A164722 -id:A164722 - cp's OEIS frontend

A164788 Numbers such that the sum of the distinct prime factors is a cube.

1, 15, 45, 75, 135, 183, 225, 285, 295, 354, 357, 375, 405, 429, 510, 549, 583, 675, 708, 799, 855, 910, 943, 1020, 1055, 1062, 1071, 1125, 1215, 1266, 1287, 1416, 1425, 1454, 1475, 1527, 1530, 1634, 1647, 1820, 1875, 2025, 2040, 2124, 2499, 2532, 2550, 2565

Offset: 1

Jonathan Vos Post, Aug 26 2009

This is the 3rd row of the infinite array A(k,n) = n-th positive integer such that the sum of the distinct prime factors is of the form j^k for integers j, k. The 2nd row is A164722.

If k >= 1 and p = (2*k)^3 - 5 is prime (see A200957) then 5*p is a term. - Marius A. Burtea, Jun 30 2019

			a(2) = 15 because 15 = 3 * 5, the sum of distinct prime factors being 3+5 = 8 = 2^3. a(5) = 183 = 3 * 61 because 3 + 61 = 64 = 4^3. a(7) = 285 because 285 = 3 * 5 * 19 and 3 + 5 + 19 = 27 = 3^3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000578, A008472, A164722.

[n:n in [1..2600]| IsPower(&+PrimeDivisors(n), 3)]; // Marius A. Burtea, Jun 30 2019

Select[Range[3000],IntegerQ[Surd[Total[Transpose[FactorInteger[#]][[1]]],3]]&] (* Harvey P. Dale, Jun 21 2013 *)

{n such that A008472(n) = k^3 for k an integer}. {n such that A008472(n) is in A000578}.

More terms from Jon E. Schoenfield, May 27 2010

A227476 Numbers whose sum of semiprime divisors (A076290) is a positive square.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 138, 169, 225, 243, 256, 289, 306, 343, 361, 426, 512, 516, 529, 625, 644, 675, 729, 841, 918, 961, 975, 1002, 1024, 1032, 1125, 1140, 1146, 1150, 1220, 1230, 1288, 1305, 1316, 1331, 1369, 1681, 1849, 2025

Offset: 1

Michel Lagneau, Jul 13 2013

nonn

Except for the number 1, the sequence A195942 (Zeroless prime powers (excluding primes)) is a subsequence of this sequence because the set of divisors of the numbers of the form p^m with p prime and m >= 2 contains only one semiprime divisor, p^2.

The subset of the nonprime powers is {138, 225, 306, 426, 516, 644, 675, 918, ...}.

			138 is in the sequence because the divisors of 138 are {1, 2, 3, 6, 23, 46, 69, 138} and the sum of the semiprime divisors is 2*3 + 2*23 + 3*23 = 11^2.

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

Cf. A001358, A025475, A076290, A164722.

semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[2000], (s = semipSigma[#]) > 0 && IntegerQ @ Sqrt[s] &] (* Amiram Eldar, May 10 2020 *)

isok(n) = issquare(s = sumdiv(n, d, d*(bigomega(d)==2))) && (s>0); \\ Michel Marcus, Sep 16 2017

Definition corrected by Michel Marcus, Sep 16 2017

A228181 Numbers k such that sum of square of prime divisors of k equals sum of prime divisors of k+1.

Original entry on oeis.org

12, 27, 385, 1120, 4840, 9936, 14500, 29440, 95795, 105875, 178904, 223155, 341248, 343343, 754985, 830908, 1059630, 1841049, 2408832, 2949375, 3564704, 4934358, 5368792, 5500312, 6695000, 6805372, 8332831, 8846656, 10126336, 12956040, 13157235, 17254600

Offset: 1

Michel Lagneau, Aug 15 2013

nonn

Numbers k such that A005063(k) = A008472(k+1).

			The prime divisors of 9936 are {2, 3, 23} and the prime divisors of 9937 are {19, 523} => 2^2 + 3^2 + 23^2 = 19 + 523 = 542, hence 9936 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..333 (terms below 10^10)

Cf. A000961, A005063, A008472, A164722.

[k:k in [2..2500000]| &+PrimeDivisors(k+1) eq &+[PrimeDivisors(k)[i]^2: i in [1..#PrimeDivisors(k)]]]; // Marius A. Burtea, Feb 18 2020

fQ[n_] := Plus @@ (First@# & /@ FactorInteger[n]^2) == Plus @@ (First@# & /@ FactorInteger[n + 1]); Select[ Range@ 100000, fQ]

A165346 Numbers such that the sum of the distinct prime factors is a fourth power.

Original entry on oeis.org

1, 39, 55, 66, 117, 132, 158, 198, 264, 275, 316, 351, 396, 507, 528, 594, 605, 632, 726, 792, 1053, 1056, 1095, 1188, 1255, 1264, 1375, 1452, 1491, 1506, 1521, 1584, 1782, 2112, 2130, 2178, 2211, 2376, 2528, 2904, 3012, 3025, 3111, 3159, 3168, 3285, 3363

Offset: 1

Jonathan Vos Post, Sep 15 2009

nonn

This is the 4th row of the infinite array A(k,n) = n-th positive integer such that the sum of the distinct prime factors is of the form j^k for integers j, k. The 2nd row is A164722 (hence the current sequence is a proper subset of A164722). The 3rd row is A164788. The smallest integers whose sum of distinct prime factors is 4^4 are {1255, 1506, 3012, ...}. The smallest integers whose sum of distinct prime factors is 5^4 are {9255, 21455, ...}. The smallest integers whose sum of distinct prime factors is 6^4 are {6455, 7746, ...}. The smallest integers whose sum of distinct prime factors is 7^4 are {4798, 9596, ...}.

			a(2) = 39, because 39 = 3*13, and 3+13 = 16 = 2^4.
a(7) = 158, because 158 = 2*79, and 2+79 = 81 = 3^4.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A000583, A008472, A164722, A164788.

A008472 := proc(n) add( p, p = numtheory[factorset](n)) ; end: isA000583 := proc(n) iroot(n,4,'exct') ; exct ; end: A165346 := proc(n) if n = 1 then 1; else for a from procname(n-1)+1 do if isA000583(A008472(a)) then RETURN(a); fi; od: fi; end: seq(A165346(n),n=1..80) ; # R. J. Mathar, Sep 20 2009

a165346[n_] := Select[Range@n, IntegerQ[Power[Plus @@ Transpose[FactorInteger[#]][[1]], 1/4]] &]; a165346[3400] (* Michael De Vlieger, Jan 06 2015 *)

isok(n) = my(f=factor(n)); ispower(vecsum(f[,1]),4); \\ Michel Marcus, Jan 06 2015

{n such that A008472(n) = k^4 for k an integer}.

{n such that A008472(n) is in A000583}.

More terms from R. J. Mathar, Sep 20 2009

A194196 Numbers k such that the sum of the divisors of k and the sum of the distinct prime divisors of k are both a square.

Original entry on oeis.org

1, 66, 94, 1092, 1146, 1416, 1491, 1782, 2130, 2159, 2805, 3012, 3531, 4836, 8736, 9065, 9911, 12532, 13156, 15960, 16194, 24096, 25866, 27652, 29316, 29484, 30942, 34162, 34782, 34860, 37736, 37884, 38232, 38688, 40257, 41331, 48204, 51460, 54162, 54411

Offset: 1

Michel Lagneau, Aug 18 2011

nonn

			94 is in the sequence because the distinct prime divisors are {2,47} -> sum = 7^2, and the divisors are {1,2,47,94} -> sum = 12^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A000203, A006532, A008472, A164722.

isA006532 :=proc(n) issqr(numtheory[sigma](n)) ; end proc:
A008472 := proc(n) add(d, d=numtheory[factorset](n)) ; end proc:
isA164722 :=proc(n) issqr(A008472(n)) ; end proc:
for n from 1 to 50000 do if isA006532(n) and isA164722(n) then printf("%d,",n); end if; end do; # R. J. Mathar, Aug 18 2011

isok(k) = my(f=factor(k)); issquare(sigma(f)) && issquare(vecsum(f[,1])); \\ Michel Marcus, Dec 05 2020

{A006532 intersection A164722}.

A227478 Numbers k such that both the sum of the semiprime divisors of k and the sum of the prime divisors of k are squares.

Original entry on oeis.org

1146, 2874, 9870, 33220, 34353, 43140, 50694, 52290, 66440, 86280, 94350, 100804, 101097, 103059, 106140, 121540, 125070, 127897, 132880, 139908, 156870, 172560, 183475, 191140, 193410, 201608, 208692, 212280, 243080, 248378, 265760, 276094, 279816, 303291

Offset: 1

Michel Lagneau, Jul 13 2013

nonn

The sequence is infinite: if a number of the form p(1) * p(2) * ... * p(i)^2 * p(i+1) * ... * p(m) is in the sequence where p(1), ..., p(m) are primes, then the numbers p(1) * p(2) * ... * p(i)^q * p(i+1) * ... * p(m) are also in the sequence for q = 3, 4, ... For example, the infinite subsequence 33220, 66440, 132880, ... contains the numbers of the form 2^q * 5 * 11 * 151 for q = 2, 3, 4, ... where 2+5+11+151 = 169 = 13^2 and 2*2 + 2*5 + 2*11 + 2*151 + 5*11 + 5*151 + 11*151 = 2809 = 53^2.

In this sequence, the corresponding pairs of squares are (961, 196), (2401, 484), (900, 64), (2809, 169), (4900, 361), (7225, 729), (2304, 100), (1521, 100), (2809, 169), (7225, 729), (1225, 64), (3721, 121), (12100, 289), (4900, 361), (2704, 100), (7225, 169), (8100, 400), (2916, 169), (2809, 169), (12769, 225), (1521, 100), (7225, 729), (8464, 225), (13225, 529), (5329, 121), (3721, 121), (1369, 64), (2704, 100), (7225, 169), (13689, 289), (2809, 169), (3364, 100), (12769, 225), (12100, 289), ...

			1146 = 2*3*191 is in the sequence because the divisors are {1, 2, 3, 6, 191, 382, 573, 1146}, so the sum of the semiprime divisors is 6 + 382 + 573 = 961 = 31^2 and the sum of the prime divisors is 2 + 3 + 191 = 196 = 14^2.

Cf. A001358, A008472, A076920, A164722, A227476.

with(numtheory):for n from 2 to 310000 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if sqrt(s1)=floor(sqrt(s1)) and sqrt(s2)=floor(sqrt(s2)) then printf(`%d, `,n):else fi:od:

Rest@ Select[Range[3*10^5], AllTrue[{DivisorSum[#, # &, PrimeOmega@ # == 2 &], DivisorSum[#, # &, PrimeQ]}, IntegerQ@ Sqrt@ # &] &] (* Michael De Vlieger, Sep 15 2017 *)

A367949 Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors (sopf) of a(n) + a(n + 1) is a perfect square.

Original entry on oeis.org

1, 13, 15, 24, 4, 10, 18, 21, 7, 32, 14, 25, 3, 11, 17, 22, 6, 8, 20, 19, 9, 5, 23, 16, 12, 2, 26, 29, 27, 28, 38, 54, 40, 52, 42, 50, 44, 48, 46, 66, 51, 41, 53, 39, 55, 37, 57, 35, 31, 61, 33, 59, 58, 34, 60, 72, 45, 47, 65, 67, 88, 70, 62, 30, 36, 56, 76, 79, 104, 80

Offset: 1

Giorgos Kalogeropoulos and Eric Angelini, Dec 05 2023

nonn

			a(1) + a(2) =  1 + 13 = 14 whose sopf is  9, a perfect square.
a(2) + a(3) = 13 + 15 = 28 whose sopf is  9, a perfect square.
a(7) + a(8) = 18 + 21 = 39 whose sopf is 16, a perfect square.
a(8) + a(9) = 21 +  7 = 28 whose sopf is  9, a perfect square.

Éric Angelini, Sums of distinct prime factors, Personal blog, December 2023.

Cf. A008472, A164722.

a[1]=1;a[n_]:=a[n]=(k=1;While[MemberQ[ar=Array[a,n-1],k] ||!IntegerQ@Sqrt@Total[First/@FactorInteger[k+a[n-1]]],k++];k);Array[a, 70]

A380953 Numbers m such that the sum of its distinct prime factors and the sum of its nonprime divisors are both squares.

Original entry on oeis.org

1, 323, 3887, 5183, 149903, 311790, 777923, 1327103, 6718463, 12446783, 14605487, 16402499, 20373435, 28128270, 30856494, 33144430, 37058230, 37380745, 68661901, 86755609, 139557721, 159954570, 221294682, 222538813, 229159043, 269108440, 360590058, 412621345

Offset: 1

Michel Lagneau, Feb 09 2025

nonn

Or numbers m such that A008472(m) and (A000203(m) - A008472(m)) are both squares.

			s1 is the sum of the prime factors, s2 is the sum of the nonprime divisors.
+----------------------------+-------------------------+-----+-------+
|    m   |  prime factors    |   nonprimedivisors      |  s1 |  s2   |
+----------------------------+-------------------------+-----+-------+
|    323 | {17, 19}          | {1, 323}                | 6^2 |  18^2 |
+----------------------------+-------------------------+-----+-------+
|   3887 | {13, 23}          | {1, 169, 299, 3887}     | 6^2 |  66^2 |
+----------------------------+-------------------------+-----+-------+
|   5183 | {71, 73}          | {1, 5183}               |12^2 |  72^2 |
+----------------------------+-------------------------+-----+-------+
| 149903 | {13, 887}         | {1, 169, 11531, 149903} |30^2 | 402^2 |
+----------------------------+-------------------------+-----+-------+

Cf. A000203, A008472, A164722.

with(numtheory):nn:=10^8:print(1):
for m from 2 to nn do:
 d:=factorset(m):n0:=nops(d):s:=sum('d[i]', 'i'=1..n0):
    if issqr(s) and issqr(sigma(m)-s) then print(m):
     else
    fi:
 od:

isok(m) = my(f=factor(m), s=sum(k=1, #f~, f[k,1])); issquare(s) && issquare(sigma(f)-s); \\ Michel Marcus, Feb 09 2025

More terms from Jinyuan Wang, Feb 11 2025

cp's OEIS Frontend

A164788 Numbers such that the sum of the distinct prime factors is a cube.

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A227476 Numbers whose sum of semiprime divisors (A076290) is a positive square.

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A228181 Numbers k such that sum of square of prime divisors of k equals sum of prime divisors of k+1.

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A165346 Numbers such that the sum of the distinct prime factors is a fourth power.

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A194196 Numbers k such that the sum of the divisors of k and the sum of the distinct prime divisors of k are both a square.

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A227478 Numbers k such that both the sum of the semiprime divisors of k and the sum of the prime divisors of k are squares.

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A367949 Lexicographically earliest sequence of distinct positive integers such that the sum of the distinct prime factors (sopf) of a(n) + a(n + 1) is a perfect square.

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