Zak Seidov - cp's OEIS frontend

A386295 Primes p such that p+1 is a triprime and 2*p+1 is prime.

11, 29, 41, 113, 173, 281, 641, 653, 761, 1901, 2273, 2693, 2741, 3413, 3593, 5441, 6053, 6113, 6521, 6581, 7121, 7841, 9293, 9473, 10253, 10733, 12101, 12821, 14081, 14621, 15233, 16493, 19301, 19373, 19553, 19913, 20441, 20693, 21341, 21701, 22433, 24473, 27281, 27581, 27893, 28793, 28901

Offset: 1

Zak Seidov and Robert Israel, Jul 17 2025

nonn

Sophie Germain primes of the form p*q*r - 1, where p, q and r are primes.

Except for 11, all terms == 5 (mod 12).

			a(3) = 41 is a term because it is prime, 41 + 1 = 42 = 2 * 3 * 7 is a triprime, and 41 * 2 + 1 = 83 is prime.

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

Cf. A014612.

Intersection of A005384 and A063639.

select(p -> isprime(p) and isprime(2*p+1) and numtheory:-bigomega(p+1) = 3, [seq(i,i=3..30000,2)]);

s= {}; Do[p = Prime[k]; If[3 == PrimeOmega[p + 1] && PrimeQ[2*p +1], AppendTo[s, p]], {k, 2000}];s

A384940 Odd semiprimes interleaved with even semiprimes.

Original entry on oeis.org

9, 4, 15, 6, 21, 10, 25, 14, 33, 22, 35, 26, 39, 34, 49, 38, 51, 46, 55, 58, 57, 62, 65, 74, 69, 82, 77, 86, 85, 94, 87, 106, 91, 118, 93, 122, 95, 134, 111, 142, 115, 146, 119, 158, 121, 166, 123, 178, 129, 194, 133, 202, 141, 206, 143, 214, 145, 218, 155, 226, 159, 254, 161, 262, 169, 274, 177

Offset: 1

Zak Seidov and Robert Israel, Jun 13 2025

			a(3) = A046315(2) = 15 is the second odd semiprime.
a(4) = A100484(2) = 6 is the second even semiprime.

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

Cf. A001358, A046315, A100484

A:= select(t -> numtheory:-bigomega(t)=2, [seq(i,i=1..1000,2)]):
B:= select(t -> numtheory:-bigomega(t)=2, [seq(i,i=2..1000,2)]):
seq(op([A[i],B[i]]),i=1..min(nops(A),nops(B)));

from math import isqrt
from sympy import primepi, primerange, prime
def A384940(n):
    if n&1:
        m = n+1>>1
        def bisection(f,kmin=0,kmax=1):
            while f(kmax) > kmax: kmax <<= 1
            kmin = kmax >> 1		
            while kmax-kmin > 1:
                kmid = kmax+kmin>>1
                if f(kmid) <= kmid:
                    kmax = kmid
                else:
                    kmin = kmid
            return kmax
        def f(x): return int(m+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
        return bisection(f,m,m)
    else:
        return prime(n>>1)<<1 # Chai Wah Wu, Jun 17 2025

a(2*k-1) = A046315(k).

a(2*k) = A100484(k).

A383665 a(n) is the least number k such that k, k - s and k + s all have n prime divisors, counted with multiplicity, where s is the sum of the decimal digits of k.

Original entry on oeis.org

15, 102, 204, 408, 3078, 14496, 88448, 128768, 6857312, 111411968, 844844000, 6059394048, 13384999936, 948305874880, 6373064359936, 186505184249928

Offset: 2

Zak Seidov and Robert Israel, May 04 2025

k - s is always divisible by 9, so a(1) does not exist, and a(2) = 15 is the only semiprime k such that k, k - s and k + s are all semiprimes.

			a(4) = 204 because 204 has digit sum 6, 204 - 6 = 198 = 2 * 3^2 * 11, 204 = 2^2 * 3 * 17 and 204 + 6 = 210 = 2 * 3 * 5 * 7 all have 4 prime divisors, counted with multiplicity, and 204 is the least number that works.

Cf. A001222, A007953, A062028, A066568, A381851, A382996.

f:= proc(n) uses priqueue; local pq, t,x,s,p,i;
      initialize(pq);
      insert([-2^n, 2$n], pq);
      do
        t:= extract(pq);
        x:= -t[1];
        s:= convert(convert(x,base,10),`+`);
        if numtheory:-bigomega(x-s) = n and numtheory:-bigomega(x+s) = n then return x fi;
        p:= nextprime(t[-1]);
        for i from n+1 to 2 by -1 while t[i] = t[-1] do
          insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
        od;
      od;
end proc:
map(f, [$2..14]);

generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, my(s=sumdigits(m*q)); if(bigomega(m*q+s) == k && bigomega(m*q-s) == k, listput(list, m*q))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, May 24 2025

A001222(a(n)) = A001222(A062028(a(n))) = A001222(A066568(a(n))) = n.

a(15) from Michael S. Branicky, May 08 2025

a(16)-a(17) from Daniel Suteu, May 24 2025

A383469 Semiprimes k such that A001358(k) + k and A001358(k) - k are also semiprimes.

Original entry on oeis.org

4, 6, 46, 55, 87, 93, 94, 206, 298, 326, 362, 382, 394, 542, 562, 629, 758, 841, 866, 926, 993, 1046, 1079, 1147, 1167, 1234, 1317, 1469, 1477, 1561, 1774, 1895, 2047, 2227, 2245, 2515, 2638, 2705, 2894, 2902, 3007, 3031, 3063, 3202, 3269, 3409, 3453, 3466, 3487, 3809, 3891, 4058, 4174, 4207

Offset: 1

Zak Seidov and Robert Israel, Apr 27 2025

nonn

			a(3) = 46 is a term because 46 = 2 * 23 is a semiprime, A001358(46) = 141, and 141 - 46 = 95 = 5 * 19 and 141 + 46 = 187 = 11 * 17 are semiprimes.

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

Cf. A001358, A383468.

k:= 0: K:= NULL: count:= 0:
for s from 1 while count < 100 do
  if numtheory:-bigomega(s) = 2 then
    k:= k+1;
    if andmap(t -> numtheory:-bigomega(t) = 2, [k, s-k, s+k]) then
      K:= K, k; count:= count+1;
    fi
  fi;
od:
K;

A001358(a(k)) = A383468(k).

A383468 Semiprimes s = A001358(k) such that k, s - k and s + k are also semiprimes.

Original entry on oeis.org

10, 15, 141, 166, 274, 298, 299, 687, 995, 1115, 1227, 1299, 1345, 1891, 1945, 2194, 2661, 2998, 3093, 3287, 3566, 3781, 3902, 4174, 4262, 4497, 4798, 5378, 5414, 5758, 6609, 7094, 7666, 8354, 8434, 9566, 10041, 10342, 11051, 11091, 11486, 11582, 11702, 12279, 12574, 13154, 13346, 13387, 13466

Offset: 1

Zak Seidov and Robert Israel, Apr 27 2025

nonn

Except for a(1) = 10 = A001358(4), s and k always have different parities.

			a(3) = 141 is a term because 141 = 3 * 47 = A001358(46) is a semiprime and 46 = 2 * 23, 141 - 46 = 95 = 5 * 19 and 141 + 46 = 187 = 11 * 17 are all semiprimes.

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

Cf. A001358. A383469.

k:= 0: R:= NULL: count:= 0:
for s from 1 while count < 100 do
  if numtheory:-bigomega(s) = 2 then
    k:= k+1;
    if andmap(t -> numtheory:-bigomega(t) = 2, [k, s-k, s+k]) then
      R:= R, s; count:= count+1;
    fi
  fi;
od:
R;

a(n) = A001358(A383469(n)).

A382982 Primes of the form Sum_{i=j..k} prime(i)^prime(i).

Original entry on oeis.org

31, 826699, 303160419086407

Offset: 1

Zak Seidov and Robert Israel, Apr 11 2025

nonn

Primes that are sums of some number of consecutive terms of A051674.

			a(1) = 31 = 2^2 + 3^3 = Sum_{i=1..2} prime(i)^prime(i).
a(2) = 826699 = Sum_{i=1..4} prime(i)^prime(i).
a(3) = 303160419086407 = Sum_{i=4..6} prime(i)^prime(i).
a(4) = Sum_{i=1..24} prime(i)^prime(i) has 174 digits.
a(5) = Sum_{i=20..34} prime(i)^prime(i) has 298 digits.
a(6) = Sum_{i=30..38} prime(i)^prime(i) has 361 digits.
a(7) = Sum_{i=38..48} prime(i)^prime(i) has 524 digits.
a(8) = Sum_{i=46..84} prime(i)^prime(i) has 1142 digits.
a(9) = Sum_{i= 7..85} prime(i)^prime(i) has 1161 digits.

Robert Israel, Table of n, a(n) for n = 1..7

Cf. A051674, A061789, A340392.

select(isprime, [seq(seq(add(ithprime(i)^ithprime(i),i=j..k),j=1..k-1),k=1..76)]);

A382933 Numbers k such that k, 2m +- 3 and 3m +- 2 are all semiprimes.

Original entry on oeis.org

451, 707, 871, 1313, 1537, 1819, 1921, 1969, 2155, 2195, 2533, 2599, 2885, 2993, 3265, 3817, 3883, 3953, 3997, 4069, 4105, 4385, 4555, 4607, 5599, 5755, 5771, 6155, 6415, 6773, 7157, 7453, 7979, 8185, 8213, 8251, 8321, 8333, 8399, 8531, 9055, 9077, 9167, 9335, 9647, 9953, 9977, 10121, 10537

Offset: 1

Zak Seidov and Robert Israel, Apr 15 2025

nonn

All terms are odd.

The first term that is a square is a(241) = 49729 = 223^2.

			a(3) = 871 because 871 = 13 * 67,
2 * 871 - 3 = 1739 = 37 * 47,
2 * 871 + 3 = 1745 = 5 * 349,
3 * 871 - 2 = 2611 = 7 * 373, and
3 * 871 + 2 = 2615 = 5 * 523 are all semiprimes.

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

Cf. A001358.

filter:= m -> andmap(t -> numtheory:-bigomega(t)=2, [m,2*m-3,2*m+3,3*m-2,3*m+2]):
select(filter, [seq(i,i=1..20000,2)]);

s = {}; Do [ If [ {2, 2, 2, 2, 2} == PrimeOmega [{m, 2*m - 3, 2*m
+ 3, 3*m - 2, 3*m + 2}], AppendTo [s, m]], {m, 5, 10^4, 2}]; s

A382765 Primes that can be expressed using exactly one of each of the prime digits 2, 3, 5, 7, using concatenation and the arithmetic operations +,-,*,/,^.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 103, 107, 113, 139, 149, 151, 163, 167, 173, 181, 197, 211, 223, 227, 229, 257, 263, 311, 353, 359, 367, 373, 379, 389, 479, 569, 571, 643, 691, 727, 733, 751, 877, 1019, 1091, 1699, 2239, 2357, 2749, 2753

Offset: 1

Zak Seidov and Robert Israel, Apr 10 2025

Concatenation is only allowed for individual digits, not expressions, so 37 - 2^5 = 5 is allowed but (7 + 2 - 5)3 = 43 is not allowed.

			a(1) = 2 = (3 - 2) * (7 - 5).
a(2) = 3 = 2 + 3 + 5 - 7.
a(3) = 5 = 37 - 2^5.
a(4) = 7 = 2 + 3 - 5 + 7.
a(83) = 4747561509941 = 7 ^ (3 * 5) - 2.

Robert Israel, Table of n, a(n) for n = 1..83 (full sequence)
Robert Israel, Maple code
Robert Israel, An expression for each term

Cf. A382901.

Maple
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See link.
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A382901 Semiprimes that can be expressed using at most one of each of the semiprime digits 4, 6, 9 using concatenation and the arithmetic operations +, -, *, /, ^.

Original entry on oeis.org

4, 6, 9, 10, 15, 33, 46, 49, 55, 58, 65, 69, 94, 469, 649, 694, 4087, 4105

Offset: 1

Zak Seidov and Robert Israel, Apr 08 2025

Concatenation is not allowed with expressions, only digits, e.g. (9-6)4 = 34 is not allowed.

			46, 49, 69, 94, 469, 649 and 694 are concatenations.
10 = 4 + 6.
15 = 6 + 9.
33 = 4 * 6 + 9.
55 = 46 + 9.
58 = 6 * 9 + 4.
65 = 69 - 4.
4087 = 4^6 - 9.
4105 = 4^6 + 9.

A2:= select(t -> numtheory:-bigomega(t)=2, map(t -> (t[1]+t[2],10*t[1]+t[2]),convert(combinat:-permute([4,6,9],2),set))):
A3:= select(t -> t::integer and numtheory:-bigomega(t) = 2, map(t -> (t[1]+t[2]+t[3],t[1]+t[2]-t[3],t[1]*t[2]+t[3],t[1]*t[2]-t[3],100*t[1]+10*t[2]+t[3],10*t[1]+t[2]-t[3],10*t[1]+t[2]+t[3],t[1]^t[2]+t[3],t[1]^t[2]-t[3],
(10*t[1]+t[2])/t[3]), convert(combinat:-permute([4,6,9]),set))):
sort(convert({4,6,9,op(A2),op(A3)},list));

A382264 Semiprimes that are the sum of the m-th prime and the m-th semiprime for some m.

Original entry on oeis.org

6, 9, 14, 25, 38, 55, 86, 122, 141, 158, 178, 185, 218, 262, 301, 326, 446, 466, 537, 634, 695, 723, 758, 785, 866, 878, 886, 895, 898, 921, 993, 1006, 1041, 1047, 1077, 1099, 1126, 1138, 1154, 1198, 1214, 1219, 1234, 1262, 1366, 1466, 1535, 1679, 1706, 1751, 1774, 1779, 1822, 1977, 2026, 2173

Offset: 1

Zak Seidov and Robert Israel, Mar 19 2025

nonn

Semiprimes in A133796.

Corresponding m's are A092021.

			a(4) = 25 is a term, because 25 = 5^2 is a semiprime and 25 = 11 + 14 where 11 is the 5th prime and 14 is the 5th semiprime.

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

Cf. A092021, A133796, A382186.

N:= 10000: # for terms where the m-th prime and m-th semiprime are <= N
P:= select(isprime, [2, seq(i, i=3..N, 2)]): nP:= nops(P):
S:= NULL:
for i from 1 to nP while P[i]^2 <= N do
  jmax:= ListTools:-BinaryPlace(P, N/P[i]);
  S:= S, op(P[i..jmax] *~ P[i]);
od:
S:= sort([S]):
m:= min(nP, nops(S)):
select(t -> numtheory:-bigomega(t)=2, P[1..m] + S[1..m]);

a(n) = A133796(A092021(n)).

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User: Zak Seidov

A386295 Primes p such that p+1 is a triprime and 2*p+1 is prime.

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A384940 Odd semiprimes interleaved with even semiprimes.

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A383665 a(n) is the least number k such that k, k - s and k + s all have n prime divisors, counted with multiplicity, where s is the sum of the decimal digits of k.

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A383469 Semiprimes k such that A001358(k) + k and A001358(k) - k are also semiprimes.

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A383468 Semiprimes s = A001358(k) such that k, s - k and s + k are also semiprimes.

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A382982 Primes of the form Sum_{i=j..k} prime(i)^prime(i).

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A382933 Numbers k such that k, 2*m +- 3 and 3*m +- 2 are all semiprimes.

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A382765 Primes that can be expressed using exactly one of each of the prime digits 2, 3, 5, 7, using concatenation and the arithmetic operations +,-,*,/,^.

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A382933 Numbers k such that k, 2m +- 3 and 3m +- 2 are all semiprimes.