A048653 -id:A048653 - cp's OEIS frontend

A048646 Primes p such that the decimal digits of p^2 can be partitioned into two or more nonzero squares.

7, 13, 19, 37, 41, 107, 191, 223, 379, 487, 997, 1063, 1093, 1201, 1301, 1907, 2029, 3019, 3169, 3371, 5081, 5099, 5693, 6037, 9041, 9619, 9721, 9907, 10007, 11681, 12227, 12763, 17393, 18493, 19013, 19213, 19219, 21059, 21157, 21193, 25931

Offset: 1

Felice Russo

			7 is present because 7^2=49 can be partitioned into two squares 4 and 9; 13^2 = 169 = 16_9; 37^2 = 1369 = 1_36_9.
997^2 = 994009 = 9_9_400_9, 1063^2 = 1129969 = 1_12996_9, 997 and 1063 are primes, so 997 and 1063 are in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A048375.

Cf. A010051, intersection of A048653 and A000040.

a048646 n = a048646_list !! (n-1)
a048646_list = filter ((== 1) . a010051') a048653_list
-- Reinhard Zumkeller, Apr 17 2015

from math import isqrt
from sympy import primerange
def issquare(n): return isqrt(n)**2 == n
def ok(n, c):
    if n%10 in {2, 3, 7, 8}: return False
    if issquare(n) and c > 1: return True
    d = str(n)
    for i in range(1, len(d)):
        if d[i] != '0' and issquare(int(d[:i])) and ok(int(d[i:]), c+1):
            return True
    return False
def aupto(lim): return [p for p in primerange(1, lim+1) if ok(p*p, 1)]
print(aupto(25931)) # Michael S. Branicky, Jul 10 2021

Corrected and extended by Naohiro Nomoto, Sep 01 2001

"Nonzero" added to definition by N. J. A. Sloane, May 08 2021

A128783 Numbers whose square is a nontrivial concatenation of other squares.

Original entry on oeis.org

7, 10, 12, 13, 19, 20, 21, 30, 35, 37, 38, 40, 41, 44, 50, 57, 60, 65, 70, 80, 90, 95, 97, 100, 102, 105, 107, 108, 110, 112, 119, 120, 121, 125, 129, 130, 138, 140, 150, 160, 170, 180, 190, 191, 200, 201, 204, 205, 209, 210, 212, 220, 223, 230, 240, 250

Offset: 1

Jonathan R. Love (japanada11(AT)yahoo.ca), Apr 07 2007

a(n) = sqrt(A019547(n)); the sequence A019547 gives the squares themselves and this sequence gives the numbers whose squares yield the numbers in A019547. All multiples of 10 are included, because for any value of x, (x*10)^2 = x^2*100, which is x^2 followed by two zeros. Every digit except 6 is used as the last digit of a number in this sequence before 50. 6 is not used as the last digit of a number in this sequence until 306.

			13 is included because 13^2 = 169, which includes 16 and 9, two perfect squares.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A019547.

Cf. A010052, A000290; A048653 is a subsequence.

a128783 n = a128783_list !! (n-1)
a128783_list = filter (f . show . (^ 2)) [1..] where
   f zs = g (init $ tail $ inits zs) (tail $ init $ tails zs)
   g (xs:xss) (ys:yss)
     | a010052 (read xs) == 1 = a010052 (read ys) == 1 || f ys || g xss yss
     | otherwise              = g xss yss
   g   = False
-- Reinhard Zumkeller, Oct 11 2011

from math import isqrt
def issquare(n): return isqrt(n)**2 == n
def ok(n, c):
    if n%10 in { 2, 3, 7, 8}: return False
    if issquare(n) and c > 1: return True
    d = str(n)
    for i in range(1, len(d)):
        if issquare(int(d[:i])) and ok(int(d[i:]), c+1): return True
    return False
print([r for r in range(251) if ok(r*r, 1)]) # Michael S. Branicky, Jul 10 2021

Missing terms 205 and 209 inserted by Reinhard Zumkeller, Oct 11 2011

A344045 Primes p such that the decimal digits of p^2 can be partitioned into two or more squares.

Original entry on oeis.org

7, 13, 19, 37, 41, 97, 107, 191, 223, 379, 397, 487, 509, 701, 997, 1049, 1063, 1093, 1201, 1301, 1801, 1907, 2011, 2029, 3019, 3169, 3319, 3371, 3767, 4013, 4451, 5009, 5011, 5081, 5099, 5693, 6037, 6397, 7001, 8009, 9041, 9521, 9619, 9721, 9907, 10007

Offset: 1

Harvey P. Dale, May 09 2021

Similar to A048646 except that here zeros are permitted as squares.

			97 is a term because 97 is a prime and 97^2 = 9409 which can be partitioned into 9, 4, 0, and 9, each of which is a square.

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A048646, A048653, A128783.

tmsQ[n_]:=Total[Boole[AllTrue[Sqrt[#],IntegerQ]&/@Rest[Table[FromDigits/@ TakeList[IntegerDigits[n^2],q],{q,Flatten[Permutations/@ IntegerPartitions[ IntegerLength[ n^2]],1]}]]]]>0; Select[Prime[ Range[ 3000]],tmsQ]

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A048646 Primes p such that the decimal digits of p^2 can be partitioned into two or more nonzero squares.

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A128783 Numbers whose square is a nontrivial concatenation of other squares.

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A344045 Primes p such that the decimal digits of p^2 can be partitioned into two or more squares.

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