A155045 -id:A155045 - cp's OEIS frontend

A355773 Numbers all of whose divisors are members of A333369.

1, 3, 5, 7, 9, 13, 15, 17, 19, 31, 35, 37, 39, 51, 53, 57, 59, 71, 73, 79, 91, 93, 95, 97, 111, 137, 139, 153, 157, 159, 173, 179, 193, 197, 221, 223, 227, 229, 317, 333, 359, 371, 379, 395, 397, 443, 449, 519, 537, 571, 579, 591, 593, 661, 663, 669, 719, 739

Offset: 1

Bernard Schott, Jul 18 2022

All terms are necessarily odd because 2 is not in A333369

			111 is a term since all the divisors of 111, i.e., 1, 3, 37 and 111, are in A333369.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Project Euler, Problem 520: Simbers.

Cf. A333369, A353735, A355770, A355771, A355772.
Similar sequences: A062687, A190217, A329419, A337741
.
Subsequences: A155045, A355853.

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[1000], AllTrue[Divisors[#], simQ] &] (* Amiram Eldar, Jul 19 2022 *)

issimber(m) = my(d=digits(m), s=Set(d)); for (i=1, #s, if (#select(x->(x==s[i]), d) % 2 != (s[i] % 2), return (0))); return (1); \\ A333369
isok(k) = fordiv(k, d, if (!issimber(d), return(0))); return(1); \\ Michel Marcus, Jul 19 2022

from sympy import divisors, isprime
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(740) if ok(k)]) # Michael S. Branicky, Jul 24 2022

A323391 Primes containing nonprime digits (from 1 to 9) in their decimal expansion and whose digits are distinct, i.e., consisting of only digits 1, 4, 6, 8, 9.

Original entry on oeis.org

19, 41, 61, 89, 149, 419, 461, 491, 619, 641, 691, 941, 1489, 4691, 4861, 6481, 6491, 6841, 8419, 8461, 8641, 8941, 9461, 14869, 46819, 48619, 49681, 64189, 64891, 68491, 69481, 81649, 84691, 84961, 86491, 98641

Offset: 1

Bernard Schott, Jan 13 2019

There are only 36 terms in this sequence, which is a finite subsequence of A152313.
Two particular examples:
6481 is also the smallest prime formed from the concatenation of two consecutive squares.
81649 is the only prime containing all the nonprime positive digits such that every string of two consecutive digits is a square.

			14869 is the smallest prime that contains all the nonprime positive digits; 98641 is the largest one.

Chris K. Caldwell and G. L. Honaker, Jr., 81649, Prime Curios!

Subsequence of A152313. Subsequence of A029743. Subsequence of A155024 (with distinct nonprime digits but with 0) and of A034844.

Cf. A029743 (with distinct digits), A124674 (with distinct prime digits), A155045 (with distinct odd digits), A323387 (with distinct square digits), A323578 (with distinct digits for which parity of digits alternates).

Select[Union@ Flatten@ Map[FromDigits /@ Permutations@ # &, Rest@ Subsets@ {1, 4, 6, 8, 9}], PrimeQ] (* Michael De Vlieger, Jan 19 2019 *)

isok(p) = isprime(p) && (d=digits(p)) && vecmin(d) && (#Set(d) == #d) && (#select(x->isprime(x), d) == 0); \\ Michel Marcus, Jan 14 2019

A355853 Primes in A333369.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 31, 37, 53, 59, 71, 73, 79, 97, 137, 139, 157, 173, 179, 193, 197, 223, 227, 229, 317, 359, 379, 397, 443, 449, 571, 593, 661, 719, 739, 751, 881, 883, 887, 937, 953, 971, 1009, 1117, 1151, 1171, 1223, 1229, 1447, 1511, 1579, 1597, 1663, 1667, 1669

Offset: 1

Bernard Schott, Jul 19 2022

			443 is prime and 443 has two 4's and one 3 in its decimal expansion, hence 443 is a term.

Intersection of A000040 and A333369.
Subsequence of A355773.
Supersequence of A155045.
Similar sequences: A002385, A004023.

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Prime[Range[300]], simQ] (* Amiram Eldar, Jul 19 2022 *)

issimber(m) = my(d=digits(m), s=Set(d)); for (i=1, #s, if (#select(x->(x==s[i]), d) % 2 != (s[i] % 2), return (0))); return (1); \\ A333369
isok(m) = isprime(m) && issimber(m); \\ Michel Marcus, Jul 19 2022

from itertools import count, islice
from sympy import isprime
def A355853_gen(startvalue=1): # generator of terms
    return filter(lambda n:not any((str(n).count(d)^int(d))&1 for d in set(str(n))) and isprime(n),count(max(startvalue,1)))
A355853_list = list(islice(A355853_gen(),30)) # Chai Wah Wu, Jul 21 2022

Extended by Michel Marcus, Jul 19 2022

A323579 Primes formed by using the four terminal digits of multidigit primes and whose digits are distinct, i.e., consisting of only digits 1, 3, 7, 9.

Original entry on oeis.org

3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173, 9371

Offset: 1

Bernard Schott, Jan 24 2019

There are only 31 terms in this sequence, which is a finite subsequence of A091633 and of A155045.

719 is also the third factorial prime belonging to A055490.

			1973 and 9371 are respectively the smallest and the largest primes formed with the four digits that can end multidigit primes.

Chris K. Caldwell and G. L. Honaker, Jr., 9371, Prime Curios!

Subsequence of A091633 and hence of A030096.

Cf. A029743 (with distinct digits), A124674 (with distinct prime digits), A155024 (with distinct nonprime digits but with 0), A155045 (with distinct odd digits), A323387 (with distinct square digits), A323391 (with distinct nonprime digits), A323578 (with distinct digits for which parity of digits alternates).

With[{w = Select[Range@ 10, GCD[#, 10] == 1 &]}, Select[FromDigits /@ Permutations[w, Length@ w], PrimeQ]] (* Michael De Vlieger, Feb 03 2019 *)
Select[FromDigits/@Flatten[Permutations/@Subsets[{1,3,7,9}],1],PrimeQ]//Union (* Harvey P. Dale, Apr 20 2025 *)

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A355773 Numbers all of whose divisors are members of A333369.

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A323391 Primes containing nonprime digits (from 1 to 9) in their decimal expansion and whose digits are distinct, i.e., consisting of only digits 1, 4, 6, 8, 9.

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A355853 Primes in A333369.

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A323579 Primes formed by using the four terminal digits of multidigit primes and whose digits are distinct, i.e., consisting of only digits 1, 3, 7, 9.

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