A364961 Odd numbers k such that A005940(k) is either k itself or its descendant in Doudna-tree, A005940.

1, 3, 5, 25, 45, 49, 40131, 50575, 79625, 1486485, 1872507, 3403125

Offset: 1

Author

Antti Karttunen, Aug 14 2023

Keywords

• nonn
• hard
• more

Comments

Odd numbers k such that A252464(k) is equal to A364569(k).

Apparently, A364960 without the even terms of A029747.

Note that 1, 25, 45, 49 are so far the only known integers x for which A005940(x) = A003961(x).

Examples

			Term (and its factorization)            A005940(term) (and its factorization)
      1                              -> 1
      3                              -> 3
      5                              -> 5
     25 = 5^2                        -> 49        = 7^2
     45 = 3^2 * 5                    -> 175       = 5^2 * 7
     49 = 7^2                        -> 121       = 11^2
  40131 = 3^2 * 7^3 * 13             -> 100847877 = 3 * 13^2 * 19^3 * 29
  50575 = 5^2 * 7 * 17^2             -> 22467159  = 3^3 * 11^2 * 13 * 23^2
  79625 = 5^3 * 7^2 * 13             -> 787365187 = 7 * 19^3 * 23^2 * 31
1486485 = 3^3 * 5 * 7 * 11^2 * 13    -> 25468143451205
                                        = 5 * 7 * 13 * 17^3 * 19 * 23 * 29^2 * 31
1872507 = 3 * 7 * 13 * 19^3          -> 240245795625
                                        = 3 * 5^4 * 11 * 17 * 23 * 31^3,
3403125 = 3^2 * 5^5 * 11^2           -> 2394659631669305
                                        = 5 * 7^3 * 11 * 13^2 * 17^5 * 23^2.
See also examples in A364959.
		

Crossrefs

Odd terms in A364960.

Cf. A003961, A005940, A029747, A156552, A252464, A364569, A364570.

Cf. also A364956 (odd terms there), A364959, A364962.

Programs

PARI

Abincompreflen(n, m) = { my(x=binary(n), y=binary(m), u=min(#x, #y)); for(i=1, u, if(x[i]!=y[i], return(i-1))); (u); };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A364569(n) = Abincompreflen(A156552(n), (n-1));
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = if(1==n,0,(bigomega(n) + A061395(n) - 1));
isA364961(n) = ((n%2)&&(A252464(n)==A364569(n)));

PARI

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA364961(n) = if(!(n%2),0,my(k=A005940(n)); while(k>n, k = A252463(k)); (k==n));