The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A289506

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A289506

Write n as a product of primes p_{s_1}*p_{s_2}*p_{s_3}*... where p_i denotes the i-th prime; then a(n) = s_1^2 + s_2^2 + s_3^2 + ...
(history; published version)

#24 by Alois P. Heinz at Sat Aug 05 19:05:14 EDT 2017

STATUS

editing

approved

#23 by Alois P. Heinz at Sat Aug 05 19:05:10 EDT 2017

KEYWORD

easy,nonn,look,easy,new

STATUS

approved

editing

#22 by Alois P. Heinz at Sat Aug 05 19:04:21 EDT 2017

STATUS

editing

approved

#21 by Alois P. Heinz at Sat Aug 05 19:04:17 EDT 2017

LINKS

Alois P. Heinz, <a href="/A289506/b289506.txt">Table of n, a(n) for n = 1..20000</a>

MAPLE

# second Maple program:

a:= n-> add(numtheory[pi](i[1])^2*i[2], i=ifactors(n)[2]):

seq(a(n), n=1..80); # Alois P. Heinz, Aug 05 2017

STATUS

approved

editing

#20 by N. J. A. Sloane at Wed Aug 02 11:33:31 EDT 2017

STATUS

editing

approved

#19 by N. J. A. Sloane at Wed Aug 02 11:32:00 EDT 2017

NAME

The sum of squares of the elements of Write n as a finite multiset product of positive integers, the multiset primes p_{s_1}*p_{e_js_2} being indexed by n = Product_j *p_{e_js_3}, *... where p_{e_j} is i denotes the e_ji-th prime; then a(n) = s_1^2 + s_2^2 + s_3^2 + ...

COMMENTS

When gcd_j(e_s_j) = 1, a(n) is the modulus of the determinant whose first row consists of the e_s_j, and whose remaining rows form a lattice basis for the space of integer solutions of Sum_j e_s_jx_j = 0. See A289507.

See also A289507, where terms are (Sum_j e_s_j^2)/gcd_j(e_s_j) rather than Sum_j e_s_j^2 (this sequence).

FORMULA

Also a(n) = Sum_j e_s_j^2, where the multiset of e_s_j's is the multiset of k's, each with multiplicity r_k.

STATUS

proposed

editing

Discussion

Wed Aug 02

11:33

N. J. A. Sloane: I rewrote the definition to make it clearer and simpler (I hope).  e_i was a misleading symbol. since normally one writes n = Prod p_i^e_i.

#18 by Michel Marcus at Wed Jul 19 12:01:10 EDT 2017

STATUS

editing

proposed

Discussion

Thu Jul 27

09:44

Christopher J. Smyth: Ready for approval now? Unless someone wants to improve my Maple!

#17 by Michel Marcus at Wed Jul 19 12:01:03 EDT 2017

COMMENTS

Compare A056239, where the same encoding for integer multisets ('Heinz encoding') is used, but where A056239(n) is the sum, rather than the sum of squares, of the elements of the corresponding multiset (partition).

FORMULA

For n = Product_k p_k^{r_k}, a(n) = Sum_k k^2 * r_k.

Also a(n) = Sum_j e_j^2, where the multiset of e_j's is the multiset of k's, each with multiplicity r_k.

EXAMPLE

Further a(p_k) = k^2 and a(2^r) = r.

PROG

(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, primepi(f[k, 1])^2*f[k, 2]); \\ Michel Marcus, Jul 19 2017

STATUS

proposed

editing

#16 by Jon E. Schoenfield at Thu Jul 13 21:32:06 EDT 2017

STATUS

editing

proposed

#15 by Jon E. Schoenfield at Thu Jul 13 21:32:02 EDT 2017

FORMULA

For n = Product_k p_k^{r_k}, a(n) = Sum_k k^2 . * r_k.

EXAMPLE

For n = 12 = 2^2 . * 3 = p_1 . * p_1 . * p_2, the multiset is {1,1,2} and so a(12) = 1^2 + 1^2 + 2^2 = 6.

STATUS

proposed

editing