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A387291 Integers x such that there exist two numbers y,z with x <= y <= z such that psi(x) = psi(y) = psi(z) = (x + y + z)/2.

6, 8, 16, 18, 28, 32, 44, 54, 64, 70, 105, 110, 128, 150, 162, 165, 182, 200, 238, 240, 256, 280, 310, 315, 364, 382, 468, 486, 512, 520, 585, 590, 644, 735, 750, 780, 790, 795, 800, 1000, 1024, 1034, 1162, 1246, 1260, 1274, 1410, 1434, 1456, 1458, 1472, 1540, 1575

Offset: 1

S. I. Dimitrov, Aug 25 2025

The numbers x, y and z form a psi-amicable triple.

			28 is in the sequence since psi(28) = psi(33) = psi(35) = 48 = (28 + 33 + 35)/2.

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A385852, A386726, A386901, A386933.

A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

Original entry on oeis.org

2, 5, 21, 129, 69, 1, 51, 23991, 171, 1371, 3, 322141431, 1431357020859

Offset: 0

Om S. M. Yadav, Aug 18 2025

Similar to A227547, primes are added in successive manner except that here the sequence breaks if an even-indexed term is not prime and considers preceding even-indexed prime as the last term of the sequence. For example, a(2) = 21 [21, 23, 26, 31, 38, 49] but since 49 is not prime, last two terms (38 and 49) are omitted leaving 31 as last term in the sequence.

a(12) is the last term, because b(j) is always divisible by 11 for some j in {2, 4, 6, 8, 10, 14, 16, 18, 22, 24, 26}. - Pontus von Brömssen, Aug 19 2025

			a(n) = k, b(m+1) = b(m) + prime(m); b(1) = k
For n = 0, a(0) = 2; b(m+1) = b(m) + prime(m): [2]
For n = 1, a(1) = 5; b(m+1) = b(m) + prime(m): [5, 7(5+2)]
For n = 2, a(2) = 21; b(m+1) = b(m) + prime(m): [21, 23(21+2), 26(23+3), 31(26+5)]
For n = 3, a(3) = 129; b(m+1) = b(m) + prime(m): [129, 131(129+2), 134(131+3), 139(134+5), 146(139+7), 157(146+11)]
For n = 4, a(4) = 69; b(m+1) = b(m) + prime(m): [69, 71(69+2), 74(71+3), 79(74+5), 86(79+7), 97(86+11), 110(97+13), 127(110+17)]
For n = 5, a(5) = 1; b(m+1) = b(m) + prime(m): [1, 3(1+2), 6(3+3), 11(6+5), 18(11+7), 29(18+11), 42(29+13), 59(42+17), 78(59+19), 101(78+23)]
For a(n), even-indexed term is prime. e.g. for a(3) = 129 [129, 131, 134, 139, 146, 157], even indexed terms 131, 139, 157 are primes.

Cf. A014284, A092163, A109723, A227547.

a(n) = my(vp=concat(2, vector(n+1, i, sum(k=1, 2*i+1, prime(k)))), v=concat(vector(n, i, 1), 0), k=1); while (apply(ispseudoprime, vector(n+1, i, vp[i]+k)) != v, k++); k; \\ Michel Marcus, Aug 19 2025

a(11) from Michel Marcus, Aug 19 2025

a(12) from Pontus von Brömssen, Aug 19 2025

A387199 Numbers which are not themselves palindromes, but a single swap of two digits creates a palindrome.

Original entry on oeis.org

110, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 211, 220, 221, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 311, 322, 330, 331, 332, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399, 411, 422

Offset: 1

James S. DeArmon, Aug 21 2025

Might be called "single-transposition palindromes".

Leading zeros are not allowed in either the initial number or the resultant palindrome.

			110 is a term since a swap of the second and third digits yields the palindrome 101.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A002113.

def pal(s): return s == s[::-1]
def swaps(s): yield from (t for i in range(len(s)-1) for j in range(i+1, len(s)) if (t:=s[:i]+s[j]+s[i+1:j]+s[i]+s[j+1:])[0]!='0')
def ok(n): return not pal(s:=str(n)) and any(pal(t) for t in swaps(s))
print([k for k in range(425) if ok(k)]) # Michael S. Branicky, Aug 22 2025

More terms from Michael S. Branicky, Aug 22 2025

A386933 Integers z such that there exist two integers 0

Original entry on oeis.org

81900, 161700, 163800, 175350, 245700, 261660, 323400, 327600, 350700, 409500, 485100, 490770, 491400, 499380, 523320, 526050, 526260, 573300, 646800, 647010, 655200, 671370, 701400, 702450, 737100, 784980, 808500, 819000, 876750, 970200, 971880, 981540, 982800, 990150, 998760

Offset: 1

S. I. Dimitrov, Aug 09 2025

The numbers x, y and z form a psi-amicable triple.

			163800 is in the sequence since psi(158340) = psi(161700) = psi(163800) = 564480 = 158340 + 161700 + 163800. Other examples: (322140, 322140, 323400), (14127960, 14224980, 14224980).

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A125492, A323330, A385852, A386901.

A386901 Integers y such that there exist two integers 0

Original entry on oeis.org

80850, 158340, 161070, 161700, 232050, 242550, 316680, 322140, 323400, 404250, 464100, 474810, 475020, 483210, 485100, 485940, 565950, 633360, 641550, 644280, 646800, 662340, 696150, 727650, 791700, 805350, 808500, 963270, 966420, 967890, 970200, 971880

Offset: 1

S. I. Dimitrov, Aug 07 2025

The numbers x, y and z form a psi-amicable triple.

			158340 is in the sequence since psi(150150) = psi(158340) = psi(175350) = 483840 = 150150 + 158340 + 175350. Other examples: (232050, 232050, 261660), (7091700, 7098630, 7098630).

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A125491, A323329, A385852, A386933.

A385852 Integers x such that there exist two integers 0

Original entry on oeis.org

79170, 150150, 158340, 161070, 232050, 237510, 300300, 316680, 322140, 395850, 450450, 464100, 468930, 474810, 475020, 483210, 554190, 570570, 600600, 622440, 633360, 641550, 644280, 696150, 712530, 750750, 791700, 805350, 937860, 949620, 950040, 963270, 966420

Offset: 1

S. I. Dimitrov, Aug 07 2025

The numbers x, y and z form a psi-amicable triple according to Dimitrov's definition.

			79170 is in the sequence since psi(79170) = psi(80850) = psi(81900) = 241920 = 79170 + 80850 + 81900. Other examples: (161070, 161070, 161700), (7063980, 7112490, 7112490).

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A125490, A323329, A386901, A386933.

A386525 a(n) is the least k such that at least n terms of A063655 starting from index k are strictly decreasing.

Original entry on oeis.org

1, 5, 13, 37, 122, 3004, 26283, 53411, 109453, 4117156, 16831081

Offset: 1

Richard S. Chang, Jul 24 2025

Lai and Reinfeld conjecture that the sequence is infinite.

			The first 40 values of A063655 are: 2, 3, 4, 4, 6, 5, 8, 6, 6, 7, 12, 7, 14, 9, 8, 8, 18, 9, 20, 9, 10, 13, 24, 10, 10, 15, 12, 11, 30, 11, 32, 12, 14, 19, 12, 12, 38, 21, 16, and 13. Because the 37th term is the first of 4 strictly decreasing values and there is not previous occurrence of four decreasing values, a(3) = 37.

H. Lai and M. Reinfeld, An Introduction to LR Numbers, Girls' Angle Bulletin, Vol. 18, No. 5 (2025), 11-15.

Cf. A063655.

a(11) from Sean A. Irvine, Aug 10 2025

A386964 a(1) = prime(1) = 2, a(n) = 10*a(n-1) + (prime(n) mod 10).

Original entry on oeis.org

2, 23, 235, 2357, 23571, 235713, 2357137, 23571379, 235713793, 2357137939, 23571379391, 235713793917, 2357137939171, 23571379391713, 235713793917137, 2357137939171373, 23571379391713739, 235713793917137391, 2357137939171373917, 23571379391713739171, 235713793917137391713

Offset: 1

Michael S. Branicky, Aug 11 2025

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A007652, A276481.

a:= proc(n) option remember; `if`(n<1, 0, a(n-1)*10+irem(ithprime(n), 10)) end:
seq(a(n), n=1..21);  # Alois P. Heinz, Aug 12 2025

a[1]=2;a[n_]:=10a[n-1]+Mod[Prime[n],10];Array[a,21] (* James C. McMahon, Aug 12 2025 *)

from sympy import nextprime
from itertools import islice
def A386964(): # generator of terms
    an = pn = 2
    while True:
        yield an
        an = 10*an + (pn:=nextprime(pn))%10
print(list(islice(A386964(), 21)))

a(n) = concatenation of A007652(1)..A007652(n).

A386727 Numbers x such that there exist three integers 0

Original entry on oeis.org

3, 10, 24, 51, 78, 105, 114, 136, 186, 220, 224, 255, 322, 348, 357, 370, 435, 478, 506, 616, 642, 710, 748, 820, 861, 885, 957, 996, 1004, 1068, 1113, 1214, 1221, 1276, 1292, 1336, 1390, 1485, 1491, 1562, 1564, 1581, 1605, 1660, 1670, 1704, 1716, 1724, 1815, 1869, 1880, 1912, 1947

Offset: 1

S. I. Dimitrov, Jul 31 2025

nonn

The numbers x, y, z and t form an amicable quadruple according to Yanney’s definition.

			114 is in the sequence since sigma(114) = sigma(158) = sigma(209) = sigma(239) = 240 = (114 + 158 + 209 + 239)/3.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Benjamin Franklin Yanney, Another definition of amicable numbers and some of their relations to Dickson's amicables, Amer. Math. Monthly, Vol. 30, No. 6 (1923), 311-315.

Cf. A000203, A036471, A386726.

isok(x1) = my(s=sigma(x1), vx=select(x->(x>=x1), invsigma(s)), v=vector(4, i, vx[1])); for (i=1, #vx, v[2] = vx[i]; for (j=1, #vx, v[3] = vx[j]; for (k=1, #vx, v[4] = vx[k]; if (vecsum(v) == 3*s, return(1));););); \\ Michel Marcus, Aug 01 2025

More terms from Michel Marcus, Aug 01 2025

A386726 Numbers x such that there exist two integers 0

Original entry on oeis.org

2, 238, 280, 308, 310, 382, 790, 795, 920, 952, 1034, 1162, 1246, 1330, 1410, 1434, 2002, 2024, 2506, 2632, 2728, 2750, 2926, 3040, 3210, 3452, 3496, 3500, 3630, 4134, 4260, 4466, 4550, 4968, 5080, 5278, 5396, 5520, 5530, 5756, 6128, 6230, 6426, 6888, 7288, 7584, 7640, 7910, 7990

Offset: 1

S. I. Dimitrov, Jul 31 2025

nonn

The numbers x, y and z form an amicable triple according to Yanney's definition.

			238 is in the sequence since sigma(238) = sigma(255) = sigma(371) = 432 = (238 + 255 + 371)/2.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Benjamin Franklin Yanney, Another definition of amicable numbers and some of their relations to Dickson's amicables, Amer. Math. Monthly, Vol. 30, No. 6 (1923), 311-315.

Cf. A000203, A125490, A386727.

isok(x1) = my(s=sigma(x1), vx=select(x->(x>=x1), invsigma(s)), v=vector(3, i, vx[1])); for (i=1, #vx, v[2] = vx[i]; for (j=1, #vx, v[3] = vx[j]; if (vecsum(v) == 2*s, return(1)););); \\ Michel Marcus, Aug 01 2025

More terms from Michel Marcus, Aug 01 2025

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A387291 Integers x such that there exist two numbers y,z with x <= y <= z such that psi(x) = psi(y) = psi(z) = (x + y + z)/2.

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A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

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A387199 Numbers which are not themselves palindromes, but a single swap of two digits creates a palindrome.

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A386933 Integers z such that there exist two integers 0

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A386901 Integers y such that there exist two integers 0

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A385852 Integers x such that there exist two integers 0

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A386525 a(n) is the least k such that at least n terms of A063655 starting from index k are strictly decreasing.

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A386964 a(1) = prime(1) = 2, a(n) = 10*a(n-1) + (prime(n) mod 10).

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A386727 Numbers x such that there exist three integers 0

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