Daniel Starodubtsev - cp's OEIS frontend

A345338 Integers whose Reverse And Add trajectory reaches its first prime after a record number of iterations (at least one iteration must be performed).

1, 5, 181, 10031, 1001320

Offset: 1

Daniel Starodubtsev, Jun 14 2021

a(6) > 10^9 (if it exists).

All numbers whose trajectory reaches a multiple of 3 or 11 before reaching a prime will never reach a prime.

			a(3) = 181 because it takes 3 iterations (181 -> 362 -> 625 -> 1151 (prime)) to reach a prime, which is more than any smaller number.

Cf. A056964.

f(n) = my(t=n, c=1); while(!isprime(t+=fromdigits(Vecrev(digits(t)))), if(gcd(t, 33)>1, return(0)); c++); c;
lista(nn) = my(m); for(k=1, nn, if(f(k)>m, print1(k, ", "); m=f(k))); \\ Jinyuan Wang, Jun 15 2021

from sympy import isprime
def ra(n): s = str(n); return int(s) + int(s[::-1])
def afind(limit):
    record = 0
    for k in range(limit+1):
        m, i = ra(k), 1
        while not isprime(m) and m%3 != 0 and m%11 != 0: m = ra(m); i += 1
        if isprime(m) and i > record: record = i; print(k, end=", ")
afind(1234567) # Michael S. Branicky, Jul 03 2021

A335787 Emirps containing only the digits 1 and 3.

Original entry on oeis.org

13, 31, 113, 311, 113131, 131311, 1131131, 1133131, 1133333, 1311311, 1313311, 1331333, 1333313, 3111313, 3131113, 3133331, 3331331, 3333311, 11113111, 11131111, 11311133, 11313311, 11331311, 11333131, 13131133, 13133311, 31111313, 31311113, 33111311, 33113131

Offset: 1

Daniel Starodubtsev, Jun 23 2020

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A006567 and A032917.

Subsequence of A020451.

Select[Flatten[Table[FromDigits[#, 10] & /@ Tuples[{1, 3}, n], {n, 8}]], # != (r = IntegerReverse[#]) && PrimeQ[#] && PrimeQ[r] &] (* Amiram Eldar, Jun 23 2020 after Vincenzo Librandi at A032917 *)

A334170 Emirps containing only the digits 1 and 2.

Original entry on oeis.org

111211, 112111, 12122221, 12222121, 1111122121, 1212211111, 11121211121, 11212221121, 12111212111, 12112221211, 111122221121, 111122222111, 111122222221, 111211221221, 111212212211, 111212222111, 111222212111, 111222221111, 112111211221, 112122112211, 112122211121

Offset: 1

Daniel Starodubtsev, Apr 17 2020

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A007931 and A006567.

Subsequence of A020450.

Union @ Flatten @ Table[FromDigits /@ Select[Tuples[{1, 2}, n] ,PrimeQ @ (m = FromDigits[#]) && # != (r = Reverse[#]) && PrimeQ @ FromDigits[r] &], {n, 12}] (* Amiram Eldar, Apr 18 2020 *)
Table[Select[FromDigits/@Tuples[{1,2},n],!PalindromeQ[#]&&AllTrue[ {#,IntegerReverse[ #]},PrimeQ]&],{n,12}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 08 2020 *)

isok(p) = if (isprime(p), my(d=digits(p), rd=Vecrev(d)); (vecsort(d,,8) == [1,2]) && isprime(fromdigits(rd)) && (rd != d));
lista(nn) = {for (n=1, nn, my(vf = vector(n, k, 1)); for (i=1, 2^n-1, my(vpos = select(x->(x==1), Vecrev(binary(i)), 1), nvf = vf); for (i=1, #vpos, nvf[vpos[i]] = 2;); my(x = eval(Str(1, fromdigits(Vecrev(nvf)), 1))); if (isok(x), print1(x, ", "));););} \\ Michel Marcus, Apr 18 2020

A333523 Number of iterations of Reverse And Add needed to reach a number divisible by n (or 0 if such a number is never reached).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 61, 1, 3, 8, 34, 22, 8, 17, 2, 8, 119, 14, 1, 17, 7, 110, 7, 12, 33, 34, 158, 28, 12, 1, 60, 11, 12, 50, 79, 7, 129, 64, 13, 42, 1, 4, 89, 131, 8, 14, 81, 30, 19, 125, 12, 1, 88, 13, 33, 67, 232, 26, 27, 24, 123, 59, 1, 24, 59, 36, 206, 148, 12, 217, 90, 97

Offset: 1

Daniel Starodubtsev, Mar 26 2020

If n is a palindrome > 0, a(n) = 1. See A002113.

a(n) > 0 for n < 10000.

			a(12) = 3, because 12 takes 3 iterations (12 -> 33 -> 66 -> 132) to become 132, which is divisible by 12.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002113, A016016, A056964.

radd(n) = fromdigits(Vecrev(digits(n)))+n; \\ A056964
a(n) = {my(i=1, k=n, x); while((x=radd(n)) % k, i++; n=x); i;} \\ Michel Marcus, Apr 11 2020

A333475 Numbers k such that S(2^k) is a perfect square, where S(t) is the sum of decimal digits of t.

Original entry on oeis.org

0, 2, 16, 22, 36, 78, 104, 110, 118, 130, 176, 186, 194, 200, 216, 240, 270, 276, 320, 358, 364, 376, 440, 558, 576, 602, 608, 612, 614, 620, 630, 700, 872, 884, 894, 918, 972, 1144, 1174, 1192, 1216, 1536, 1566, 1610, 1658, 1798, 1882, 2000, 2312, 2630, 2928, 3042, 3540, 3648, 3744, 3750, 3774

Offset: 1

Daniel Starodubtsev, Mar 23 2020

Numbers k such that A000079(k) is in A028839.

All terms are even. - Robert Israel, Mar 24 2020

			16 is in the sequence, because S(2^16) = S(65536) = 25 is a perfect square.

Robert Israel, Table of n, a(n) for n = 1..209

Cf. A000079, A007953, A028839.

sd:= n -> convert(convert(n,base,10),`+`):
select(t -> issqr(sd(2^t)), [$0..10000]); # Robert Israel, Mar 24 2020

isok(k) = issquare(sumdigits(2^k)); \\ Michel Marcus, Mar 23 2020

A333474 Numbers k such that 2^k + 1 is divisible by the sum of its decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 9, 98, 135, 200, 665, 1782, 4230, 4521, 6815, 17010, 19635, 30338, 31365, 35427, 49555, 96619, 102897, 157850, 193734, 273050, 393225, 449217, 477333, 483310, 493350, 534465, 661815, 918918, 947925, 1050858, 1114690, 1134945, 1204686, 1350990, 1428105

Offset: 1

Daniel Starodubtsev, Mar 23 2020

Numbers k such that A000051(k) is in A005349.

			9 is in the sequence, because 2^9 + 1 = 513 is divisible by 5 + 1 + 3.

Cf. A000051, A005349, A095412.

isok(k) = my(x=2^k+1); !(x % sumdigits(x)); \\ Michel Marcus, Mar 23 2020

print([i for i in range(5000) if (2**i+1)%sum([int(i) for i in str(2**i+1)]) == 0])

More terms from Giovanni Resta, Mar 23 2020

A309527 Numbers k such that 6^k + 17 is prime.

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 19, 27, 79, 198, 565, 787, 2183, 3811, 4748, 6210, 7887, 8965, 13303, 20125, 23433, 28797

Offset: 1

Daniel Starodubtsev, Aug 06 2019

a(20) > 14000. - Daniel Starodubtsev, Apr 17 2020

			3 is in the sequence because 6^3 + 17 = 233, which is prime.

Cf. A013600, A013607, A059614, A145106, A182331, A217351, A217352.

lista(nn)=for(k=0,nn,if(ispseudoprime(6^k+17),print1(k", ")))

a(17)-a(18) from Daniel Starodubtsev, Mar 16 2020
a(19) from Daniel Starodubtsev, Apr 17 2020
a(20)-a(22) from Michael S. Branicky, Mar 14 2023

A309726 Numbers k such that k^2 - 12 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 25, 29, 35, 41, 49, 53, 59, 61, 79, 85, 91, 95, 97, 103, 107, 113, 119, 121, 137, 139, 145, 149, 163, 169, 173, 179, 181, 185, 191, 205, 209, 227, 233, 235, 245, 251

Offset: 1

Daniel Starodubtsev, Aug 14 2019

nonn

All terms are odd and not divisible by 3.

			11 is in the sequence because 11^2 - 12 = 109, which is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028870, A028873, A028876, A028879, A028882, A028885, A186815, A090696, A296507.

Cf. A056927.

Select[Range[5,301,2],PrimeQ[#^2-12]&] (* Harvey P. Dale, Dec 23 2019 *)

select(n->isprime(n^2-12), [1..1000]) \\ Andrew Howroyd, Aug 14 2019

If A056927(k) = 12, then k is a term. - A.H.M. Smeets, Aug 15 2019

A309489 Numbers k such that the sum of digits in odd places is equal to the sum of digits in even places in k!.

Original entry on oeis.org

11, 18, 20, 25, 27, 28, 32, 35, 41, 43, 51, 57, 60, 68, 72, 74, 80, 102, 105, 107, 113, 121, 122, 138, 140, 145, 156, 161, 166, 171, 228, 233, 245, 282, 301, 307, 308, 311, 315, 329, 333, 335, 340, 347, 349, 351, 353, 366, 386, 412, 420, 454, 469, 478

Offset: 1

Daniel Starodubtsev, Aug 04 2019

Numbers k such that A000142(k) is a term of A135499. - Michel Marcus, Aug 04 2019

Daniel Starodubtsev, Table of n, a(n) for n = 1..300

Cf. A000142 (n!), A135499.

aQ[n_] := Total[(d = IntegerDigits[n!])[[1;;-1;;2]]] == Total[d[[2;;-1;;2]]]; Select[Range[500], aQ] (* Amiram Eldar, Aug 04 2019 *)
sdoeQ[n_]:=With[{nf=IntegerDigits[n!]},Total[Take[nf,{1,-1,2}]]==Total[Take[nf,{2,-1,2}]]]; Select[Range[500],sdoeQ] (* Harvey P. Dale, Jun 04 2025 *)

isok(k) = {my(d = digits(k!), so=0, se=0); for (i=1, #d, if (i%2, so += d[i], se += d[i])); so == se; } \\ Michel Marcus, Aug 04 2019

def c(n): s = str(n); return sum(map(int, s[::2])) == sum(map(int, s[1::2]))
def afind(limit):
    k, factk = 1, 1
    while k <= limit:
        if c(factk): print(k, end=", ")
        k += 1; factk *= k
afind(500) # Michael S. Branicky, Nov 18 2021

A309393 Let f(n) be equal to n + S(n) + S(S(n)) ... + S(S(S..(n))), where the last term is less than 10 and S(n) is the sum of digits. This is the sequence of numbers k such that the equation f(x) = k has a record number of solutions.

Original entry on oeis.org

1, 30, 66, 204, 819, 70032, 3000000000096

Offset: 1

Daniel Starodubtsev, Jul 28 2019

Conjecture from Daniel Starodubtsev and Dmitry Petukhov, Nov 19 2019: (Start)
a(8) = 20000000000000046;
a(9) = 8900000000000000000000127. (End)

			a(4) = 204, because 204 = f(179) = f(185) = f(191) = f(201), which has more solutions than any smaller number.

Cf. A007953, A006064.

T = 0*Range[10^5]; f[n_] := Block[{x=n, s=n}, While[x >= 10, x = Plus@@ IntegerDigits[x]; s += x]; s]; Do[v = f[i]; If[v <= 10^5, T[[v]]++], {i, 10^5}]; Flatten[Position[T, #, 1, 1] & /@ Range[6]] (* Giovanni Resta, Jul 30 2019 *)

f(n) = {s=n;m=n;while(sumdigits(s)>9,s=sumdigits(s);m+=s);if(n<10,m=0);m+sumdigits(s);}
g(n) = sum(k=1,n,f(k)==n);
lista(NN) = {x=1;print1(1);for(n=2,NN,if(g(n)>x,x=g(n);print1(", ",n)))} \\ Jinyuan Wang, Jul 31 2019

a(7) from Bert Dobbelaere, Aug 15 2019

cp's OEIS Frontend

User: Daniel Starodubtsev

A345338 Integers whose Reverse And Add trajectory reaches its first prime after a record number of iterations (at least one iteration must be performed).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

A335787 Emirps containing only the digits 1 and 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A334170 Emirps containing only the digits 1 and 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A333523 Number of iterations of Reverse And Add needed to reach a number divisible by n (or 0 if such a number is never reached).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A333475 Numbers k such that S(2^k) is a perfect square, where S(t) is the sum of decimal digits of t.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A333474 Numbers k such that 2^k + 1 is divisible by the sum of its decimal digits.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Extensions

A309527 Numbers k such that 6^k + 17 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A309726 Numbers k such that k^2 - 12 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs