A121518 -id:A121518 - cp's OEIS frontend

A048261 Numbers that are the sum of the squares of distinct primes.

4, 9, 13, 25, 29, 34, 38, 49, 53, 58, 62, 74, 78, 83, 87, 121, 125, 130, 134, 146, 150, 155, 159, 169, 170, 173, 174, 178, 179, 182, 183, 194, 195, 198, 199, 203, 204, 207, 208, 218, 222, 227, 231, 243, 247, 252, 256, 289, 290, 293, 294, 298, 299, 302, 303

Offset: 1

Jud McCranie

nonn

17163 is the largest of 2438 positive integers that can't be expressed as the sum of squares of distinct primes. See A121518. - T. D. Noe, Aug 04 2006

			13 = 2^2 + 3^2.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 17163.

T. D. Noe, Table of n, a(n) for n = 1..2000
Robert E. Dressler, Louis Pigno, and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40. MR 54 #7373.
Index entries for sequences related to sums of squares

Cf. A024450 (sum of squares of the first n primes).

nn=10; s={0}; Do[p=Prime[n]; s=Union[s,s+p^2], {n,nn}]; s=Select[s,0<#<=Prime[nn]^2&] (* T. D. Noe, Aug 04 2006 *)

It is easy to check that these 2438 numbers that are not the sum of distinct primes squared are all of the form sum_i e_i*q_i where e_i is 1 or -1 and the q_i's are distinct primes. - W. Edwin Clark, Oct 19 2003

A121571 Largest number that is not the sum of n-th powers of distinct primes.

Original entry on oeis.org

6, 17163, 1866000

Offset: 1

T. D. Noe, Aug 08 2006

As stated by Sierpinski, H. E. Richert proved a(1) = 6. Dressler et al. prove a(2) = 17163.
Fuller & Nichols prove T. D. Noe's conjecture that a(3) = 1866000. They also prove that 483370 positive numbers cannot be written as the sum of cubes of distinct primes. - Robert Nichols, Sep 08 2017
Noe conjectures that a(4) = 340250525752 and that 31332338304 positive numbers cannot be written as the sum of fourth powers of distinct primes. - Charles R Greathouse IV, Nov 04 2017

			a(1) = 6 because only the numbers 1, 4 and 6 are not the sum of distinct primes.

W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964, p. 143-144.

R. E. Dressler, Addendum to "A stronger Bertrand’s postulate with an application to partitions", Proc. Am. Math. Soc., 38 (1973), 667.
Robert E. Dressler, Louis Pigno and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40.
C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18, (2015), #15.10.5.
H. E. Richert, Über Zerfällungen in ungleiche Primzahlen, Math. Z. 52 no. 1 (1948), 342-343.

Cf. A231407 (numbers that are not the sum of distinct primes).
Cf. A121518 (numbers that are not the sum of squares of distinct primes).
Cf. A213519 (numbers that are the sum of cubes of distinct primes).
Cf. A001661 (integers instead of primes).

a(1) = A231407(3), a(2) = A121518(2438). - Jonathan Sondow, Nov 26 2013

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)).

A352600 Number of positive integers that are not the sum of n-th powers of distinct primes.

Original entry on oeis.org

3, 2438, 483370

Offset: 1

Ilya Gutkovskiy, Mar 22 2022

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

Cf. A121518, A121571, A173563, A231407.

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A048261 Numbers that are the sum of the squares of distinct primes.

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A121571 Largest number that is not the sum of n-th powers of distinct primes.

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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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A352600 Number of positive integers that are not the sum of n-th powers of distinct primes.

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