A224787 -id:A224787 - cp's OEIS frontend

A385238 Numbers k such that A224787(k) - k is a square.

8, 16, 20, 25, 95, 169, 221, 234, 295, 312, 323, 410, 543, 1027, 1681, 3071, 3419, 3721, 4183, 4352, 6649, 7448, 7979, 8188, 9047, 9200, 10108, 11203, 12769, 15732, 16240, 20303, 22819, 25351, 26291, 28769, 32761, 33728, 42880, 51198, 51338, 52206, 53613, 55303, 56800, 63731, 65567, 71531, 77550

Offset: 1

Will Gosnell and Robert Israel, Jul 28 2025

nonn

Numbers k such that the sum of the cubes of the prime factors of k, counted with multiplicity, is k plus a square.

Includes p^2 for p in A027862.

			a(3) = 20 = 2^2 * 5 is a term because 2*2^3 + 5^3 - 20 = 121 is a square.

Robert Israel, Table of n, a(n) for n = 1..2451

Cf. A027862, A224787, A386623, A386640.

filter:= proc(n) local t;
   issqr(add(t[1]^3*t[2], t=ifactors(n)[2]) - n)
end proc:
select(filter, [$1..10^5]);

A386623 Numbers k such that k - A224787(k) is a square.

Original entry on oeis.org

1, 64, 630, 1225, 1296, 1750, 1925, 2079, 3125, 3402, 3888, 7150, 11495, 13000, 16445, 16464, 17160, 17500, 25578, 25935, 26082, 27508, 36975, 39083, 42688, 47125, 55955, 57188, 61740, 66671, 85085, 88451, 99372, 104544, 111375, 120736, 122452, 128898, 137547, 141427, 145509, 146927, 152592

Offset: 1

Will Gosnell and Robert Israel, Jul 27 2025

nonn

Numbers k such that k is the sum of a square and the cubes of the prime factors of k, counted with multiplicity.
Except for a(1) = 1, all terms are the product of at least 4 (not necessarily distinct) primes.
a(1) = 1 and a(7) = 1925 have k - A224787(k) = 1.  Are there any others? - Will Gosnell and Robert Israel, Aug 01 2025

			a(3) = 630 = 2 * 3^2 * 5 * 7 is a term because 630 - 2^3 - 2 * 3^3 - 5^3 - 7^3 = 100 is a square.

Robert Israel, Table of n, a(n) for n = 1..2511

Cf. A224787, A385238, A386245, A386304, A386640.

filter:= proc(n) local t;
   issqr(n - add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

A386640 Numbers k such that k + A224787(k) is a square.

Original entry on oeis.org

1, 225, 270, 1900, 4988, 5656, 6120, 8704, 11180, 16588, 17710, 19228, 24475, 28449, 29458, 32330, 34606, 38088, 39292, 40221, 41181, 42476, 48545, 48640, 53795, 56832, 57288, 64975, 78793, 84925, 86242, 117116, 124135, 128478, 129673, 134044, 136224, 136896, 147149, 150528, 168055, 183141

Offset: 1

Will Gosnell and Robert Israel, Jul 27 2025

nonn

Numbers k such that the sum of k and the cubes of the prime factors of k, counted with multiplicity, is a square.

			a(3) = 270 = 2 * 3^3 * 5 is a term because 270 + 2^3 + 3 * 3^3 + 5^3 =  484 = 22^2 is a square.

Robert Israel, Table of n, a(n) for n = 1..3000

Cf. A224787, A385238, A386246, A386257, A386623.

filter:= proc(n) local t;
   issqr(n + add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

lim=184000;f[{p_,e_}]:=e*p^3;a224787[k_]:=If[k==1,0,Total[f/@FactorInteger[k]]];q[k_]:=IntegerQ[Sqrt[k+a224787[k]]];Select[Range[lim],q[#]&] (* James C. McMahon, Jul 30 2025 *)

A001414 Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 12, 18, 16, 67, 21, 26, 14, 71, 12, 73, 39, 13, 23, 18, 18

Offset: 1

N. J. A. Sloane

MacMahon calls this the potency of n.
Downgrades the operators in a prime decomposition. E.g., 40 factors as 2^3 * 5 and sopfr(40) = 2 * 3 + 5 = 11.
Consider all ways of writing n as a product of zero, one, or more factors; sequence gives smallest sum of terms. - Amarnath Murthy, Jul 07 2001
a(n) <= n for all n, and a(n) = n iff n is 4 or a prime.
Look at the graph of this sequence. At the lower edge of the logarithmic scatterplot there is a set of fuzzy but unmistakable diagonal fringes, sloping toward the southeast. Their spacing gradually increases, and their slopes gradually decrease; they are more distinct toward the lower edge of the range. Is any explanation known? - Allan C. Wechsler, Oct 11 2015
For n >= 2, the glb and lub are: 3 * log(n) / log(3) <= a(n) <= n, where the lub occurs when n = 3^k, k >= 1. (Jakimczuk 2012) - Daniel Forgues, Oct 12 2015
Except for the initial term, row sums of A027746. - M. F. Hasler, Feb 08 2016
Atanassov proves that a(n) <= A065387(n) - n. - Charles R Greathouse IV, Dec 06 2016
From Robert G. Wilson v, Aug 15 2022: (Start)
Differs from A337310 beginning with n at 64, 192, 256, 320, 448, 512, ..., .
The number of terms which equal k is A000607(k).
The first occurrence of k>1 is A056240(k).
The last occurrence of k>1 is A000792(k).
The Amarnath Murthy comment of Jul 07 2001 is a result of the fundamental theorem of arithmetic.
(End)

			a(24) = 2+2+2+3 = 9.
a(30) = 10: 30 can be written as 30, 15*2, 10*3, 6*5, 5*3*2. The corresponding sums are 30, 17, 13, 11, 10. Among these 10 is the least.

K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from Franklin T. Adams-Watters)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Kevin S. Brown, The Sum of the Prime Factors of N.
Es-said En-naoui, Study of the generalized Von Mangoldt function defined by L-additive function, arXiv:2301.09677 [math.GM], 2023.
Hans Havermann, Log plot of 100000 terms
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
Rafael Jakimczuk, Sum of Prime Factors in the Prime Factorization of an Integer, International Mathematical Forum, Vol. 7, No. 53 (2012), pp. 2617-2621.
Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380.
Eric Weisstein's World of Mathematics, Sum of Prime Factors.
Wikipedia, Table of prime factors.
Steve Witham, Linear-log plot (The clear upper lines are n (the primes), n/2, n/3, n/4... but there is a dark band at sqrt(n).)
Steve Witham, Log-log plot (Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3.)

Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343, A120007, A036288.
A000607(n) gives the number of values of k for which A001414(k) = n.
Cf. A036349 (indices of even terms), A356163 (their char. function), A335657 (indices of odd terms), A289142 (of multiples of 3), A373371 (their char. function).
For sum of squares of prime factors see A067666, for cubes see A224787.
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A056239 (with a(p)=primepi(p)), A059975 (with a(p)=p-1), A064097 (with a(p)=1+a(p-1)), A113177 (with a(p)=Fib(p)), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
For other completely additive sequences see the cross-references in A104244.

a001414 1 = 0
a001414 n = sum $ a027746_row n
-- Reinhard Zumkeller, Feb 27 2012, Nov 20 2011

[n eq 1 select 0 else (&+[j[1]*j[2]: j in Factorization(n)]): n in [1..100]]; // G. C. Greubel, Jan 10 2019

A001414 := proc(n) add( op(1,i)*op(2,i),i=ifactors(n)[2]) ; end proc:
seq(A001414(n), n=1..100); # Peter Luschny, Jan 17 2011

a[n_] := Plus @@ Times @@@ FactorInteger@ n; a[1] = 0; Array[a, 78] (* Ray Chandler, Nov 12 2005 *)

a(n)=local(f); if(n<1,0,f=factor(n); sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))

A001414(n) = (n=factor(n))[,1]~*n[,2] \\ M. F. Hasler, Feb 07 2009

from sympy import factorint
def A001414(n):
    return sum(p*e for p,e in factorint(n).items()) # Chai Wah Wu, Jan 08 2016

[sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0,len(factor(n)))) for n in range(1,79)] # Giuseppe Coppoletta, Jan 19 2015

If n = Product p_j^k_j then a(n) = Sum p_j * k_j.
Dirichlet g.f. f(s)*zeta(s), where f(s) = Sum_{p prime} p/(p^s-1) = Sum_{k>0} primezeta(k*s-1) is the Dirichlet g.f. for A120007. Totally additive with a(p^e) = p*e. - Franklin T. Adams-Watters, Jun 02 2006
For n > 1: a(n) = Sum_{k=1..A001222(n)} A027746(n,k). - Reinhard Zumkeller, Aug 27 2011
Sum_{n>=1} (-1)^a(n)/n^s = ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s), if Re(s)>1 and 0 if s=1 (Alladi and Erdős, 1977). - Amiram Eldar, Nov 02 2020
a(n) >= k*log(n), where k = 3/log(3). This bound is sharp. - Charles R Greathouse IV, Jul 28 2025

A134618 Numbers such that the sum of cubes of their prime factors (taken with multiplicity) is a prime.

Original entry on oeis.org

12, 28, 40, 45, 48, 52, 54, 56, 63, 75, 80, 96, 104, 108, 117, 136, 152, 153, 165, 175, 210, 224, 232, 245, 250, 261, 268, 300, 320, 325, 333, 344, 350, 363, 384, 387, 390, 399, 405, 416, 432, 462, 464, 468, 475, 477, 504, 507, 531, 536, 539, 561, 570, 584

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

			a(2) = 28, since 28 = 2*2*7 and 2^3 + 2^3 + 7^3 = 359 which is prime.

Harvey P. Dale and Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (first 1000 terms from _Harvey P. Dale_)

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

Cf. A134600, A134602, A134605, A134608, A134612, A134616, A134620.

Select[Range[600],PrimeQ[Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[#]]^3]]&] (* Harvey P. Dale, Feb 01 2013 *)

from sympy import factorint, isprime
def ok(n): return isprime(sum(p**3 for p in factorint(n, multiple=True)))
print([k for k in range(585) if ok(k)]) # Michael S. Branicky, Dec 28 2021

{k: A224787(k) in A000040}. - R. J. Mathar, Mar 25 2025

Example clarified by Harvey P. Dale, Feb 01 2013

Minor edits by Hieronymus Fischer, May 06 2013

A337395 a(n) is the largest exponent k such that the sums, with multiplicity, of the i-th powers of the prime factors of A100118(n) are all prime for i=1 to k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 9, 1, 1, 4, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Torlach Rush, Aug 25 2020

nonn

			a(4) = 2 because (2^1) + (3^1) = 5 and (2^2) + (3^2) = 13.
a(6) = 2 because (2^1) + (5^1) = 7 and (2^2) + (5^2) = 29.
a(8) = 6 because (2^1) + (2^1) + (3^1) = 7 and (2^2) + (2^2) + (3^2) = 17 and (2^3) + (2^3) + (3^3) = 43 and (2^4) + (2^4) + (3^4) = 113 and (2^5) + (2^5) + (3^5) = 307 and (2^6) + (2^6) + (3^6) = 857.

Cf. A001414, A100118.

Cf. A067666, A224787.

a(n) = {my(f=factor(n), x = 1, y = 1); while(y, if(isprime(sum(i=1, #f~, f[i, 1]^x*f[i, 2])), x++, y = 0)); return(x - 1)}
for (n = 2, 220, if(a(n) > 0, print1(a(n), ", ")))

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A385238 Numbers k such that A224787(k) - k is a square.

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A386623 Numbers k such that k - A224787(k) is a square.

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A386640 Numbers k such that k + A224787(k) is a square.

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A001414 Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).

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A134618 Numbers such that the sum of cubes of their prime factors (taken with multiplicity) is a prime.

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A337395 a(n) is the largest exponent k such that the sums, with multiplicity, of the i-th powers of the prime factors of A100118(n) are all prime for i=1 to k.

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