A025475 -id:A025475 - cp's OEIS frontend

A134605 Composite numbers such that the square root of the sum of squares of their prime factors (with multiplicity) is an integer.

16, 48, 81, 320, 351, 486, 512, 625, 1080, 1260, 1350, 1375, 1792, 1836, 2070, 2145, 2175, 2401, 2730, 2772, 3072, 3150, 3510, 4104, 4305, 4625, 4650, 4655, 4998, 5880, 6000, 6174, 6545, 7098, 7128, 7182, 7650, 7791, 7889, 7956, 9030, 9108, 9295, 9324

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(2)=48 since 48=2*2*2*2*3 and sqrt(4*2^2+3^2)=sqrt(25)=5.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376, A134600, A134602, A134608, A134611, A134616, A134618, A134620.

srssQ[n_]:=IntegerQ[Sqrt[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]^2]]]; Select[Range[10000],CompositeQ[#]&&srssQ[#]&] (* Harvey P. Dale, Jan 22 2019 *)

is(n)=my(f=factor(n)); issquare(sum(i=1,#f~,f[i,1]^2*f[i,2])) && !isprime(n) && n>1 \\ Charles R Greathouse IV, Apr 29 2015

A134608 Composite numbers such that the cube root of the sum of cubes of their prime factors is an integer.

Original entry on oeis.org

256, 588, 693, 3840, 6561, 14157, 17787, 141960, 178360, 313600, 337365, 350000, 387072, 390625, 407442, 432000, 466560, 531674, 535815, 541310, 664909, 697851, 1044582, 1262056, 1264640, 1299272, 1374327, 1547570, 1608575, 1660360

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(3)=693, since 693=3*3*7*11 and (2*3^3+7^3+11^3)^(1/3)=1728^(1/3)=12.

Hieronymus Fischer, Table of n, a(n) for n = 1..300

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134611, A134616, A134618, A134620.

criQ[n_]:=IntegerQ[Surd[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]^3],3]]; Select[Range[1670000],CompositeQ[#] && criQ[#]&] (* Harvey P. Dale, Sep 19 2021 *)

lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); if (ispower(s, 3), print1(i, ", "));););} \\ Michel Marcus, Apr 14 2013

Minor edits by Hieronymus Fischer, Apr 20 2013

A134602 Composite numbers such that the square mean of their prime factors is a nonprime integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

Original entry on oeis.org

378, 455, 527, 918, 1265, 1615, 2047, 2145, 2175, 2345, 2665, 3713, 3835, 4207, 4305, 4633, 5000, 5117, 5382, 6061, 6678, 6887, 6965, 7055, 7267, 7327, 7497, 7685, 7791, 8470, 8785, 8918, 9641, 10205, 10545, 10647, 11137, 11543, 11713, 13482, 14079

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

Numbers included in A134600, but not in A134601. a(1)=378 is the minimal number with this property.

Also numbers included in A134603, but not in A134604.

			a(2)=455, since 455=5*7*13 and sqrt((5^2+7^2+13^2)/3)=sqrt(81)=9.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134604, A134608, A134611, A134617, A134619, A134621.

f[{a_,b_}]:=Table[a,b];Select[Range[2,14079],!PrimeQ[#]&&!PrimeQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&&IntegerQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 08 2025 *)

Definition clarified by Hieronymus Fischer, Apr 20 2013, Jun 01 2013

A036785 Numbers divisible by the squares of two distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 900, 936, 968, 972, 980, 1000, 1008, 1044

Offset: 1

Alford Arnold

Not squarefree, not a nontrivial prime power and not in {squarefree} times {nontrivial prime powers}.

Numbers k such that A056170(k) > 1. The asymptotic density of this sequence is 1 - (6/Pi^2) * (1 + A154945) = 0.05668359058... - Amiram Eldar, Nov 01 2020

CRC Standard Mathematical Tables and Formulae, 30th ed., (1996) page 102-105.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A005117, A025475, A056170, A154945.
Equivalent sequence for 3 distinct primes: A318720.
Cf. A085986, A338539, A339245 (subsequences).
Subsequence of A038838.

Select[Range@ 1050, And[Length@ # > 1, Total@ Boole@ Map[# > 1 &, #[[All, -1]]] > 1] &@ FactorInteger@ # &] (* Michael De Vlieger, Apr 25 2017 *)
dstdpQ[n_]:=Length[Select[Sqrt[#]&/@Divisors[n],PrimeQ]]>1; Select[ Range[ 1100],dstdpQ] (* Harvey P. Dale, Jan 15 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4));#f>1&&f[2]>1 \\ Charles R Greathouse IV, Nov 15 2012

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2000

New name from Charles R Greathouse IV, Nov 15 2012

A014673 Smallest prime factor of greatest proper divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 2, 1, 3, 1, 2, 7, 11, 1, 2, 5, 13, 3, 2, 1, 3, 1, 2, 11, 17, 7, 2, 1, 19, 13, 2, 1, 3, 1, 2, 3, 23, 1, 2, 7, 5, 17, 2, 1, 3, 11, 2, 19, 29, 1, 2, 1, 31, 3, 2, 13, 3, 1, 2, 23, 5, 1, 2, 1, 37, 5, 2, 11, 3, 1, 2, 3, 41, 1, 2, 17, 43, 29, 2, 1, 3, 13, 2, 31, 47

Offset: 1

Reinhard Zumkeller, Jun 24 2003

nonn

For n > 1: a(n) = 1 iff n is prime; a(A001358(n)) = A084127(n); a(A025475(n)) = A020639(A025475(n)). [corrected by Peter Munn, Feb 19 2017]
When n is composite, this is the 2nd factor when n is written as a product of primes in nondecreasing order. For example, 12 = 2*2*3, so a(12) = 2. - Peter Munn, Feb 19 2017
For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(2) = 7. - Peter Munn, Jul 12 2019

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

Cf. A001222, A001358, A020639, A025475, A027746, A032742, A054576, A084127, A085392, A085393, A115561, A117357, A117358, A281889.

PrimeFactors[ n_ ] := Flatten[ Table[ # [ [ 1 ] ], {1} ] & /@ FactorInteger[ n ] ]; f[ n_ ] := Block[ {gpd = Divisors[ n ][ [ -2 ] ]}, If[ gpd == 1, 1, PrimeFactors[ gpd ][ [ 1 ] ] ] ]; Table[ If[ n == 1, 1, f[ n ] ], {n, 1, 95} ]
(* Second program: *)
Table[If[Or[PrimeQ@ n, n == 1], 1, FactorInteger[n/SelectFirst[Prime@ Range@ PrimePi[Sqrt@ n], Divisible[n, #] &]][[1, 1]] ], {n, 94}] (* Michael De Vlieger, Aug 14 2017 *)

lpf(n)=if(n>1,factor(n)[1,1],1)
a(n)=lpf(n/lpf(n)) \\ Charles R Greathouse IV, May 09 2013

a(n)=if(n<4||isprime(n),return(1)); my(f=factor(n)); if(f[1,2]>1, f[1,1], f[2,1]) \\ Charles R Greathouse IV, May 09 2013

(define (A014673 n) (A020639 (/ n (A020639 n)))) ;; Code for A020639 given under that entry - Antti Karttunen, Aug 12 2017

a(n) = A020639(A032742(n)).
A117357(n) = A020639(A054576(n)); A117358(n) = A032742(A054576(n)) = A054576(n)/A117357(n). - Reinhard Zumkeller, Mar 10 2006
If A001222(n) >= 2, a(n) = A027746(n,2), otherwise a(n) = 1. - Peter Munn, Jul 13 2019

A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

Original entry on oeis.org

2, 0, 2, 3, 0, 2, 4, 3, 4, 8, 0, 1, 8, 14, 1, 7, 7, 4, 25, 2, 15, 15, 17, 16, 10, 45, 2, 44, 20, 26, 18, 0, 2, 28, 52, 36, 42, 32, 45, 45, 47, 19, 30, 106, 36, 35, 4, 114, 28, 135, 89, 42, 87, 42, 34, 66, 192, 106, 56, 23, 39, 37, 165, 49, 37, 262, 58, 160, 22

Offset: 1

Jon Perry, Mar 07 2002

Does this sequence have any terms appearing infinitely often? In particular, are {2, 5, 11, 32, 77} the only zeros? As an example, {121, 122, 123, 124, 125} is an interval containing no primes, corresponding to a(11) = 0. - Gus Wiseman, Dec 02 2024

			The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 667 terms from Lei Zhou)

For primes between nonsquarefree numbers we have A236575.
For composite instead of prime we have A378456.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A080101 counts prime powers between primes.
A246547 lists the non prime prime powers, differences A053707.
A246655 lists the prime powers not including 1, complement A361102.
Cf. A001597, A024619, A031218, A046933, A276781, A345531, A366833, A377051, A377057, A377282, A377286-A377288.

t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* T. D. Noe, May 21 2013 *)
nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] (* Michael De Vlieger, Oct 26 2023 *)

a(n) = A000720(A025475(n+3)) - A000720(A025475(n+2)). - David Wasserman, Dec 20 2002

More terms from David Wasserman, Dec 20 2002

Definition clarified by N. J. A. Sloane, Oct 27 2023

A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

Original entry on oeis.org

1, 4, 9, 12, 18, 24, 34, 47, 60, 79, 117, 178, 198, 206, 215, 244, 311, 402, 465, 614, 782, 1078, 1109, 1234, 1890, 1939, 1961, 2256, 2290, 3149, 3377, 3460, 3502, 3722, 3871, 4604, 4694, 6634, 8073, 8131, 8793, 12370, 12661, 14482, 14990, 15912, 17140, 19166

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    34: {1,7}
    47: {15}
    60: {1,1,2,3}
    79: {22}
   117: {2,2,6}
   178: {1,24}
   198: {1,2,2,5}
   206: {1,27}
   215: {3,14}
   244: {1,1,18}

For compression instead of sorted firsts we have A376308.
For run-lengths instead of sorted firsts we have A376309.
For run-sums instead of sorted firsts we have A376310.
The version for squarefree numbers is the unsorted version of A376311.
The unsorted version is A376341.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A078147, A110969, A120430, A174965, A373948, A375706.

q=Differences[Select[Range[100],PrimePowerQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, -5, 15, -34, 63, -97, 115, -54, -251, 1184, -3536, 8736, -18993, 37009, -64545, 98442, -121393, 82008, 147432, -860818, 2710023, -7110594, 17077281, -38873146, 85085287, -179965647, 367885014, -725051280, 1372311999, -2481473550, 4257624252

Offset: 0

Gus Wiseman, Oct 22 2024

sign

			The sixth differences of A000961 begin: -5, 10, -9, 1, 6, -10, 16, -18, ..., so a(6) = -5.

The version for primes is A007442, noncomposites A030016, composites A377036.
For squarefree numbers we have A377041, nonsquarefree A377049.
This is the first column of the array A377051.
For antidiagonal-sums we have A377052, absolute A377053.
For positions of first zeros we have A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A000961 lists the powers of primes, differences A057820.
A001597 lists perfect-powers, complement A007916.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A025475, A053707, A093555, A174965, A175804, A246655, A361102, A376340, A376596, A377056.

q=Select[Range[100],#==1||PrimePowerQ[#]&];
Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]/2}]

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*q(k)

A134621 Numbers such that the arithmetic mean of the 4th power of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

15, 28, 39, 48, 51, 65, 68, 76, 77, 85, 87, 93, 111, 119, 133, 141, 143, 148, 155, 161, 175, 187, 189, 209, 212, 215, 221, 225, 235, 244, 275, 287, 295, 301, 315, 316, 320, 323, 329, 355, 393, 403, 404, 411, 428, 437, 447, 451, 455, 481, 505, 508, 515, 517

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(3)=39, since 39=3*13 and (3^4+13^4)/2=14321 which is prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134600, A134602, A134605, A134608, A134613, A134617, A134619.

Minor edits by Hieronymus Fischer, May 06 2013

A303554 Union of the prime powers (p^k, p prime, k >= 0) and numbers that are the product of 2 or more distinct primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

nonn

			42 is in the sequence because 42 = 2*3*7 (3 distinct prime factors).
81 is in the sequence because 81 = 3^4 (4 prime factors, 1 distinct).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Complement of A126706.
Union of A005117 and A246547.
Union of A000469 and A246655.
Union of A000961 and A120944.
Cf. A025475.

Select[Range[110], PrimePowerQ[#] || SquareFreeQ[#] &]
Select[Range[110], PrimeNu[#] == 1 || PrimeNu[#] == PrimeOmega[#] &]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A303554(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 19 2024

cp's OEIS Frontend

A134605 Composite numbers such that the square root of the sum of squares of their prime factors (with multiplicity) is an integer.

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A134608 Composite numbers such that the cube root of the sum of cubes of their prime factors is an integer.

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A134602 Composite numbers such that the square mean of their prime factors is a nonprime integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

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A036785 Numbers divisible by the squares of two distinct primes.

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A014673 Smallest prime factor of greatest proper divisor of n.

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A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

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A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

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A377054 First term of the n-th differences of the powers of primes. Inverse zero-based binomial transform of A000961.

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