A252666 -id:A252666 - cp's OEIS frontend

A252668 Let k be the smallest number such that s(k) = odd part of digital sum of 5^k is a multiple of prime(n); then a(n)=k, if s(k) = prime(n). Otherwise, or if there is no such k, a(n)=0.

1, 2, 5, 4, 14, 6, 7, 16, 21, 23, 24, 0, 0, 32, 19, 20, 22, 186, 177, 26, 29, 27, 61, 236, 34, 0, 36, 78, 54, 0, 41, 87, 43, 44, 188, 0, 55, 118, 229, 66, 59, 70, 69, 60, 58, 0, 279, 147, 81, 610, 74, 325, 85, 101, 75, 179, 0, 369, 100, 97, 0, 91, 193, 95, 205

Offset: 3

Vladimir Shevelev, Dec 20 2014

We conjecture that k in the definition exists for every n>=3.
a(n)=0 for n=14,15,28,32,38,48,59,63,69,76,91,...
Note that we can continue the series of sequences A252666, A252668, ... by changing 2^k in the definition to 5^k, 7^k, 11^k, ..., prime(i)^k, ... .
Let the position of the first zero in the sequence corresponding to prime(i) be u(i). Then we call v(i)=u(i)-3 the exponential digital index (EDI) of prime(i). It is clear that in the case of i=2, prime(i)=3 and EDI(3)=0.
EDI(p) shows how many consecutive primes, beginning with 5, we obtain in the considered sequence corresponding to prime p.

			If n=4, evidently, k=2, since 5^2=25, s(2)= 2+5 = 7 = prime(4). So a(4)=2.
If n=14, then k=57, but s(57)>prime(14)=43, so a(14)=0 (the equation s(x)=43 has the smallest solution x=107).

Cf. A251964, A252280, A252281, A252282, A252283, A252666.

s(k) = my(sd = sumdigits(5^k)); sd/2^valuation(sd, 2);
a(n) = {p = prime(n); k = 1; while ((sk=s(k)) % p, k++); if (sk == p, k, 0);} \\ Michel Marcus, Dec 29 2014

More terms from Peter J. C. Moses, Dec 20 2014

A252670 a(n) is exponential digital index of prime(n).

Original entry on oeis.org

22, 0, 11, 4, 2, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 1, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 5, 0, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Vladimir Shevelev, Dec 20 2014

For the definition of the exponential digital index of a prime p (EDI(p)), see comment in A252668.

Is 22 the maximal term? Is 11 the second maximal term?

Cf. A000040, A251964, A252280, A252281, A252282, A252283, A252666, A252668.

s(k, pp) = my(sd = sumdigits(pp^k)); sd/2^valuation(sd, 2);
f(n, pp) = {my(p = prime(n), k = 1); while ((sk=s(k, pp)) % p, k++); if (sk == p, k, 0); }
a(n) = {my(pp=prime(n), j=3); while (f(j,pp), j++); j - 3;} \\ Michel Marcus, Dec 09 2018

a(n)=0, iff n is not in A251964; a(n)=1, iff n is in A251964, but is not A252280;

a(n)=2, iff n is in A251964 & A252980, but is not in A252981, a(n)>=3, iff n is in A251964 & A252980 & A252281.

More terms from Michel Marcus, Dec 09 2018

A252732 In view of their definitions, let us refer to A251964 as sequence "5", A252280 as sequence "7", and similarly define sequence "prime(n)"; a(n) is the third term of the intersection of sequences "5", ..., "prime(n)".

Original entry on oeis.org

7, 7, 7, 7, 421, 2311, 43321, 59730109, 537052693

Offset: 3

Vladimir Shevelev, Dec 21 2014

Is this sequence finite?

Up to n=13, the first two terms of the intersection of sequences "5", ..., "prime(n)" are 2 and 5 respectively.

Cf. A251964, A252280, A252281, A252282, A252283, A252666, A252668, A252670.

s[p_, k_] := Module[{s = Total[IntegerDigits[p^k]]}, s/2^IntegerExponent[s, 2]]; f[p_,q_] := Module[{k = 1}, While[ ! Divisible[s[p, k], q], k++]; k]; okQ[p_,q_] := s[p, f[p,q]] == q; okpQ[p_,nbseq_] := Module[{ans=True}, Do[If[!okQ[p,Prime[k+2]], ans=False; Break[]],{k,1,nbseq}]; ans]; a[n_]:=Module[{c=0, p=2},While[c<3 , If[okpQ[p,n],c++];p=NextPrime[p]];NextPrime[p,-1]]; Array[a,6] (* Amiram Eldar, Dec 09 2018 *)

s(p, k) = my(s=sumdigits(p^k)); s >> valuation(s, 2);
f(p, vp) = my(k=1); while(s(p,k) % vp, k++); k;
isok(p, vp) = s(p, f(p, vp)) == vp;
isokp(p, nbseq) = {for (k=1, nbseq, if (! isok(p, prime(k+2)), return (0));); return (1);}
a(n) = {my(nbpok = 0); forprime(p=2, oo, if (isokp(p, n), nbpok ++); if (nbpok == 3, return (p)););} \\ Michel Marcus, Dec 09 2018

More terms from Peter J. C. Moses, Dec 21 2014

a(10)-a(11) from Michel Marcus, Dec 09 2018

cp's OEIS Frontend

A252668 Let k be the smallest number such that s(k) = odd part of digital sum of 5^k is a multiple of prime(n); then a(n)=k, if s(k) = prime(n). Otherwise, or if there is no such k, a(n)=0.

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A252670 a(n) is exponential digital index of prime(n).

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A252732 In view of their definitions, let us refer to A251964 as sequence "5", A252280 as sequence "7", and similarly define sequence "prime(n)"; a(n) is the third term of the intersection of sequences "5", ..., "prime(n)".

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