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# Maple code based on R. J. Mathar's code for A171797:

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(Python)

def a(n):

s = str(n)

e = sum(1 for c in s if c in "02468")

return int(str(e) + str(len(s)) + str(len(s)-e))

print([a(n) for n in range(55)]) # Michael S. Branicky, Mar 29 2022

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A modified Sisyphus function: a(n) = concatenation of (number of even digits in n) (number of even digits in n) (number of odd digits in n).

110, 101, 11, 110, 101, 11, 110, 101, 11, 110, 101, 11, 110, 101, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 11, 121, 22, 121, 22, 121, 22, 121, 22, 121, 22, 220, 211, 121, 220, 211, 121, 220, 211, 121, 220, 211, 121, 220, 211, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 121, 121, 22, 121, 22, 121, 22, 121, 22, 121, 22, 220, 211, 121, 220, 211, 121, 220, 211, 121, 220, 211, 121, 220, 211, 211, 202, 211121, 121, 22, 121, 22, 121

If we start with n and repeatedly apply the map i -> a(i), we eventually reach 312 132 (see A073054).

11 has 2 digits, both odd, so a(11)=20222 (leading zeros are omitted).

12 has 2 digits, one even and one odd, so a(12)=211121. Then a(211121) = 312132.

Maple code based on R. J. Mathar's code for A171797:

nevenDgs := proc(n) local a, d; a := 0 ; for d in convert(n, base, 10) do if type(d, 'even') then a :=a +1 ; end if; end do; a ; end proc:

cat2 := proc(a, b) local ndigsb; ndigsb := max(ilog10(b)+1, 1) ; a*10^ndigsb+b ; end:

catL := proc(L) local a, i; a := op(1, L) ; for i from 2 to nops(L) do a := cat2(a, op(i, L)) ; end do; a; end proc:

A055642 := proc(n) max(1, ilog10(n)+1) ; end proc:

A308003 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n2, n1, n1-n2]) ; end proc:

seq(A308003(n), n=0..80) ;

nonn,base,easy,changed

Corrected the definition of the sequence.

Modified the terms in the sequence with the corrected definition.

Corrected by Matthew E. Coppenbarger.

Reason: The original Sisyphus function allowed leading digits to be 0. The cited reference defined the modified Sisyphus function to avoid that. The first component should be (the number of digits of n) - this term is never zero.

Removed program: I'm not confident with changing the code, so I deleted it until someone else can provide the correction.

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A modified Sisyphus function: a(n) = concatenation of (number of even digits in n) (number of even digits in n) (number of odd digits in n).

110, 11, 101, 110, 11, 101, 110, 11, 101, 110, 11, 101, 110, 11, 121, 22, 121, 22, 121, 22, 121, 22, 121, 22, 101, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 220, 121, 211, 220, 121, 211, 220, 121, 211, 220, 121, 211, 220, 121, 121, 22, 121, 22, 121, 22, 121, 22, 121, 22, 211, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 220, 121, 211, 220, 121, 211, 220, 121, 211, 220, 121, 211, 220, 121, 121, 22, 121, 22, 121211, 211, 202, 211

If we start with n and repeatedly apply the map i -> a(i), we eventually reach 132 312 (see A073054).

11 has 2 digits, both odd, so a(11)=22 (leading zeros are omitted)202.

12 has 2 digits, one even and one odd, so a(12)=121211. Then a(121211) = 132312.

Maple code based on R. J. Mathar's code for A171797:

nevenDgs := proc(n) local a, d; a := 0 ; for d in convert(n, base, 10) do if type(d, 'even') then a :=a +1 ; end if; end do; a ; end proc:

cat2 := proc(a, b) local ndigsb; ndigsb := max(ilog10(b)+1, 1) ; a*10^ndigsb+b ; end:

catL := proc(L) local a, i; a := op(1, L) ; for i from 2 to nops(L) do a := cat2(a, op(i, L)) ; end do; a; end proc:

A055642 := proc(n) max(1, ilog10(n)+1) ; end proc:

A308003 := proc(n) local n1, n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n2, n1, n1-n2]) ; end proc:

seq(A308003(n), n=0..80) ;

Corrected the definition of the sequence.

Modified the terms in the sequence with the corrected definition.

Corrected by Matthew E. Coppenbarger.

Reason: The original Sisyphus function allowed leading digits to be 0. The cited reference defined the modified Sisyphus function to avoid that. The first component should be (the number of digits of n) - this term is never zero.

Removed program: I'm not confident with changing the code, so I deleted it until someone else can provide the correction.

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seq(A308003(n), n=10..80) ;

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N. J. A. Sloane, <a href="/A308003/b308003_1.txt">Table of n, a(n) for n = 0..28000</a>