A061868 -id:A061868 - cp's OEIS frontend

A061268 Numbers k such that k^2 has property that the sum of its digits and the product of its digits are nonzero squares.

1, 2, 3, 12, 21, 122, 212, 221, 364, 463, 518, 537, 543, 589, 661, 715, 786, 969, 1111, 1156, 1354, 1525, 1535, 1608, 1617, 1667, 1692, 1823, 1941, 2166, 2235, 2337, 2379, 2515, 2943, 2963, 3371, 3438, 3631, 3828, 4018, 4077, 4119, 4271, 4338, 4341, 4471

Offset: 1

Amarnath Murthy, Apr 24 2001

See A061267 for the corresponding squares (the so-called ultrasquares). - M. F. Hasler, Oct 25 2022

			212^2 = 44944, 4+4+9+4+4 = 25 = 5^2 and 4*4*9*4*4 = 2304 = 48^2.

Amarnath Murthy, Infinitely many common members of the Smarandache Additive as well as multiplicative square sequence, (To be published in Smarandache Notions Journal).
Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

Cf. A061267 (the corresponding squares), A053057 (squares with square digit sum), A053059 (squares with square product of digits).

Sequence A061868 allows digit products = 0.

select( {is_A061268(n)=vecmin(n=digits(n^2))&&issquare(vecprod(n))&&issquare(vecsum(n))}, [1..4567]) \\ M. F. Hasler, Oct 25 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 11 2001

A345336 Prime numbers p such that the sum and the product of digits of p^2 are both squares.

Original entry on oeis.org

2, 3, 101, 103, 257, 283, 347, 401, 463, 491, 499, 509, 571, 599, 653, 661, 743, 751, 797, 1013, 1021, 1031, 1039, 1103, 1201, 1229, 1237, 1301, 1381, 1399, 1427, 1453, 1499, 1553, 1571, 1597, 1667, 1733, 1741, 1759, 1823, 2003, 2011

Offset: 1

Tanya Khovanova, Jun 14 2021

Primes in A061868.

			101^2 = 10201. The sum of the digits is 4, the product is 0: both are squares. Thus, 101 is in the sequence.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A061868.

filter:= proc(n) local L;
  if not isprime(n) then return false fi;
  L:= convert(n^2,base,10);
  issqr(convert(L,`+`)) and issqr(convert(L,`*`))
end proc:
select(filter, [$1..10000]); # Robert Israel, Jun 17 2021

Select[Range[3000], PrimeQ[#] && IntegerQ[Sqrt[Total[IntegerDigits[#^2]]]] && IntegerQ[Sqrt[Times @@ IntegerDigits[#^2]]] &]
Select[Prime[Range[3500]],With[{c=IntegerDigits[#^2]},AllTrue[{Sqrt[Total[c]],Sqrt[Times@@c]},IntegerQ]]&] (* Harvey P. Dale, Feb 05 2025 *)

isok(p) =if (isprime(p), my(d=digits(p^2)); issquare(vecsum(d)) && issquare(vecprod(d))); \\ Michel Marcus, Jun 14 2021

from numbthy import isprime
counter = 1
for p in range (2,1090821):
    if isprime(p) and (counter <= 10000):
        pp_product = 1
        pp_sum = 0
        for digit in range (0, len(str(p*p))):
            pp_product *= int(str(p*p)[digit])
            pp_sum += int(str(p*p)[digit])
        if pow(int(pp_product**0.5),2) == pp_product:
            if pow(int(pp_sum**0.5),2) == pp_sum:
                print(counter, p)
                counter += 1; # Karl-Heinz Hofmann, Jun 17 2021

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A061268 Numbers k such that k^2 has property that the sum of its digits and the product of its digits are nonzero squares.

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A345336 Prime numbers p such that the sum and the product of digits of p^2 are both squares.

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