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 A351327 Numbers whose trajectory under iteration of the product of squares of nonzero digits map includes 1. 2
 1, 5, 10, 11, 15, 25, 50, 51, 52, 100, 101, 105, 110, 111, 115, 125, 150, 151, 152, 205, 215, 250, 251, 255, 357, 375, 455, 500, 501, 502, 510, 511, 512, 520, 521, 525, 537, 545, 552, 554, 573, 735, 753, 1000, 1001, 1005, 1010, 1011, 1015, 1025, 1050, 1051 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS To determine whether a given number k is a term of this sequence, start with k, take the square of the product of its nonzero digits, apply the same process to the result, and continue until 1 is reached or a loop is entered. If 1 is reached, k is a term of this sequence. Every power 10^k is a term of this sequence. If k is a term, the numbers obtained by inserting zeros anywhere in k are terms. If k is a term, the numbers obtained by inserting ones anywhere in k are terms. If k is a term, each distinct permutation of the digits of k gives another term. If k is a term, the number of iterations required to converge to 1 is less than or equal to 3 (conjectured). From Michael S. Branicky, Feb 07 2022: (Start) The product of squares of nonzero digits map, f, has fixed points given in A115385. The map f has (at least) the following cycles: - 324, 576, 44100, 256, 3600; - 11664, 20736, 63504, 129600; - 15876, 2822400, 65536, 7290000; - 5308416, 8294400; - 49787136000000, 64524128256, 849346560000, 386983526400, 55725627801600. (End) LINKS Luca Onnis, On a variant of the happy numbers and their generalizations, arXiv:2203.03381 [math.GM], 2022. EXAMPLE 255 is a term of the sequence: the square of the product of its nonzero digits is (2*5*5)^2=2500, the square of the product of its nonzero digits is (2*5)^2=100, and the square of the product of its nonzero digits is 1^2=1. 2 is not a term of the sequence because its trajectory under the map is 2 -> 4 -> 16 -> 36 -> 324 -> 576 -> 44100 -> 256 -> 3600 -> 324 (reached earlier), so it enters a loop and never reaches 1. MAPLE b:= proc() false end: q:= proc(n) local m, s; m, s:= n, {}; do if m=1 then return true elif m in s or b(m) then b(n):= true; return false else s, m:= {s[], m}, mul(max(1, i)^2, i=convert(m, base, 10)) fi od end: select(q, [\$1..2000])[]; # Alois P. Heinz, Feb 11 2022 MATHEMATICA Select[Range, FixedPoint[ Product[ReplaceAll[0 -> 1][IntegerDigits[#]][[i]]^2, {i, 1, Length[ReplaceAll[0 -> 1][IntegerDigits[#]]]}] &, #, 10] == 1 &] PROG (Python) from math import prod def psd(n): return prod(int(d)**2 for d in str(n) if d != "0") def ok(n): seen = set() while n not in seen: # iterate until fixed point or in cycle seen.add(n) n = psd(n) return n == 1 def aupto(n): return [k for k in range(1, n+1) if ok(k)] print(aupto(1205)) # Michael S. Branicky, Feb 07 2022 (PARI) f(n) = vecprod(apply(d -> if (d, d^2, 1), digits(n))) is(n) = { my (m=f(n)); while (1, if (n==1, return (1), n==m, return (0), n=f(n); m=f(f(m)))) } \\ Rémy Sigrist, Feb 11 2022 CROSSREFS Cf. A007770, A051801, A115385. Sequence in context: A275200 A225838 A036788 * A136811 A130228 A136826 Adjacent sequences: A351324 A351325 A351326 * A351328 A351329 A351330 KEYWORD nonn,base,changed AUTHOR Luca Onnis, Feb 07 2022 STATUS approved

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Last modified April 1 09:50 EDT 2023. Contains 361688 sequences. (Running on oeis4.)