A252281 -id:A252281 - cp's OEIS frontend

A252283 Intersection of A251964, A252280 and A252281.

2, 5, 7, 241, 383, 421, 523, 947, 971, 1013, 1031, 1033, 1123, 1973, 2207, 2311, 2837, 2927, 4373, 4721, 5507, 6301, 8011, 8297, 9319, 10141, 12413, 14071, 14081, 17957, 18311, 18353, 19163, 21013, 21401, 22501, 22901, 28211, 30103, 32027, 37699, 38083, 40507, 42797, 43321

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 16 2014

For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let, for the first time, s(p,k) be divisible by 5 for k=k_1, be divisible by 7 for k=k_2 and
  be divisible by 11 for k=k_3.
Sequence lists primes p for which s(p,k_1)=5, s(p,k_2)=7 and  s(p,k_3)=11.
Consider also sequence which lists primes p with s(p,k_1)=5, s(p,k_2)=7, s(p,k_3)=11 and s(p,k_4)=13; sequence which lists primes p with s(p,k_1)=5, s(p,k_2)=7, s(p,k_3)=11, s(p,k_4)=13 and s(p,k_5)=17; etc. Then it seems that we will be eventually left with 2 and 5.
For example, for s(p,k_1)=5, s(p,k_2)=7,
s(p,k_3)=11, s(p,k_4)=13, s(p,k_5)=17 and
s(p,k_6)=19, the known terms of the sequence are 2, 5, 2311, 4721, 43321.
A weaker conjecture: {2,5} is the intersection of all such sequences.

Cf. A221858, A225039, A225093, A251964, A252280, A252281, A252282.

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; Select[Range[2400],  PrimeQ[#] && okQ[#, 5] && okQ[#, 7] && okQ[#, 11] &] (* Amiram Eldar, Dec 08 2018 *)

s(p,k) = my(s=sumdigits(p^k)); s >> valuation(s, 2);
f5(p) = my(k=1); while(s(p,k) % 5, k++); k;
isok5(p) = s(p, f5(p)) == 5;
f7(p) = my(k=1); while(s(p,k) % 7, k++); k;
isok7(p) = s(p, f7(p)) == 7;
f11(p) = my(k=1); while(s(p,k) % 11, k++); k;
isok11(p) = s(p, f11(p)) == 11;
lista(nn) = forprime(p=2, nn, if (isok5(p) && isok7(p) && isok11(p), print1(p, ", "))); \\ Michel Marcus, Dec 08 2018

More terms from Michel Marcus, Dec 08 2018

A225039 a(1)=2, a(2)=3, for n>=3, a(n) is the n-th number which is obtained by application Eratosthenes-like sieve to sequence: odd part of digit sum of 5^m, m>=1.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 19, 23, 17, 29, 59, 61, 31, 67, 37, 41, 79, 89, 83, 53, 103, 109, 137, 149, 151, 127, 167, 43, 211, 191, 199, 97, 181, 197, 193, 241, 269, 113, 233, 257, 139, 311, 317, 293, 283, 263, 349, 409, 173, 47, 353, 419, 431, 389, 401, 439, 463, 461

Offset: 1

Vladimir Shevelev, Apr 25 2013

We conjecture that every term is prime; moreover, we conjecture that the sequence is a permutation of the sequence of all primes.

For comparison, if in the definition we replace 5^m with 13^m, then we obtain a sequence containing 25. - Vladimir Shevelev, Dec 17 2014

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A000040, A221858, A225017, A225040, A225093, A251964, A252280, A252281, A252282, A252283.

Flatten[{{2,3,5}, DeleteDuplicates[Select[Map[#/(2^IntegerExponent[#,2] 5^IntegerExponent[#,5])&[Total[IntegerDigits[5^#]]]&, Range[2,199]], PrimeQ]]}] (* Peter J. C. Moses, Apr 25 2013 *)

More terms from Peter J. C. Moses, Apr 25 2013

A225093 a(1)=2, a(2)=3, for n>=3, a(n) is the n-th number which is obtained by application of Eratosthenes-like sieve to A225091.

Original entry on oeis.org

2, 3, 7, 13, 5, 11, 31, 43, 37, 29, 73, 79, 19, 41, 97, 59, 103, 127, 71, 157, 17, 163, 89, 23, 107, 211, 181, 241, 199, 131, 67, 101, 61, 271, 277, 149, 313, 307, 47, 367, 173, 397, 331, 409, 179, 197, 191, 457, 251, 499, 239, 233, 487, 139, 547, 523, 571, 151

Offset: 1

Vladimir Shevelev, Apr 27 2013

nonn

We conjecture that every term is prime; moreover, we conjecture that the sequence is a permutation of the sequence of all primes. - Vladimir Shevelev, Dec 17 2014

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Cf. A000040, A221858, A225017, A225039, A225040, A225091, A251964, A252280, A252281, A252282, A252283.

Flatten[{{2,3}, DeleteDuplicates[Select[Map[#/(2^IntegerExponent[#,2])&[Total[IntegerDigits[7^#]]]&, Range[200]], PrimeQ]]}] (* Peter J. C. Moses, Apr 27 2013 *)

More terms from Peter J. C. Moses, Apr 27 2013

A252282 Intersection of A251964 and A252280.

Original entry on oeis.org

2, 5, 7, 11, 19, 37, 41, 61, 71, 73, 101, 109, 127, 163, 181, 211, 241, 271, 307, 313, 383, 421, 433, 523, 541, 587, 601, 613, 631, 811, 947, 971, 983, 1013, 1031, 1033, 1063, 1123, 1153, 1171, 1201, 1229, 1303, 1423, 1483, 1531, 1621, 1973, 2053, 2113, 2207, 2311, 2341

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 16 2014

For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let, for the first time, s(p,k) be divisible of 5 for k=k_1 and be divisible of 7 for k=k_2.

Sequence lists primes p for which s(p,k_1)=5 and s(p,k_2)=7.

Cf. A221858, A225039, A225093, A251964, A252280, A252281.

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; Select[Range[2400],  PrimeQ[#] && okQ[#, 5] && okQ[#, 7] &] (* Amiram Eldar, Dec 08 2018 *)

s(p,k) = my(s=sumdigits(p^k)); s >> valuation(s, 2);
f5(p) = my(k=1); while(s(p,k) % 5, k++); k;
isok5(p) = s(p, f5(p)) == 5;
f7(p) = my(k=1); while(s(p,k) % 7, k++); k;
isok7(p) = s(p, f7(p)) == 7;
lista(nn) = forprime(p=2, nn, if (isok5(p) && isok7(p), print1(p, ", "))); \\ Michel Marcus, Dec 08 2018

More terms from Michel Marcus, Dec 08 2018

A252666 Let k be the smallest number such that s(k) = odd part of digital sum of 2^k is multiple of prime(n); then a(n)=k, if s(k) = prime(n). Otherwise, or if there is no such k, a(n)=0.

Original entry on oeis.org

5, 4, 7, 8, 47, 12, 42, 19, 20, 24, 23, 28, 31, 76, 35, 34, 38, 39, 48, 52, 116, 63, 0, 140, 68, 81, 66, 89, 96, 198, 97, 92, 415, 94, 0, 229, 119, 121, 539, 132, 133, 146, 292, 162, 158, 148, 651, 164, 153, 167, 196, 165, 757, 205, 201, 0, 184, 175

Offset: 3

Vladimir Shevelev, Dec 20 2014

We conjecture that k in the definition exists for every n>=3.

a(n)=0 for n = 25, 37, 58, ... .

			If n=3, evidently, k=5, since 2^5=32, s(5)= 3+2 = 5 = prime(3). So a(3)=5.
If n=25, then k=387, but s(387)>prime(25)=97, so a(25)=0 (the equation s(x)=97 has the smallest solution x=517).

Cf. A251964, A252280, A252281, A252282, A282283.

s(k) = my(sd = sumdigits(2^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

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.

Original entry on oeis.org

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

A252283 Intersection of A251964, A252280 and A252281.

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A225039 a(1)=2, a(2)=3, for n>=3, a(n) is the n-th number which is obtained by application Eratosthenes-like sieve to sequence: odd part of digit sum of 5^m, m>=1.

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A225093 a(1)=2, a(2)=3, for n>=3, a(n) is the n-th number which is obtained by application of Eratosthenes-like sieve to A225091.

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A252282 Intersection of A251964 and A252280.

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A252666 Let k be the smallest number such that s(k) = odd part of digital sum of 2^k is 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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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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