editing

approved

a(n)=a(n') if respective digits of n and n' differ by 0 or 5. In particular, a(10k+m) = a(10k+m+5) if 0 <= m <= 4.

(PARI) A169916(n)={u=vector(#n=digits(n), i, 1); n=apply(d->n+d*u, n)%10; sum(i=0, 2*#n-2, sum(j=max(1, #n-i), min(2*#n-1-i, #n), n[2*#n-i-j][j])%10*10^i)} \\ M. F. Hasler, Mar 26 2015

approved

editing

David Applegate, _Marc LeBrun (mlb(AT)well.com) _ and N. J. A. Sloane, Jul 20 2010

_David Applegate (david(AT)research.att.com), _, Marc LeBrun (mlb(AT)well.com) and N. J. A. Sloane, Jul 20 2010

David Applegate (david(AT)research.att.com), Marc LeBrun (mlb(AT)well.com) and _N. J. A. Sloane (njas(AT)research.att.com), _, Jul 20 2010

<a href="/Sindx_index/Ca.html#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>

<a href="/Sindx_Ca.html#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>

nonn,base,new

Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be addition mod 10.

0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 220, 242, 264, 286, 208, 220, 242, 264, 286, 208, 440, 462, 484, 406, 428, 440, 462, 484, 406, 428, 660, 682, 604, 626, 648, 660, 682, 604, 626, 648, 880, 802, 824, 846, 868, 880, 802, 824, 846, 868, 0, 22, 44, 66, 88, 0, 22, 44, 66, 88, 220, 242

0,2

The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.

<a href="Sindx_Ca.html#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>

a(16) = 16*16 = 242:

....16

....16

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....72 (6*6 = 6+6 mod 10 = 2, 6*1 = 6+1 mod 10 = 7)

...27.

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...242

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The four versions are A059729, A169916, A169917, A169918.

nonn,base,new

David Applegate (david(AT)research.att.com), Marc LeBrun (mlb(AT)well.com) and N. J. A. Sloane (njas(AT)research.att.com), Jul 20 2010

approved