A221858 -id:A221858 - cp's OEIS frontend

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.

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

A251964 For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let k_1 be the smallest k such that s(p,k) is divisible by 5. Sequence lists primes p for which s(p,k_1)=5.

Original entry on oeis.org

2, 5, 7, 11, 19, 23, 37, 41, 61, 71, 73, 101, 109, 113, 127, 131, 163, 179, 181, 211, 229, 241, 251, 271, 307, 311, 313, 383, 389, 401, 421, 433, 449, 479, 521, 523, 541, 557, 569, 571, 587, 601, 613, 631, 659, 677, 751, 811, 827, 839, 857, 929, 947, 971, 977

Offset: 1

Vladimir Shevelev, Dec 11 2014

Let p be a prime other than 3. If p is not in the sequence, then either s(p,k_1) >= 25 or k_1 does not exist. We conjecture that k_1=k_1(p) exists for every prime p.

			For p=7, s(p,1) = 7, s(p,2) = 4+9 = 13, s(p,3) = (3+4+3)/2 = 5. So 7 is a term.
For p=13, s(p,1) = 1, s(p,2) = 1, s(p,3) = 19, s(p,4) = 11, s(p,5) = 25. So 13 is not in the sequence.

Cf. A221858, A225039, A225093.

s[p_, k_] := Module[{s = Total[IntegerDigits[p^k]]}, s/2^IntegerExponent[s, 2]]; f5[p_] := Module[{k = 1}, While[! Divisible[s[p, k], 5], k++]; k]; ok5Q[p_] := s[p, f5[p]] == 5; Select[Range[1000], PrimeQ[#] && ok5Q[#] &] (* Amiram Eldar, Dec 07 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;
lista5(nn) = forprime(p=2, nn, if (isok5(p), print1(p, ", "))); \\ Michel Marcus, Dec 07 2018

More terms from Peter J. C. Moses, Dec 11 2014

A252280 For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let k_1 be the smallest k such that s(p,k) is divisible by 7. Sequence lists primes p for which s(p,k_1)=7.

Original entry on oeis.org

2, 5, 7, 11, 19, 29, 31, 37, 41, 43, 59, 61, 71, 73, 79, 97, 101, 103, 107, 109, 127, 137, 149, 151, 163, 167, 181, 193, 197, 211, 223, 233, 239, 241, 257, 271, 277, 293, 307, 313, 331, 347, 359, 373, 383, 397, 409, 419, 421, 431, 433, 467, 487, 491, 509, 523

Offset: 1

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

For s(p,k_1)=5, see A251964.

Cf. A221858, A225039, A225093, A251964.

s[p_, k_] := Module[{s = Total[IntegerDigits[p^k]]}, s/2^IntegerExponent[s, 2]]; f7[p_] := Module[{k = 1}, While[! Divisible[s[p, k], 7], k++]; k]; ok7Q[p_] := s[p, f7[p]] == 7; Select[Range[1000],  PrimeQ[#] && ok7Q[#] &] (* Amiram Eldar, Dec 07 2018*)

s(p,k) = my(s=sumdigits(p^k)); s >> valuation(s, 2);
f7(p) = my(k=1); while(s(p,k) % 7, k++); k;
isok7(p) = s(p, f7(p)) == 7;
lista7(nn) = forprime(p=2, nn, if (isok7(p), print1(p, ", "))); \\ Michel Marcus, Dec 07 2018

A252281 For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let k_1 be the smallest k such that s(p,k) is divisible by 11. Sequence lists primes p for which s(p,k_1)=11.

Original entry on oeis.org

2, 5, 7, 13, 23, 29, 31, 43, 47, 53, 59, 79, 83, 97, 137, 139, 173, 191, 227, 239, 241, 257, 263, 281, 317, 331, 337, 349, 353, 359, 373, 383, 421, 439, 443, 449, 461, 463, 467, 479, 499, 509, 523, 547, 557, 563, 569, 593, 599, 607, 619, 641, 643, 653, 659

Offset: 1

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

For s(p,k_1)=5 and s(p,k_1)=7 see A251964 and A252280 respectively.

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

s[p_, k_] := Module[{s = Total[IntegerDigits[p^k]]}, s/2^IntegerExponent[s, 2]]; f11[p_] := Module[{k = 1}, While[! Divisible[s[p, k], 11], k++]; k]; ok11Q[p_] := s[p, f11[p]] == 11; Select[Range[1000], PrimeQ[#] && ok11Q[#] &] (* Amiram Eldar, Dec 07 2018 *)

s(p,k) = my(s=sumdigits(p^k)); s >> valuation(s, 2);
f11(p) = my(k=1); while(s(p,k) % 11, k++); k;
isok11(p) = s(p, f11(p)) == 11;
lista11(nn) = forprime(p=2, nn, if (isok11(p), print1(p, ", "))); \\ Michel Marcus, Dec 07 2018

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

A252283 Intersection of A251964, A252280 and A252281.

Original entry on oeis.org

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

A225091 The odd part of the digit sum of 7^n.

Original entry on oeis.org

1, 7, 13, 5, 7, 11, 7, 25, 31, 7, 43, 49, 37, 13, 29, 1, 13, 29, 73, 79, 19, 41, 97, 85, 73, 97, 7, 91, 133, 121, 59, 115, 103, 127, 71, 157, 17, 115, 65, 17, 71, 37, 17, 169, 175, 163, 187, 175, 17, 89, 23, 217, 49, 55, 217, 107, 211, 181, 241, 211, 199, 205

Offset: 0

Vladimir Shevelev, Apr 27 2013

Conjecture: the sequence contains all primes > 3.

Peter J. C. Moses, Table of n, a(n) for n = 0..1999

Cf. A221858, A225017, A225039, A225040.

read(transforms) :
A225091 := proc(n)
    A000265(digsum(7^n)) ;
end proc: # R. J. Mathar, May 05 2013

Map[#/(2^IntegerExponent[#,2])&[Total[IntegerDigits[7^#]]]&, Range[0,99]] (* Peter J. C. Moses, Apr 27 2013 *)

a(n) = my(s = sumdigits(7^n)); s >> valuation(s, 2); \\ Michel Marcus, Dec 19 2018

a(n) = A000265(A066003(n)). - R. J. Mathar, May 05 2013

A225430 a(n) is the smallest m for which there appears for the first time a prime p which equals odd part of sum of digits of m^2.

Original entry on oeis.org

4, 7, 8, 17, 43, 83, 167, 314, 707, 836, 6833, 8167, 21886, 41833, 89437, 134164, 947617, 987917, 3143167, 3162083, 9272917, 24060133, 60827617, 434738887, 529027313, 2641873937, 5383305583, 14141757313

Offset: 1

Vladimir Shevelev, May 07 2013

The corresponding primes are 7, 13, 5, 19, 11, 31,...

			The sequence of odd parts of sums of digits of n^2 begins 1, 1, 9, 7, 7, 9, 13, 5,... . The first appearances of primes are on places 4, 7, 8,..., so a(1) = 4, a(2) = 7, a(3)= 8, etc.

Cf. A221858, A225017, A225039, A225040, A225093.

Sort[Map[#[[1]][[2]]&,SplitBy[Sort[Select[Map[{(#/2^IntegerExponent[#,2])&[Total[IntegerDigits[#^2]]],#}&,Range[1000000]],PrimeQ[#[[1]]]&]],First]]] (* Peter J. C. Moses, May 09 2013 *)

A225516 Primes corresponding to terms of A225430.

Original entry on oeis.org

7, 13, 5, 19, 11, 31, 17, 37, 43, 23, 29, 61, 67, 73, 79, 41, 47, 97, 103, 53, 109, 59, 127, 139, 71, 151, 157, 163

Offset: 1

Vladimir Shevelev, May 09 2013

nonn

It is a result of application of Eratosthenes-like sieve to sequence of odd parts of sums of digits of n^2 which begins 1,1,9,7,7,9,13,5,... .

Conjecture: the sequence is a permutation of all primes > 3.

Cf. A221858, A225017, A225039, A225040, A225093, A225430.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A251964 For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let k_1 be the smallest k such that s(p,k) is divisible by 5. Sequence lists primes p for which s(p,k_1)=5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A252280 For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let k_1 be the smallest k such that s(p,k) is divisible by 7. Sequence lists primes p for which s(p,k_1)=7.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

A252281 For a prime p, denote by s(p,k) the odd part of the digital sum of p^k. Let k_1 be the smallest k such that s(p,k) is divisible by 11. Sequence lists primes p for which s(p,k_1)=11.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

A252282 Intersection of A251964 and A252280.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A252283 Intersection of A251964, A252280 and A252281.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A225091 The odd part of the digit sum of 7^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica