A048261 -id:A048261 - cp's OEIS frontend

A005063 Sum of squares of primes dividing n.

0, 4, 9, 4, 25, 13, 49, 4, 9, 29, 121, 13, 169, 53, 34, 4, 289, 13, 361, 29, 58, 125, 529, 13, 25, 173, 9, 53, 841, 38, 961, 4, 130, 293, 74, 13, 1369, 365, 178, 29, 1681, 62, 1849, 125, 34, 533, 2209, 13, 49, 29, 298, 173, 2809, 13, 146, 53, 370, 845, 3481, 38, 3721

Offset: 1

N. J. A. Sloane

nonn

The set of these terms apart from 0 is A048261. - Bernard Schott, Feb 10 2022

Inverse Möbius transform of n^2 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000290, A005066, A005071, A005075, A005079, A005083, A059841, A079978.
Cf. A067666, A081403, A048261. - Franklin T. Adams-Watters, May 03 2009
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), this sequence (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

A005063 := proc(n)
        add(d^2, d= numtheory[factorset](n)) ;
end proc;
seq(A005063(n),n=1..40) ; # R. J. Mathar, Nov 08 2011

a[n_] := Total[FactorInteger[n][[All, 1]]^2]; a[1]=0; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 20 2017 *)
Array[DivisorSum[#, #^2 &, PrimeQ] &, 61] (* Michael De Vlieger, Jul 11 2017 *)

a(n)=local(fm,t);fm=factor(n);t=0;for(k=1,matsize(fm)[1],t+=fm[k,1]^2);t \\ Franklin T. Adams-Watters, May 03 2009

a(n) = vecsum(apply(x->x^2, factor(n)[, 1])); \\ Michel Marcus, Sep 19 2020

from sympy import primefactors
def a(n): return sum(p**2 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005063 n) (if (= 1 n) 0 (+ (A000290 (A020639 n)) (A005063 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^2.
G.f.: Sum_{k>=1} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005066(n) + 4*A059841(n).
a(n) = A005079(n) + A005083(n) + 4*A059841(n).
a(n) = A005071(n) + A005075(n) + 9*A079978(n).
(End)
Dirichlet g.f.: primezeta(s-2)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p^2. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^2 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Franklin T. Adams-Watters, May 03 2009

A121518 Numbers that are not the sum of the squares of distinct primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86

Offset: 1

T. D. Noe, Aug 04 2006

There are 2438 terms, the largest of which is 17163.

See A048261.

T. D. Noe, Table of n, a(n) for n = 1..2438

s={0}; Do[p=Prime[n]; s=Union[s,s+p^2], {n,PrimePi[140]}]; s=Select[s,0<#

Complement of A048261.

A088910 Conjectured minimal required number k of terms in a representation n=sum_(i=1..k)e_i*(p_i)^2 by distinct primes p_i, where e_i is 1 or -1.

Original entry on oeis.org

4, 3, 4, 4, 1, 2, 5, 5, 4, 1, 4, 4, 3, 2, 4, 3, 2, 5, 5, 4, 3, 2, 4, 3, 2, 1, 4, 4, 3, 2, 3, 5, 4, 3, 2, 4, 3, 4, 3, 3, 2, 5, 5, 4, 3, 2, 4, 3, 2, 1, 4, 4, 3, 2, 3, 5, 4, 3, 2, 4, 5, 4, 3, 3, 4, 3, 5, 4, 3, 4, 3, 3, 2, 3, 2, 4, 3, 4, 3, 4, 4, 3, 4, 3, 5, 4, 4, 3, 4, 5, 5, 4, 3, 4, 4, 3, 2, 3, 4, 4, 3, 4, 5, 5, 4

Offset: 0

Hugo Pfoertner, Oct 24 2003

nonn

It is conjectured that all sequence terms are <=5. The terms with a(n)=5 were provided by W. Edwin Clark.

			The following are representation with the minimal number of terms:
  a(0)=4: 0=7^2-11^2-17^2+19^2;
  a(1)=3: 1=7^2+11^2-13^2;
  a(4)=1: 4=2^2;
  a(5)=2: 5=3^2-2^2;
  a(6)=5: 6=-(2^2)+3^2+7^2+11^2-13^2.

Robert E. Dressler, Louis Pigno, and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40.
Hugo Pfoertner, Conjectured minimal representations of n by squaresof distinct primes (Table for n<=400).

Cf. A088934 (maximum required prime in representation), A048261, A088908, A088909.

A185077 Numbers such that the largest prime factor equals the sum of the squares of the other prime factors.

Original entry on oeis.org

78, 156, 234, 290, 312, 468, 580, 624, 702, 742, 936, 1014, 1160, 1248, 1404, 1450, 1484, 1872, 2028, 2106, 2320, 2496, 2808, 2900, 2968, 3042, 3744, 4056, 4212, 4498, 4640, 4992, 5194, 5616, 5800, 5936, 6084, 6318, 7250, 7488, 8112, 8410, 8424, 8715, 8996, 9126, 9280, 9962

Offset: 1

Michel Lagneau, Feb 18 2011

nonn

Observation : it seems that the prime divisors of a majority of numbers n are of the form {2, p, q} with q = 2^2 + p^2, but there exists more rarely numbers with more prime divisors (examples : 8715 = 3*5*7*83; 153230 = 2*5*7*11*199).

Terms which are odd: 8715, 26145, 41349, 43575, 61005, 61971, 78435, ..., . - Robert G. Wilson v, Jul 02 2014

			8996 is in the sequence because the prime divisors are {2, 13, 173} and 173 = 13^2 + 2^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..725 from Robert Israel)

Cf. A071140.

See also the related sequences A048261, A121518.

filter:= proc(n)
local F,f,x;
F:= numtheory:-factorset(n);
f:= max(F);
evalb(f = add(x^2,x=F minus {f}));
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 02 2014

Reap[Do[p = First /@ FactorInteger[n]; If[p[[-1]] == Plus@@(Most[p]^2), Sow[n]], {n, 9962}]][[2, 1]]
lpfQ[n_]:=With[{f=FactorInteger[n][[;;,1]]},Total[Most[f]^2]==Last[f]]; Select[Range[10000],lpfQ] (* Harvey P. Dale, Jul 28 2024 *)

isok(n) = {my(f = factor(n)); f[#f~, 1] == sum(i=1, #f~ - 1, f[i, 1]^2);} \\ Michel Marcus, Jul 02 2014

Corrected by T. D. Noe, Feb 18 2011

A244637 Primes that are the sum of the squares of distinct primes.

Original entry on oeis.org

13, 29, 53, 83, 173, 179, 199, 227, 293, 347, 367, 373, 419, 439, 463, 467, 487, 491, 541, 563, 569, 587, 607, 613, 617, 641, 653, 659, 709, 727, 733, 751, 809, 823, 827, 829, 853, 857, 877, 881, 919, 953, 971, 977, 991, 997, 1013, 1019, 1021, 1039, 1049

Offset: 1

Michel Marcus, Jul 03 2014

nonn

Primes in A048261.
Provide the prime factors of A185077.
A045637 is a subsequence.
There are only 368 primes not in this sequence, the largest being 12601. - Robert Israel, Jul 04 2014

			13 is in the sequence since it is prime and 13 = 2^2 + 3^2 (2 and 3 are distinct primes).

Vincenzo Librandi, Table of n, a(n) for n = 1..2870

Cf. A045637, A048261, A121518, A185077.

nn=10;s={0};Do[p=Prime[n];s=Union[s,s+p^2],{n,nn}];Select[s,(0<#<=Prime[nn]^2)&&PrimeQ[#]&] (* Michel Lagneau, Jul 03 2014 *)

A287965 Smallest number which can be represented as the sum of distinct squares of primes in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

4, 410, 1014, 1494, 1685, 2188, 2335, 2573, 2717, 2863, 3054, 3389, 3224, 3654, 3534, 4014, 4232, 4183, 4254, 4064, 4589, 4618, 4544, 4593, 4903, 5193, 5503, 5215, 5579, 5433, 5455, 5673, 5962, 5983, 6158, 6178, 5744, 5864, 5984, 5913, 6223, 6273, 6678, 6393, 6442, 6513, 6870, 6535, 7038, 7015

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

nonn

It appears that 1275 is the first k for which a(k) = 0. - Robert Israel, Oct 14 2024

			a(2) = 410 because 410 = 7^2 + 19^2 = 11^2 + 17^2 and this is the smallest number that can be written as the sum of distinct squares of primes in 2 different ways.

Robert Israel, Table of n, a(n) for n = 1..1274
Index entries for sequences related to sums of squares

Cf. A001248, A048261, A097563, A111900, A121518.

N:= 100: # to try with primes up to N
P:= select(isprime, [2,seq(i,i=3..N,2)]):
nP:= nops(P):
S:= mul(1+x^(P[i]^2), i=1..nP):
M:= 100: # for a(1) .. a(M)
V:= Vector(M): count:= 0:
for i from 4 to N^2 while count < M do
  r:= coeff(S,x,i);
  if r >= 1 and r <= M and V[r] = 0 then count:= count+1; V[r]:= i; fi
od:
convert(V,list); # Robert Israel, Oct 14 2024

A111900(a(n)) = n.

A281477 Expansion of Sum_{k>=1} x^(prime(k)^2)/(1 + x^(prime(k)^2)) * Product_{k>=1} (1 + x^(prime(k)^2)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Ilya Gutkovskiy, Jan 27 2017

nonn

Total number of parts in all partitions of n into distinct squares of primes (A001248).

			a(38) = 3 because we have [25, 9, 4].

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example
Index entries for related partition-counting sequences

Cf. A001248, A024938, A048261, A111900, A121518, A281449, A281542, A281668.

Primes:= select(isprime, [$1..20]):
g:= add(x^(p^2)/(1+x^(p^2)),p=Primes)*mul(1+x^(p^2),p=Primes):
S:= series(g, x, 20^2+1):
seq(coeff(S,x,n),n=1..20^2); # Robert Israel, Feb 08 2017

nmax = 125; Rest[CoefficientList[Series[Sum[x^Prime[k]^2/(1 + x^Prime[k]^2), {k, 1, nmax}] Product[1 + x^Prime[k]^2, {k, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{k>=1} x^(prime(k)^2)/(1 + x^(prime(k)^2)) * Product_{k>=1} (1 + x^(prime(k)^2)).

A287962 Positive numbers that are the sum of the squares of distinct Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 4, 5, 9, 10, 13, 14, 25, 26, 29, 30, 34, 35, 38, 39, 64, 65, 68, 69, 73, 74, 77, 78, 89, 90, 93, 94, 98, 99, 102, 103, 169, 170, 173, 174, 178, 179, 182, 183, 194, 195, 198, 199, 203, 204, 207, 208, 233, 234, 237, 238, 242, 243, 246, 247, 258, 259, 262, 263, 267, 268, 271, 272, 441, 442, 445, 446, 450

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

nonn

Index entries for sequences related to sums of squares

Cf. A000119, A003995, A007598, A048261, A280933.

nmax = 450; f[x_] := Product[1 + x^Fibonacci[k]^2, {k, 2, 10}]; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]] // Rest

A351326 a(n) is the least number k such that k and all larger numbers can be expressed as the sum of n-th powers of distinct primes.

Original entry on oeis.org

7, 17164, 1866001

Offset: 1

Ilya Gutkovskiy, Mar 24 2022

C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.

Cf. A048261, A121571, A213519, A334360.

a(n) = A121571(n) + 1.

cp's OEIS Frontend

A005063 Sum of squares of primes dividing n.

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A121518 Numbers that are not the sum of the squares of distinct primes.

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A088910 Conjectured minimal required number k of terms in a representation n=sum_(i=1..k)e_i*(p_i)^2 by distinct primes p_i, where e_i is 1 or -1.

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A185077 Numbers such that the largest prime factor equals the sum of the squares of the other prime factors.

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A244637 Primes that are the sum of the squares of distinct primes.

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A287965 Smallest number which can be represented as the sum of distinct squares of primes in exactly n ways, or 0 if no such integer exists.

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A281477 Expansion of Sum_{k>=1} x^(prime(k)^2)/(1 + x^(prime(k)^2)) * Product_{k>=1} (1 + x^(prime(k)^2)).

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