This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Naohiro Nomoto has authored 383 sequences. Here are the ten most recent ones:
a(57)=2 because there are two such partitions of 228: {43,47,67,71}, {43,53,61,71}.
In the first partition (i.e., p_1 = 43, p_2 = 47, p_3 = 67, p_4 = 71), (43,47) and (67,71) are prime pairs with difference 4 (since p_2 - p_1 = p_4 - p_3 = 4), and they are consecutive among such pairs (i.e., there does not exist any intervening pair of primes with difference 4). It is also true that (43,67) and (47,71) are prime pairs with difference 24 (since p_3 - p_1 = p_4 - p_2 = 24), and they are consecutive among such pairs (i.e., no intervening pair of primes with difference 24 exists).
Similarly, in the second partition (i.e., p_1 = 43, p_2 = 53, p_3 = 61, p_4 = 71), (43,53) and (61,71) are consecutive pairs of prime numbers with difference 10: p_2 - p_1 = p_4 - p_3 = 10, and (43,61) and (53,71) are consecutive pairs of prime numbers with difference 18: p_3 - p_1 = p_4 - p_2 = 18.
chk(s, t, d)={forprime(p=s, t, if(isprime(p+d), return(0))); 1}
a(n) = {my(s=0); forprime(p=3, n, if(isprime(2*n-p), forprime(q=p+1, n, if(isprime(2*n-q) && chk(p+1, 2*n-q-1, q-p) && chk(p+1,q-1,2*n-q-p), s++)))); s} \\ Andrew Howroyd, Feb 03 2023
For k = 26, a = 10, b = 15, c = 5, d = 20, e = 3, f = 22, and 10*15 = 5*26 + 20, 5*20 = 3*26 + 22, 10 + 15 = 5 + 20 = 3 + 22.
432 is a nonsquare, m = 45, a = 16, b = 27, c = 9, 432 = 16*27 = 9*45 + 27, 27^2 = 16*45 + 9. Therefore 432 is a term.
4800 is the sum of divisors of 1512 and 2058, and rad(1512) = rad(2058) = 42, hence 4800 is in the sequence with j=1512 and k=2058.
218 is the sum of proper divisors of 250 and 160, and rad(250) = rad(160) = 10, hence 218 is in the sequence with j=250 and k=160. Other examples of n and j, k: For n = 189648, j = 95832, k = 85536. For n = 720240, j = 288120, k = 246960. For n = 119967120, j = 38755080, k = 34398000. For n = 129705984, j = 71614464, k = 60424704.
For n = 36, m = 24, 36 is the sum of the proper divisors of 24, and rad(36) = rad(24) = 6.
rad[n_] := Times @@ (First@ # & /@ FactorInteger@ n); f[n_] := Block[{sd = DivisorSigma[1, n] - n}, If[ rad[n] == rad[sd], sd, 0]]; k = 1; lst = {}; While[k < 1000001, a = f@ k; If[a > 0, AppendTo[lst, a]]; k++]; Sort@ lst (* Robert G. Wilson v, Feb 28 2015 *)
For n=2, all primes have 2 divisors and satisfy sigma(x)-x=1, so a(2) = 1. For n=4, 27 and 35 have 4 divisors and the sum of their proper divisors is 13 for both (1+3+9 and 1+5+7). For n=6, 98 and 175 have 6 divisors and the sum of their proper divisors is 73 for both (1+2+7+14+49 and 1+5+7+25+35). For n=8, 104 and 110 have 8 divisors and the sum of their proper divisors is 106 for both (1+2+4+8+13+26+52 and 1+2+5+10+11+22+55). For n=9, 163^2*167^2 and 61^2*353^2 have 9 divisors and the sum of their proper divisors is 9064940 for both. For n=10, 7203 and 7857 have 10 divisors and the sum of their proper divisors is 4001 for both. For n=12, 276 and 306 have 12 divisors and the sum of their proper divisors is 396 for both.
546^3 = 338 * 546 * 882, 546^3 + 1 = 337 * 547 * 883. 25632^3 = 15842 * 25632 * 41472, 25632^3 + 1 = 15841 * 25633 * 41473.
F:=Fibonacci;; List([1..30], n-> 2*F(2*n)*F(2*n-1) ); # G. C. Greubel, Jul 15 2019
F:=Fibonacci; [2*F(2*n)*F(2*n-1): n in [1..30]]; // G. C. Greubel, Jul 15 2019
with(combinat); A240836:=n->2*fibonacci(2*n)*fibonacci(2*n-1); seq(A240836(n), n=1..30); # Wesley Ivan Hurt, Apr 13 2014
Table[2Fibonacci[2n]Fibonacci[2n-1], {n, 30}] (* Wesley Ivan Hurt, Apr 13 2014 *)
vector(30, n, f=fibonacci; 2*f(2*n)*f(2*n-1)) \\ G. C. Greubel, Jul 15 2019
f=fibonacci; [2*f(2*n)*f(2*n-1) for n in (1..30)] # G. C. Greubel, Jul 15 2019
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