Randy L. Ekl - cp's OEIS frontend

A378949 Numbers with monotonically increasing digits, increasing by only 0 or 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 22, 23, 33, 34, 44, 45, 55, 56, 66, 67, 77, 78, 88, 89, 99, 111, 112, 122, 123, 222, 223, 233, 234, 333, 334, 344, 345, 444, 445, 455, 456, 555, 556, 566, 567, 666, 667, 677, 678, 777, 778, 788, 789, 888, 889, 899, 999

Offset: 1

Randy L. Ekl, Dec 18 2024

			33 is a term since the digits are monotonically increasing and their difference is 0.
34 is also a term since the digits are monotonically increasing and their difference is 1.
35 is not a term since the difference in consecutive digits is not 0 or 1.
32 is not a term since the digits are decreasing.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A378808, A378774.

extend:= proc(k) local m,d;
  m:= 10^ilog10(k);
  d:= floor(k/m);
  if d = 1 then 10*m+k else (d-1)*10*m+k, d*10*m+k fi
end proc:
R:= $1..9:
A:= [R]:
for i from 2 to 5 do
  A:= map(extend,A);
  R:= R, op(sort(A));
od:
R; # Robert Israel, Jan 18 2025

Select[Range[1000],SubsetQ[{0, 1}, Union@ Differences@ IntegerDigits[#]] &] (* James C. McMahon, Dec 21 2024 *)

from itertools import count, islice
def bgen(last, d):
    if d == 0: yield tuple(); return
    t = (1, 9) if last == None else (last, min(last+1, 9))
    for i in range(t[0], t[1]+1): yield from ((i, )+r for r in bgen(i, d-1))
def agen(): # generator of terms
    yield from (int("".join(map(str, i))) for d in count(1) for i in bgen(None, d))
print(list(islice(agen(), 62))) # Michael S. Branicky, Dec 18 2024

Offset corrected by James C. McMahon, Dec 21 2024

A378775 Prime numbers with monotonically decreasing digits, differing by at most 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 43, 211, 433, 443, 877, 887, 2111, 2221, 3221, 5443, 8887, 9887, 22111, 33211, 43321, 54443, 65543, 76543, 98887, 99877, 322111, 332221, 443221, 444443, 766543, 888887, 988877, 2221111, 3221111, 3222211, 3222221, 3333221, 4322221, 4433333, 4443221

Offset: 1

Randy L. Ekl, Dec 06 2024

			211 is a term since 211 is a prime number, the digits of 211 are monotonically decreasing, and the difference between consecutive digits is at most 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Primes in A378808.

Cf. A378774, A028864.

extend:= proc(x) local d,s,i;
  d:= ilog10(x);
  s:= floor(x/10^d);
  seq(10^(d+1)*i+x, i=s .. min(9,s+1))
end proc:
R:= 2,3,5,7: count:= 4:
M:= [1,3,7,9];
for d from 2 while count < 100 do
  M:= map(extend,M):
  S:= sort(select(isprime,M));
  count:= count+nops(S);
  R:= R,op(S);
od:
R; # Robert Israel, Feb 09 2025

Select[Prime[Range[312218]],ContainsOnly[Drop[IntegerDigits[#],-1]-Rest[IntegerDigits[#]],{0,1}]&] (* James C. McMahon, Dec 21 2024 *)

isok(p) = if (isprime(p), my(d=digits(p), dd = vector(#d-1, k, d[k+1]-d[k])); (#dd==0) || ((vecmin(dd)>=-1) && (vecmax(dd)<=0))); \\ Michel Marcus, Dec 09 2024

A378774 Prime numbers with monotonically increasing digits, increasing by only 0 or 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 67, 89, 223, 233, 677, 1123, 1223, 2333, 4567, 7789, 8999, 23333, 45667, 45677, 55667, 67777, 67789, 77899, 78889, 112223, 344567, 445567, 555677, 556789, 566677, 567899, 666667, 788999, 1112333, 2222333, 3445567, 3445667, 3455567, 3456667, 4455667, 4456789, 4556777

Offset: 1

Randy L. Ekl, Dec 06 2024

			223 is a term since the digits of 223 are monotonically increasing, consecutive digits differ by at most 1, and 223 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A028864, A378775.

extend:= proc(x) local d,s,i;
  d:= ilog10(x);
  s:= floor(x/10^d);
  seq(10^(d+1)*i+x, i=max(1,s-1) .. s)
end proc:
R:= 2,3,5,7: count:= 4:
M:= [1,3,7,9];
for d from 2 while count < 100 do
  M:= map(extend,M):
  S:= sort(select(isprime,M));
  count:= count+nops(S);
  R:= R,op(S);
od:
R; # Robert Israel, Feb 09 2025

Select[Prime[Range[319629]], ContainsOnly[Rest[IntegerDigits[#]]-Drop[IntegerDigits[#], -1], {0, 1}]&] (* James C. McMahon, Dec 21 2024 *)

isok(p) = if (isprime(p), my(d=digits(p), dd = vector(#d-1, k, d[k+1]-d[k])); (#dd==0) || ((vecmin(dd)>=0) && (vecmax(dd)<=1))); \\ Michel Marcus, Dec 09 2024

A378808 Numbers with monotonically decreasing digits, decreasing by only 0 or 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 32, 33, 43, 44, 54, 55, 65, 66, 76, 77, 87, 88, 98, 99, 100, 110, 111, 210, 211, 221, 222, 321, 322, 332, 333, 432, 433, 443, 444, 543, 544, 554, 555, 654, 655, 665, 666, 765, 766, 776, 777, 876, 877, 887, 888, 987, 988, 998, 999

Offset: 1

Randy L. Ekl, Dec 07 2024

			32 is a term since it has monotonically decreasing digits whose difference is at most 1.
33 is a term since it also has monotonically decreasing digits whose difference is at most 1.

Cf. A009996, A064222, A378775.

Select[Range[999],SubsetQ[{0,1},-Differences[IntegerDigits[#]]] &] (* Stefano Spezia, Dec 08 2024 *)

from itertools import count, islice
def bgen(last, d):
    if d == 0: yield tuple(); return
    t = (1, 9) if last == None else (max(0, last-1), last)
    for i in range(t[0], t[1]+1): yield from ((i,)+r for r in bgen(i, d-1))
def agen(): # generator of terms
    yield from (int("".join(map(str, i))) for d in count(1) for i in bgen(None, d))
print(list(islice(agen(), 62))) # Michael S. Branicky, Dec 08 2024

A305579 Square array read by antidiagonals upwards in which row k has k as its first term and each subsequent term is the least possible value such that the sum of any 2 or more terms does not equal a prime.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 5, 6, 87, 5, 5, 7, 8, 87, 6, 7, 11, 83, 10, 1151, 7, 8, 9, 29, 235, 12, 5371, 8, 8, 10, 79, 215, 395, 14, 199276, 9, 10, 13, 12, 131, 511, 5275, 16, 32281747, 10, 11, 12, 37, 14, 196, 8729, 76128, 18, 16946784207, 11, 11, 13, 14, 67, 16, 3983, 20526, 9782734, 20

Offset: 1

Randy L. Ekl and Robert G. Wilson v, Jun 05 2018

Rows which appear to have consecutive even numbers are for k = 2, 6, 8, 14, 18, 20, 26, 36, 44, 48, 50, 54,56, 68, 74, 78, 86, 96, 114, ..., .

Conjecture: these row terms are a proper subset of A005843.

			Row 1 is A133660 and is a good illustration of the definition.
Array begins:
============================================================================
k\n|  1   2   3   4    5     6      7       8         9           10
---|------------------------------------------------------------------------
1  |  1,  3,  5, 87, 113, 1151,  5371, 199276, 32281747, 16946784207, ..., ;
2  |  2,  4,  6,  8,  10,   12,    14,     16,       18,          20, ..., ;
3  |  3,  5,  7, 83, 235,  395,  5275,  76128,  9782734, ..., ;
4  |  4,  5, 11, 29, 215,  511,  8729,  20526,  9745499, ..., ;
5  |  5,  7,  9, 79, 131,  196,  3983,  16380,   270270, ..., ;
6  |  6,  8, 10, 12,  14,   16,    18,     20,       22,          24, ..., ;
7  |  7,  8, 13, 37,  67, 1087,  5128, 137886,  6353767, ..., ;
8  |  8, 10, 12, 14,  16,   18,    20,     22,       24,          26, ..., ;
9  |  9, 11, 13, 71, 112,  281,  1952, 147630,  1729159, ..., ;
10 | 10, 11, 14, 25,  94,  756,  2394,  28480,  1466566, ..., ;
11 | 11, 13, 14, 25, 109,  559,  2719,  57985,  2589731, ..., ;
12 | 12, 13, 14, 37,  79,  673,  2929, 113256,  9708060, ..., ;
13 | 13, 14, 19, 31,  97,  882,  2028, 161340,  3635970, ..., ;
14 | 14, 16, 18, 20,  22,   24,    26,     28,       30,          32, ..., ;
15 | 15, 17, 18, 31, 137,  502,  7983, 599346, 27105801, ..., ;
16 | 16, 17, 18, 47, 107,  395,  6480,  91140,   467730, ..., ;
17 | 17, 18, 21, 31,  77,  637,  3609,  77910,   652680, ..., ;
18 | 18, 20, 22, 24,  26,   28,    30,     32,       34,          36, ..., ;
19 | 19, 20, 25, 30,  61,  235,  2965,   4415,   394170,     5769540, ..., ;
20 | 20, 22, 24, 26,  28,   30,    32,     34,       36,          38, ..., ;
21 | 21, 23, 25, 47,  73,  797, 20419, 235665,      ..., ;
22 | 22, 23, 27, 42,  69,  462,   672,    783,    71652,      935298, ..., ;
23 | 23, 25, 26, 37,  73, 1555,  4219, 196260,  3698520, ..., ;
24 | 24, 25, 26, 31, 193,  504,  3756,  91831,  7703843, ..., ;
25 | 25, 26, 29, 31,  39,  750,  4350,  85830,   661350, ..., ;
26 | 26, 28, 30, 32,  34,   36,    38,     40,       42,          44, ..., ;
27 | 27, 28, 29, 35, 232,  888,  5670, 134400,  4058376, ..., ;
28 | 28, 29, 34, 53,  59, 1045,  3696, 249240,  9475589, ..., ;
29 | 29, 31, 33, 55,  57,  674,  6581, 126272,  2549747, ..., ;
30 | 30, 32, 33, 52,  60,   63,    90,    120,      150,         180, ..., ;
31 | 31, 32, 33, 54,  90,  714,  9450, 188850,  2598573, ..., ;
32 | 32, 33, 37, 45, 138,  597,  2703, 101055,  2754885, ..., ;
33 | 33, 35, 37, 47, 133,  555,  4155, 332885,  3090195, ..., ;
34 | 34, 35, 41, 43,  77,  594,  2940,  35700,  2323246, ..., ;
35 | 35, 37, 39, 43, 210, 1061, 10125, 372955, 30373014, ..., ;
36 | 36, 38, 40, 42,  44,   46,    48,     50,       52,          54, ..., ;
37 | 37, 38, 39, 47,  48,  631,  8862, 124851,  4972506, ..., ;
..., etc.

Cf. A005843, A052349, A133660, A133661, first column: A000027.

(* first do *) Needs["Combinatorica`"] (* then *) lst = {k}; g[k_] := Block[{j = 1, l = 2^Length@lst}, While[j < l && !PrimeQ[Plus @@ NthSubset[j, lst] + k], j++ ]; If[j == l, False, True]]; f[n_] := Block[{k = lst[[-1]] + 1}, While[PrimeQ@k || g[k] == True, k++; k++ ]; AppendTo[lst, k]; k]; Do[ Print@ f@ n, {n, 10}] (* Robert G. Wilson v, Jun 05 2018 *)

A280636 Semiprime numbers whose digit string can be partitioned into three parts such that the product of the first two parts equals the third part.

Original entry on oeis.org

111, 122, 133, 155, 166, 177, 326, 339, 515, 717, 818, 2918, 3721, 5315, 6742, 7214, 7642, 9327, 9763, 11111, 11333, 11555, 12929, 13333, 13535, 13565, 13791, 13939, 14114, 14141, 14242, 14545, 15115, 15151, 15353, 15454, 15757, 16161, 16565

Offset: 1

Randy L. Ekl, Jan 06 2017

Could be called semiprime area numbers. If the first set of digits of the number is considered the length, the second set of digits of the number is considered the width, and the final set of digits of the number is considered the area, then length * width = area.

			2918 is semiprime (2*1459) and an area number (2*9 = 18).
6742 is semiprime (2*3371) and an area number (6*7 = 42).
Leading zeros are not allowed.

Cf. A088294, A280635.

A001358 INTERSECT A280635.

A280635 Numbers whose digit string can be partitioned into three nonempty parts such that the product of the first two parts equals the third part.

Original entry on oeis.org

111, 122, 133, 144, 155, 166, 177, 188, 199, 212, 224, 236, 248, 313, 326, 339, 414, 428, 515, 616, 717, 818, 919, 2510, 2612, 2714, 2816, 2918, 3412, 3515, 3618, 3721, 3824, 3927, 4312, 4416, 4520, 4624, 4728, 4832, 4936, 5210, 5315, 5420, 5525, 5630, 5735

Offset: 1

Randy L. Ekl, Jan 06 2017

Could be called "area numbers" since if the first set of digits is the length, and the second set of digits is the width, then the last set of digits is the area, with length * width = area.

			236 is in the sequence since 2*3=6. 3515 is in the sequence since 3*5=15. Leading zeros are not allowed, thus 2036 (2*03=6) is not included.

Cf. A088294, A280636.

read("transforms") : # implements digcatL
isA280635 := proc(n)
    local dgs,spl1,spl2,dgs1,dgs2,dgs3;
    dgs := convert(n,base,10) ;
    if nops(dgs) >= 3 then
        for spl1 from 1 to nops(dgs)-2 do
        for spl2 from spl1+1 to nops(dgs)-1 do
            if op(-1,dgs) <> 0 and op(spl1,dgs) <> 0 and op(spl2,dgs) <> 0 then
                dgs1 := ListTools[Reverse]([op(spl2+1..nops(dgs),dgs)]) ;
                dgs2 := ListTools[Reverse]([op(spl1+1..spl2,dgs)]) ;
                dgs3 := ListTools[Reverse]([op(1..spl1,dgs)]) ;
                if digcatL(dgs1)*digcatL(dgs2) = digcatL(dgs3) then
                    return true;
                end if
            end if;
        end do:
        end do:
        false ;
    else
        false;
    end if;
end proc:
for n from 100 do
    if isA280635(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Jan 10 2017

With[{nn = 1}, Union@ Flatten@ Table[FromDigits@ Flatten@ Map[IntegerDigits, {n, k, n k}], {n, 10^nn - 1}, {k, 10^nn - 1}]] (* Michael De Vlieger, Jan 07 2017 *)

A280406 Odd semiprimes that can be represented as 2p+3q, where p and q are primes, in an increasing number of ways.

Original entry on oeis.org

15, 25, 55, 91, 115, 187, 215, 235, 335, 403, 415, 515, 535, 655, 695, 835, 955, 1115, 1195, 1393, 1405, 1555, 1895, 1963, 2095, 2155, 2335, 2395, 2615, 2815, 2995, 3235, 3295, 3635, 3935, 3985, 4295, 4415, 4435, 4735, 4855, 4915, 5095, 5515, 5815, 6415, 6835, 7135, 7195, 7415, 8035, 8135

Offset: 1

Randy L. Ekl, Jan 02 2017

nonn

It is conjectured that all terms of this sequence above 1963 are divisible by 5. Initial search to 1000000.

Cf. A046315, A280389.

A280405 Odd semiprimes that cannot be represented as 2p+3q, where p and q are primes.

Original entry on oeis.org

9, 33, 51, 69, 87, 111, 123, 141, 159, 177, 201, 213, 237, 249, 267, 291, 303, 321, 339, 381, 393, 411, 447, 471, 489, 501, 519, 537, 573, 591, 633, 669, 681, 699, 717, 753, 771, 789, 807, 831, 843, 879, 921, 933, 951, 993

Offset: 1

Randy L. Ekl, Jan 02 2017

nonn

			33 = 3*11 is a semiprime, and cannot be represented as twice a prime plus three times a prime.  21=3*7 is a semiprime which CAN be represented in that form, i.e. 2*3+3*5, and thus is not in this sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A046315 (odd semiprimes)

Cf. A280389 (odd semiprimes which can be represented as 2p+3q, where p and q are prime)

N:= 10^4: # to get all terms <= N
Primes:= select(isprime,[2,seq(i,i=3..N/2,2)]):
Cands:= {seq(i,i=1..N,2)} minus {seq(seq(2*p+3*q,p=Primes),q=Primes)}:
sort(convert(select(numtheory:-bigomega=2, Cands),list)); # Robert Israel, Jan 09 2017

A280389 Odd semiprimes that can be represented as 2p+3q, where p and q are primes.

Original entry on oeis.org

15, 21, 25, 35, 39, 49, 55, 57, 65, 77, 85, 91, 93, 95, 115, 119, 121, 129, 133, 143, 145, 155, 161, 169, 183, 185, 187, 203, 205, 209, 215, 217, 219, 221, 235, 247, 253, 259, 265, 287, 289, 295, 299, 301, 305, 309, 319, 323, 327, 329, 335, 341, 355, 361, 365, 371, 377, 391, 395, 403

Offset: 1

Randy L. Ekl, Jan 02 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A046315 (odd semiprimes).

Cf. A280405 (odd semiprimes which can NOT be represented as 2p+3q, where p and q are prime).

N:= 10^3: # to get all terms <= N
Primes:= select(isprime, [2, seq(i, i=3..N/2, 2)]):
Cands:= select(t -> t::odd and t <= N, {seq(seq(2*p+3*q, p=Primes), q=Primes)}):
sort(convert(select(numtheory:-bigomega=2, Cands), list)); # Robert Israel, Jan 09 2017

cp's OEIS Frontend

User: Randy L. Ekl

A378949 Numbers with monotonically increasing digits, increasing by only 0 or 1.

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A378775 Prime numbers with monotonically decreasing digits, differing by at most 1.

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A378774 Prime numbers with monotonically increasing digits, increasing by only 0 or 1.

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A378808 Numbers with monotonically decreasing digits, decreasing by only 0 or 1.

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A305579 Square array read by antidiagonals upwards in which row k has k as its first term and each subsequent term is the least possible value such that the sum of any 2 or more terms does not equal a prime.

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A280636 Semiprime numbers whose digit string can be partitioned into three parts such that the product of the first two parts equals the third part.

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A280635 Numbers whose digit string can be partitioned into three nonempty parts such that the product of the first two parts equals the third part.

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Maple

Mathematica

A280406 Odd semiprimes that can be represented as 2p+3q, where p and q are primes, in an increasing number of ways.

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A280405 Odd semiprimes that cannot be represented as 2p+3q, where p and q are primes.

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