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A387198 Smallest integer that can be expressed as the sum of k different primes, for all k’s between 2 and n, with n >= 2.

2, 5, 10, 21, 28, 45, 58, 81, 106, 129, 166, 201, 238, 285, 338, 399, 440, 511, 572, 645, 718, 811, 888, 985, 1064, 1173, 1268, 1383, 1484, 1611, 1730, 1869, 1988, 2139, 2276, 2439, 2594, 2769, 2924, 3111, 3266, 3459, 3638, 3835, 4028, 4245, 4454, 4665, 4888, 5121, 5356

Offset: 1

Paolo P. Lava, Aug 21 2025

Lower bounds are listed in A007504.

			a(2) = 5 because 5 = 2 + 3;
a(3) = 10 because 10 = 3 + 7 = 2 + 3 + 5;
a(4) = 21 because 21 = 2 + 19 = 3 + 5 + 13 = 2 + 3 + 5 + 11;
a(5) = 28 because 28 = 5 + 23 = 2 + 7 + 19 = 3 + 5 + 7 + 13 = 2 + 3 + 5 + 7 + 11; etc.

David A. Corneth, Table of n, a(n) for n = 2..500
Carlos Rivera, Puzzle 1233. Sum of Primes such that... , The Prime Puzzles & Problems Connection.

Cf. A007504, A073619, A090700, A387215.

a(22) and more terms from David A. Corneth, Aug 21 2025

a(1) prepended by David A. Corneth, Aug 26 2025

A387068 Number of equivalence classes (up to homeomorphism) of finite, connected ribbon graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

Original entry on oeis.org

1, 3, 31, 1831, 462638, 243066565

Offset: 0

J. J. P. Veerman, Aug 15 2025

n is called the least separating genus.

These numbers are larger than those of A387067, because different embeddings of the same graph are counted separately. For instance, there is a ribbon graph for the bouquet of 3 circles with least separating genus 1 and a different embedding with least separating genus 2.

			For genus 0: thickened circle.
For genus 1: thickened versions of a bouquet of 2 circles, bouquet of 3 circles, 4-fold multi-edge.

C. N. Aagaard and J. J. P. Veerman, Classification of Minimal Separating Sets of Low Genus Surfaces, Topology and its Applications, Accepted, 2025.

Cf. A384639, A387066, A387067.

A387067 Number of equivalence classes (up to graph homeomorphism) of finite,connected graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

Original entry on oeis.org

1, 3, 17, 164, 3096, 111445

Offset: 0

J. J. P. Veerman, Aug 15 2025

n is called the minimal separating genus.

			For genus 0: only the circle.
For genus 1: bouquet of 2 circles, bouquet of 3 circles, 4-fold multi-edge.

C. N. Aagaard and J. J. P. Veerman, Classification of Minimal Separating Sets of Low Genus Surfaces, Topology and its Applications, Accepted, 2025.

Cf. A384639, A387066, A387068.

A387066 Number of equivalence classes (up to graph homeomorphism) of finite graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

Original entry on oeis.org

1, 4, 21, 191, 3338, 115438

Offset: 0

J. J. P. Veerman, Aug 15 2025

n is called the least separating genus.

			For genus 0: only the circle.
For genus 1: 2 circles, bouquet of 2 circles, bouquet of 3 circles, 4-fold multi-edge.

C. N. Aagaard and J. J. P. Veerman, Classification of Minimal Separating Sets of Low Genus Surfaces, Topology and its Applications, Accepted, 2025.
J. Bernhard and J. J. P. Veerman, The Topology of Surface Mediatrices, Topology and its Applications, 154, 54-68, 2007.

Cf. A384639, A387067, A387068.

A384639 Number of equivalence classes (up to graph homeomorphism) of finite graphs that have an embedding in an orientable surface of genus n which minimally separates the surface (that is, no proper subset of the embedding separates the genus n surface).

Original entry on oeis.org

1, 5, 26, 217, 3555, 118993

Offset: 0

J. J. P. Veerman, Aug 13 2025

			For genus 0: only the circle.
For genus 1: 1 circle, 2 circles, bouquet of 2 circles, bouquet of 3 circles, 4-fold multi-edge.

C. N. Aagaard and J. J. P. Veerman, Classification of Minimal Separating Sets of Low Genus Surfaces, Topology and its Applications, Accepted, 2025.
J. Bernhard and J. J. P. Veerman, The Topology of Surface Mediatrices, Topology and its Applications, 154, 54-68, 2007.

Cf. A387066, A387067, A387068.

A386935 Integers with the same arithmetic mean for divisors and anti-divisors.

Original entry on oeis.org

3, 15, 135, 376, 6956, 1913646, 1838558856

Offset: 1

Paolo P. Lava, Aug 09 2025

For the listed numbers the arithmetic means are 2, 6, 30, 90, 1064, 97128, 143824680, ...

a(8) > 10^10, if it exists. - Amiram Eldar, Aug 12 2025

			Divisors of 135 are 8: 1, 3, 5, 9, 15, 27, 45, 135. Their sum is 240 and 240/8 = 30.
Anti-divisors of 135 are 7: 2, 6, 10, 18, 30, 54, 90. Their sum is 210 and 210/7 = 30.

Cf. A000203/A000005, A066417/A066272.

Cf. A003601, A192284.

with(numtheory): P:=proc(q) local a, b, k, n, v; v:=[];
for n from 3 to q do k:=2; a:=0; b:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then b:=b+1; a:=a+k; fi; od;
if sigma(n)/tau(n)=a/b then v:=[op(v), n]; fi; od; op(v); end: P(10^4);

from itertools import count, islice
from sympy.ntheory.factor_ import divisor_sigma, antidivisors
def A386935_gen(startvalue=3): # generator of terms >= startvalue
    for k in count(max(startvalue,3)):
        if divisor_sigma(k)*len(d:=antidivisors(k))==divisor_sigma(k,0)*sum(d):
            yield k
A386935_list = list(islice(A386935_gen(),5)) # Chai Wah Wu, Aug 12 2025

a(6) from Michel Marcus, Aug 09 2025

a(7) from Amiram Eldar, Aug 10 2025

A385245 Primes that are no longer prime if in their binary representation any single bit is flipped but stay prime if a 1 bit is prepended.

Original entry on oeis.org

223, 257, 509, 787, 853, 877, 1259, 1451, 1973, 2917, 3511, 5099, 6287, 6521, 7841, 8171, 8923, 9319, 10567, 11353, 12517, 12637, 12763, 13687, 14107, 14629, 15217, 15607, 16943, 17519, 18089, 18593, 18743, 19139, 20183, 20393, 20639, 21701, 22943, 26591, 26891

Offset: 1

Alois P. Heinz, Jul 28 2025

			257 = 100000001_2 and 769 = 1100000001_2 are primes and 256, 259, 261, 265, 273, 289, 321, 385, 1 are not prime. So 257 is a term.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Set difference of A137985 and A065092.

Cf. A000040, A002808.

q:= p-> (m-> andmap(isprime, [p, 2^(m+1)+p]) and not ormap
        (i->isprime(Bits[Xor](p, 2^i)), [$0..m]))(ilog2(p)):
select(q, [$2..27000])[];

Select[Prime[Range[3000]], PrimeQ[2^BitLength[#] + #] && NoneTrue[BitXor[#, 2^Range[0, BitLength[#] - 1]], PrimeQ] &] (* Paolo Xausa, Aug 05 2025 *)

{ A137985 } minus { A065092 }.

A386597 Number of distinct values of the permanent of an n X n (0,1)-matrix with exactly three 1's in each row.

Original entry on oeis.org

1, 4, 8, 16, 29, 50, 82

Offset: 3

Robert P. P. McKone, Jul 27 2025

a(n) > A185178(n) for n >= 4.
The permanents for a(n) contain all permanents from a(n-1).
a(10) >= 121.
a(11) >= 186.
a(12) >= 276.
a(13) >= 422.
a(14) >= 638.
a(15) >= 824.

Robert P. P. McKone, Complete frequency counts for each permanent for a(3)..a(9)
Robert P. P. McKone, C++ code for enumerating and counting each permanent's frequency for a(n)
Robert P. P. McKone, Known permanents through simulation for a(10)..a(15)

Cf. A185178 (number of distinct permanents with exactly three 1's in each row and column).

A386204 Number of distinct values of the determinant of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

Original entry on oeis.org

1, 2, 3, 3, 7, 11, 7

Offset: 3

Robert P. P. McKone, Jul 15 2025

a(10) >= 25.

Robert P. P. McKone, Python code to find A386204(n) unique determinants and one example matrix
Robert P. P. McKone, One example matrix for each known determinant for a(3)-a(9)

Cf. A185178 (distinct permanents).

Cf. A001501 (number of n X n (0,1)-matrix with exactly three 1's in each row and each column).

A386247 Primes containing 000 as a substring.

Original entry on oeis.org

10007, 10009, 40009, 70001, 70003, 70009, 90001, 90007, 100003, 100019, 100043, 100049, 100057, 100069, 130003, 140009, 150001, 160001, 160009, 170003, 180001, 180007, 200003, 200009, 200017, 200023, 200029, 200033, 200041, 200063, 200087, 220009, 230003, 240007

Offset: 1

Alois P. Heinz, Jul 16 2025

Differs from A164968 first at n=10: a(10) = 100019 < 200003 = A164968(10).

Alois P. Heinz, Table of n, a(n) for n = 1..20761

Cf. A000040, A164968, A243527, A166580, A166581, A166582, A167281, A131645, A167282, A167290, A167292, A386240.

Select[Prime[Range[1230, 25000]], StringContainsQ[IntegerString[#], "000"] &] (* Paolo Xausa, Jul 19 2025 *)

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A387198 Smallest integer that can be expressed as the sum of k different primes, for all k’s between 2 and n, with n >= 2.

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A387066 Number of equivalence classes (up to graph homeomorphism) of finite graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

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A384639 Number of equivalence classes (up to graph homeomorphism) of finite graphs that have an embedding in an orientable surface of genus n which minimally separates the surface (that is, no proper subset of the embedding separates the genus n surface).

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A386935 Integers with the same arithmetic mean for divisors and anti-divisors.

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A385245 Primes that are no longer prime if in their binary representation any single bit is flipped but stay prime if a 1 bit is prepended.

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Maple

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A386597 Number of distinct values of the permanent of an n X n (0,1)-matrix with exactly three 1's in each row.

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A386204 Number of distinct values of the determinant of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

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A386247 Primes containing 000 as a substring.

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