The longest run of divisors of n greater than one has length A328457.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a (not necessarily longest) maximal run of length n is A181063.
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A199970, A328195.

tav=Table[Max@@Length/@Split[Divisors[n],#2==#1+1&],{n,10000}];
Table[If[FreeQ[tav,i],0,Position[tav,i][[1,1]]],{i,0,Max@@tav}]

a(n) = LCM(1,2,...,n) = A003418(n) if n + 1 is a prime power, otherwise a(n) = 0.

A372695 Cubefull numbers that are not prime powers.

Original entry on oeis.org

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5488, 5832, 6912, 7776, 8000, 9261, 10000, 10125, 10368, 10648, 10976, 11664, 13824, 15552, 16000, 16875, 17496, 17576, 19208, 20000, 20736, 21296, 21952, 23328, 25000

Offset: 1

Michael De Vlieger, May 14 2024

nonn

Numbers k such that rad(k)^3 | k and omega(k) > 1. In other words, numbers with at least 2 distinct prime factors whose prime power factors have exponents that exceed 2.
Proper subset of the following sequences: A001694, A036966, A126706, A286708.
Superset of A372841.
Smallest term k with omega(k) = m is k = A002110(m)^3 = A115964(m).

			Table of smallest 12 terms and instances of omega(a(n)) = m for m = 2..4
    n      a(n)
  ------------------------
    1      216 = 2^3 * 3^3
    2      432 = 2^4 * 3^3
    3      648 = 2^3 * 3^4
    4      864 = 2^5 * 3^3
    5     1000 = 2^3 * 5^3
    6     1296 = 2^4 * 3^4
    7     1728 = 2^6 * 3^3
    8     1944 = 2^3 * 3^5
    9     2000 = 2^4 * 5^3
   10     2592 = 2^5 * 3^4
   11     2744 = 2^3 * 7^3
   12     3375 = 3^3 * 5^3
  ...
   43    27000 = 2^3 * 3^3 * 5^3
  ...
  587  9261000 = 2^3 * 3^3 * 5^3 * 7^3

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A007947, A024619, A036966, A115964, A126706, A286708, A372841.

Cf. A065483, A152441.

nn = 25000; Rest@ Select[Union@ Flatten@ Table[a^5 * b^4 * c^3, {c, Surd[nn, 3]}, {b, Surd[nn/(c^3), 4]}, {a, Surd[nn/(b^4 * c^3), 5]}], Not@*PrimePowerQ]

from math import gcd
from sympy import primepi, integer_nthroot, factorint
def A372695(n):
    def f(x):
        c = n+1+x+sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length()))
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c -= integer_nthroot(z//y**4,3)[0]
        return c
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

Intersection of A036966 and A024619.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) - Sum_{p prime} 1/(p^2*(p-1)) - 1 = A065483 - A152441 - 1 = 0.0188749045... . - Amiram Eldar, May 17 2024

A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Sep 28 2024

nonn

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of first appearances (A376268):
  1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, ...

These are the sorted positions of first appearances in A053289 (union A023055).
The complement is A376519.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A052410, A069623, A093555, A174965, A216765, A303707, A305630, A305631, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, 112, 114, 128, 136, 144, 145, 162, 180, 188, 198, 216, 226, 235, 246, 264, 265, 275, 285, 295, 305, 316, 317, 325, 328, 338, 350, 360, 367, 373, 385, 406, 416, 417, 419, 431, 443

Offset: 1

Gus Wiseman, Sep 28 2024

nonn

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of latter appearances (A376519):
  8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, ...

These are the sorted positions of latter appearances in A053289 (union A023055).
The complement is A376268.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A046933, A052410, A069623, A174965, A216765, A303707, A305630, A305631, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],MemberQ[Take[q,#-1],q[[#]]]&]

A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 15, 3, 6, 4, 10, 2, 20, 2, 6, 6, 6, 6, 15, 2, 6, 6, 15, 2, 20, 2, 10, 10, 6, 2, 21, 3, 10, 6, 10, 2, 15, 6, 15, 6, 6, 2, 35, 2, 6, 10, 7, 6, 20, 2, 10, 6, 20, 2, 21, 2, 6, 10, 10, 6, 20, 2

Offset: 1

Michael De Vlieger, Oct 09 2024

For prime power p^k, a(p^k) = A010846(p^k) = A000005(p^k) = k+1. Therefore, for prime p, a(p) = A010846(p) = A000005(p) = 2.

For n in A024619, a(n) != A010846(n) and A010846(n) > A000005(n).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A001222, A007947, A010846, A024619, A068795, A376248, A376846, A376847.

with(NumberTheory):
a := n -> binomial(Omega(n) + Omega(n, distinct), Omega(n, distinct)):
seq(a(n), n = 1..79);  # Peter Luschny, Oct 25 2024

Array[Binomial[#2 + #1, #1] & @@ {PrimeNu[#], PrimeOmega[#]} &, 120]

a(n) = length of row n of A376248.

a(n) = A010846(n) - A376846(n) + A376847(n).

cp's OEIS Frontend

A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

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A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

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A378367 Greatest non prime power <= n, allowing 1.

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A222085 Sum of the least divisors of n whose LCM is equal to n.

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A268376 Numbers n for which A001222(n) > A267116(n).

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A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

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A372695 Cubefull numbers that are not prime powers.

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A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

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